tag:blogger.com,1999:blog-106143482021-10-26T23:37:09.701-07:00TGD diaryDaily musings, mostly about physics and consciousness, heavily biased by Topological Geometrodynamics background.Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger1848125tag:blogger.com,1999:blog-10614348.post-25425210082458829172021-10-26T23:35:00.004-07:002021-10-26T23:36:12.466-07:00Quantum hydrodynamics in nuclear physics and hadron physics
The field equations of TGD defining the space-time surfaces have interpretation as conservation laws for isometry charges and therefore have a hydrodynamics character. The hydrodynamic character is actually characterized in quite concrete ways (see <A HREF="http://tgdtheory.fi/public_html/articles/TGD2021.pdf">this</A>, <A HREF="http://tgdtheory.fi/public_html/articles/TGDcondmatshort.pdf">this</A>,
and <A HREF="http://tgdtheory.fi/public_html/articles/TGDhydro.pdf">this</A>).
</p><p>
Also nuclear and hadron physics suggest applications for Quantum Hydrodynamics (QHD). The basic vision about what happens in high energy nuclear and hadron collisions is that two BSFRs take place. The first BSFR creates the intermediate state with h<sub>eff</sub>>h: the entire system formed by colliding systems need not be in this state. In nuclear physics this state corresponds to a dark nucleus which decays in the next BSFR to ordinary nuclei. </p><p>The basic notions are the notion of dark matter at MB and ZEO, in particular the change of the arrow of time in BSFR.
</p><p>
<B>1. Cold fusion, nuclear tunnelling, ℏ<sub>eff</sub>, and BSFRs</B>
</p><p>
This model allows us to understand "cold fusion" in an elegant manner (see <A HREF="http://tgdtheory.fi/public_html/articles/krivit.pdf">this</A>, <A HREF="http://tgdtheory.fi/public_html/articles/proposal.pdf">this</A>, and <A HREF="http://tgdtheory.fi/public_html/articles/cfagain.pdf">this</A>). The dark protons at flux tubes associated with water and created by the Pollack effect have much smaller nuclear binding energy than ordinary nucleons. This energy is compensated to a high degree by the positive Coulomb binding energy which corresponds roughly to distance given by electron Compton length.</p><p>Dark nuclear reactions between these kinds of objects do not require large collision energy to increase the value of h<sub>eff</sub> and can take place at room temperature. After the reaction the dark nuclei can transform to ordinary nuclei and liberate the ordinary nuclear binding energy. One can say that in ordinary nuclear reactions one must get to the top of the energy hill and in "cold fusion" one already is at the top of the hill.
</p><p>
Quite generally, the mechanism creating intermediate dark regions in the system of colliding nuclei in BSFR, would be the TGD counterpart of quantum tunnelling in the description of nuclear reactions based on Schrödinger equation. This mechanism could be involved with all tunnelling phenomena.
</p><p>
<B>2. QHD and hadron physics</B>
</p><p>
Hadron physics suggests applications of QHD.
</p><p>
<B>2.1 Quark gluon plasma and QHD</B>
</p><p>
In hadron physics quark gluon plasma (see <A HREF="https://cutt.ly/xEDQNZA">this</A>) has turned out to be what it was thought to be originally. Instead of being like a gas of quarks and gluons with a relatively large dissipation, it has turned out to behave like almost perfect fluid. This means that the ratio η/s of viscosity and entropy is near to its minimal value proposed by string model based arguments to be η/s=ℏ/m.
</p><p>
To be a fluid means that the system has long range correlations whereas in gas the particles move randomly and one cannot assign to the system any velocity field or more general currents. In the TGD framework, the existence of a velocity field means at the level of the space-time geometry generalized Beltrami flow allowing to define a global coordinate varying along the flow lines (see <A HREF="http://tgdtheory.fi/public_html/articles/SCBerryTGD.pdf">this</A> and <A HREF="http://tgdtheory.fi/public_html/articles/TGD2021.pdf">this</A>). This would be a geometric property of space-time surfaces and the finite size of the space-time surface would serve as a limitation.
</p><p>
In the TGD framework the replacement ℏ→ ℏ<sub>eff</sub> requires that s increases in the same proportion. If the fluid flow is realized in terms of vortices controlled by pairs of monopole flux tubes defining their cores and Lagrangian flux tubes with gradient flow defining the exteriors of the cores, this situation is achieved.
</p><p>
In this picture entropy could but need not be associated with the monopole flux tubes with non-Beltrami flow and with non-vanishing entropy since the number of the geometric degrees of freedom is infinite which implies limiting temperature known has Hagedorn temperature T<sub>H</sub> which is about 175 MeV for hadrons, and slightly higher than pion mass. In fact, the Beltrami property holds for the flux tubes with 2-D CP<sub>2</sub> projection, which is a complex manifold for monopole flux tubes. The fluid flow associated with (controlled by) the monopole flux tubes would have non-vanishing vorticity for monopole fluxes and could dissipate.
</p><p>
The monopole flux tube at the core of the vortex could therefore serve as the source of entropy. One expects that η/s as minimal value is not affected by h→ h<sub>eff</sub>. One expects that s → (ℏ<sub>eff</sub>/ℏ)s= ns since the dimension of the extension of rationals multiplies the Galois degrees of freedom by n.
</p><p>
Almost perfect fluids are known to allow almost non-interacting vortices. For a perfect fluid, the creation of vortices is impossible due to the absence of friction at the walls. This suggests that the ordinary viscosity is not the reason for the creation of vortices, and in the TGD picture the situation is indeed this. The striking prediction is that the masses of Sun and Earth appear as basic parameters in the gravitational Compton lengths Λ<sub>gr</sub> determining ν<sub>gr</sub>= Λ<sub>gr</sub>c. </p><p>
<B>2.2 The phase transition creating quark gluon plasma</B>
</p><p>
The phase transition creating what has been called quark gluon plasma is now what it was expected to be. That the outcome behaves like almost perfect fluid was the first example. TGD leads however to a proposal that since quantum criticality is involved, phases with ℏ<sub>eff</sub>>h must be present.
</p><p>
p-Adic length scale hypothesis led to the proposal (see <A HREF="http://tgdtheory.fi/public_html/articles/tgdnewphys1.pdf">this</A> and <A HREF="http://tgdtheory.fi/public_html/articles/tgdnewphys2.pdf">this</A>) that this transition could allow production of so called M<sub>89</sub> hadrons characterized by Mersenne prime M<sub>89</sub>=2<sup>89</sup>-1 whereas ordinary hadrons would correspond to M<sub>107</sub>. The mass scale of M<sup>89</sup> hadrons would be by a factor 512 higher than that of ordinary hadrons and there are indications for the existence of scaled versions of mesons.
</p><p>
How M<sub>89</sub> hadrons could be created. The temperature T<sub>H</sub>= 175 MeV is by a factor 1/512 lower than the mass scale of M<sub>89</sub> pion. Somehow the colliding nuclei or hadrons must provide the needed energy from their kinetic energy. What certainly happens is that this energy is materialized in the ordinary nuclear reaction to ordinary pions and other mesons. The mesons should correspond to closed flux tubes assignable to circular vortices of the highly turbulent hydrodynamics flow created in the collision.
</p><p>
Could roughly 512 mesonic flux tubes reconnect to circular but flattened long flux tubes having length of M<sub>89</sub> meson, which is 512 times that of ordinary pions? I have proposed this kind of process, analogous to BEC, to be fundamental in both biology (see <A HREF="http://tgdtheory.fi/public_html/articles/darkchemi.pdf">this</A>, <A HREF="http://tgdtheory.fi/public_html/articles/bioharmony2020.pdf">this</A>, and <A HREF="http://tgdtheory.fi/public_html/articles/TIH.pdf">this</A>) and also to explain the strange findings of Eric Reiter challenging some basic assumptions of nuclear physics if taken at face value (see <A HREF="http://tgdtheory.fi/public_html/articles/unquantum.pdf">this</A>).
</p><p>
The process generating an analog of BEC would create in the first BSFR M<sub>89</sub> mesons having ℏ<sub>eff</sub>/ℏ=512. In the second BSFR the transition ℏ<sub>eff</sub>→ ℏ would take place and yield M<sup>89</sup> mesons. It would seem that part of the matter of the composite system ends up to n M<sub>89</sub> hadronic phase with 512 times higher T<sub>H</sub>. In the number theoretic picture, these BEC like states would be Galois confined states (see <A HREF="http://tgdtheory.fi/public_html/articles/GaloisTGD.pdf">this</A> and <A HREF="http://tgdtheory.fi/public_html/articles/Galoiscode.pdf">this</A>).
</p><p>
<B>2.3 Can the size of a quark be larger than the size of a hadron?</B>
</p><p>
The Compton wavelength Λ<sub>c</sub>= ℏ/m is inversely proportional to mass. This implies that the Compton length of the quark as part of the hadron is longer than the Compton length of the hadron. If one assigns to Compton length a geometric interpretation as one does in M<sup>8</sup>-H duality mapping mass shell to CD with radius given by Compton length, this sounds paradoxical. How can a part be larger than the whole? One can think of many approaches to what might look like a paradox.
</p><p>
One could of course argue that being a part in the sense of tensor product has nothing to with being a part in geometric sense. However, if one requires quantum classical correspondence (QCC), one could argue that a hadron is a small region to which much larger quark 3-surfaces are attached.
</p><p>
One could also say that Compton length characterizes the size of the MB assignable to a particle which itself has size of order CP<sub>2</sub> length scale. In this case the strange looking situation would appear only at the level of MBs and the magnetic bodies could have sizes which increase when the particle mass decreases.
</p><p>
What if one takes QCC completely seriously? One can look at the situation in ZEO.
<OL>
<LI> The size of the CD corresponds to Compton length and CDs for different particle masses have a common center and form a Russian doll-like hierarchy. One can continue the geodesic line defining point of CD associated with the hadron mass so that it intersects the CDs associated with quarks, in particular that for the lightest quark.
<LI> The distances between the quarks would define the size scale of the system in this largest CD and in the case of light hadrons containing U and D quarks it would be of the order of the Compton length of the lightest quark involved having mass about 5 MeV: this makes about .2 × 10<sup>-13</sup> m. There are indeed indications that the MB of proton has this size scale.
</OL>
One could also require that there must be a common CD based on such an identification of h<sub>eff</sub> for each particle that its size does not depend on the mass of the particles.
<OL>
<LI> Here ℏ<sub>gr</sub>= GMm/β<sub>0</sub> provides a possible solution. The size of the CD would correspond to Λ<sub>gr</sub> =GM/v<sub>0</sub> for all particles involved. One could call this size the quantum gravitational size of the particle.</p><p><LI> There is an intriguing observation related to this. To be in gravitational interaction could mean ℏ<sub>eff</sub>=ℏ<sub>gr</sub>=GMm/v<sub>0</sub> so that the size of the common CD would be given by Λ<sub>gr</sub>= GMm/v<sub>0</sub>. The minimum mass M given ℏ<sub>gr</sub>>ℏ would be M=β<sub>0</sub> M<sub>Pl</sub><sup>2</sup>/m. For protons this gives M ≥ 1.5 × 10<sup>38</sup> m<sub>p</sub>. Assuming density ρ ≈ 10<sup>30</sup>A/m<sup>3</sup>, A the atomic number, the length L for the side cube with minimal mass M is L×β<sub>0</sub>× 10<sup>2</sup>/A<sup>1/3</sup>. For β<sub>0</sub>= 2<sup>-11</sup> assignable to the Sun-Earth system, this gives L∼ 5/A<sup>1/3</sup> mm. The value of Λ<sub>gr</sub> for Earth is 4.35 mm for β<sub>0</sub>=1. The orders of magnitude are the same. Is this a mere accident?
</OL>
One solution to the problem is that the ratio
ℏ<sub>eff</sub>(H)/ℏ<sub>eff</sub>(q) is so large that the problem disappears.
<OL>
<LI> If ℏ<sub>eff</sub>(1)=ℏ, the value of ℏ<sub>eff</sub> for hadron should be so large that the geometric intuitions are respected: this would require h<sub>eff</sub>/h;≥ m<sub>H</sub>/m<sub>q</sub>. The hadrons containing u, d, and c quarks are very special.
<LI> Second option is that the value of h<sub>eff</sub> for quarks is smaller than h to guarantee that the Compton length is not larger than ℏ. The perturbation theory for states consisting of free quarks would not converge since Kähler coupling strength α<sub>K</sub> ∝ 1/ℏ<sub>eff</sub> would be too large. This would conform with the QCD view and provide a reason for color confinement. Quarks would be dark matter in a well-defined sense.
<LI> The condition would be ℏ<sub>eff</sub>(H)/ℏ<sub>eff</sub>(q)≥ m(H)/m<sub>q</sub>, where q is the lightest quark in the hadron. For heavy hadrons containing heavy quarks this condition would be rather mild. For light hadrons containing u,d, and c quarks it would be non-trivial. Ξ gives the condition ℏ/ℏ<sub>eff</sub>≥ 262. The condition could not be satisfied for too small masses of the value of ℏ= 7!ℏ<sub>0</sub>=5040ℏ<sub>0</sub> identifiable as the ratio of dark CP<sub>2</sub> deduced from p-adic mass calculations and Planck length.
</OL>Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-71896308557750556212021-10-26T21:05:00.008-07:002021-10-26T21:07:26.763-07:00The emergence of twistor spaces from M8-H duality
<B>Generalizations of M<sup>8</sup>-H duality</B>
</p><p>
It has become clear that M<sup>8</sup>-H duality generalizes and there is a connection with the twistorialization at the level of H.
</p><p>
<B>Space-time surfaces as images of associative surfaces in M<sup>8</sup></B>
</p><p>
M<sup>8</sup>-H duality would provide an explicit construction of space-time surfaces as algebraic surfaces with an associative normal space (see <A HREF="http://Tgdtheory.fi/public_html/articles/M8H1.pdf">this</A>, <A HREF="http://Tgdtheory.fi/public_html/articles/M8H1.pdf">this</A>, and <A HREF="http://Tgdtheory.fi/public_html/articles/TGD2021.pdf">this</A>). M<sup>8</sup> picture codes space-time surface by a real polynomial with rational coefficients. One cannot exclude coefficients in an extension of rationals and also analytic functions with rational or algebraic coefficients can be considered as well as polynomials of infinite degree obtained by repeated iteration giving rise algebraic numbers as extension and continuum or roots as limits of roots.
</p><p>
M<sup>8</sup>-H duality maps these solutions to H and one can consider several forms of this map. The weak form of the duality relies on holography mapping only 3-D or even 2-D data to H and the strongest form maps entire space-time surfaces to H. The twistor lift of TGD allows to identify the space-time surfaces in H as base spaces of 6-D surfaces representing the twistor space of space-time surface as an S<sup>2</sup> bundle in the product of twistor spaces of M<sup>4</sup> and CP<sub>2</sub>. These twistor spaces must have Kähler structure and only the twistor spaces of M<sup>4</sup> and CP<sub>2</sub> have it so that TGD is unique also mathematically.
</p><p>
An interesting question relates to the possibility that also 6-D commutative space-time surfaces could be allowed. The normal space of the space-time surface would be a commutative subspace of M<sup>8</sup><sub>c</sub> and therefore 2-D. Commutative space-time would be a 6-D surface X<sup>6</sup> in M<sup>8</sup>.
</p><p>
This raises the following question: Could the inverse image of the 6-D twistor-space of 4-D space-time surface X<sup>4</sup> so that X<sup>6</sup> would be M<sup>8</sup> analog of twistor lift? This requires that X<sup>6</sup>⊂ M<sup>8</sup><sub>c</sub> has the structure of an S<sup>2</sup> bundle and there exists a bundle projection X<sup>6</sup>→ X<sup>4</sup>.
</p><p>
The normal space of an associative space-time surface actually contains this kind of commutative normal space! Its existence guarantees that the normal space of X<sup>4</sup> corresponds to a point of CP<sub>2</sub>. Could one obtain the M<sup>8</sup><sub>c</sub> analog of the twistor space and the bundle bundle projection X<sup>6</sup>→ X<sup>4</sup> just by dropping the condition of associativity. Space-time surface would be a 4-surface obtained by adding the associativity condition.
</p><p>
One can go even further and consider 7-D surfaces of M<sup>8</sup> with real
and therefore well-ordered normal space. This would suggest
dimensional hierarchy: 7→ 6→ 4.
</p><p>
This leads to a possible interpretation of twistor lift of TGD at the level of M<sup>8</sup> and also about generalization of M<sup>8</sup>-H correspondence to the level of twistor lift. Also the generalization of twistor space to a 7-D space is suggestive. The following arguments representa vision about "how it must be" that emerged during the writing of this article and there are a lot of details to be checked.
</p><p>
<B>Commutative 6-surfaces and twistorial generalization of M<sup>8</sup>-H correspondence</B>
</p><p>
Consider first the twistorial generalization of M<sup>8</sup>-H correspondence.
<OL>
<LI> The complex 6-D surface X<sup>6</sup><sub>c</sub>⊂ M<sup>8</sup><sub>c</sub> has commutative normal space and thus corresponds to complexified octonionic complex numbers (z<sub>1</sub>+z<sub>2</sub>I). X<sup>6</sup><sub>c</sub> has real dimension 12 just as the product T(M<sup>4</sup>)× T(CP<sub>2</sub>) of 6-D twistor spaces of M<sup>4</sup> and CP<sub>2</sub>. It has a bundle structure with a complex 4-D base space which is mapped M<sup>4</sup>× CP<sub>2</sub> by M<sup>8</sup>H duality. The fiber has complex dimension 2 and corresponds to the dimension for the product of twistor spheres of the twistor spaces of M<sup>4</sup> and CP<sub>2</sub>.
<LI> This suggests that M<sup>8</sup>-H duality generalizes so that it maps X<sup>6</sup><sub>c</sub> ⊂ M<sup>8</sup><sub>c</sub> to T(M<sup>4</sup>)× T(CP<sub>2</sub>) . It would map the point of X<sup>6</sup><sub>c</sub> to its real projection identified as a point of T(M<sup>4</sup>). "Real" means here that the complex continuation of the number theoretical norm squared for octonions is real so that the components of M<sup>8</sup> point are either real or imaginary with respect to the commuting imaginary unit i. The complex 6-D tangent space of X<sup>6</sup><sub>c</sub> would be mapped to a point of T(CP<sub>2</sub>).
</p><p>
The beauty of this picture would be that the entire complex 6-D surface would carry physical information mapped directly to the twistor space.
</OL>
One can try to guess the form of the map of X<sup>6</sup><sub>c</sub> to the product T(M<sup>4</sup>)× T(CP<sub>2</sub>).
</p><p>
The surfaces X<sup>6</sup> have local normal space basis 1⊕ e<sub>7</sub> . The problem is that this space is invariant under SU(3) for M<sup>8</sup>-H for CP<sub>2</sub>. Could one choose the 2-D normal space to be something else without losing the duality. If e<sub>7</sub> and e<sub>1</sub> are permuted, the tangent space basis vector transforms by a phase phase factor under U(1)× U(1). The 4-D sub-basis of normal space would be now (1,e<sub>1</sub>,e<sub>7</sub>,e<sub>2</sub>). This does not affect the M<sup>8</sup>-H-duality map to CP<sub>2</sub>. The 6-D space of normal spaces would be the flag manifold SU(2)/U(1)times U(1), which is nothing but the twistor space T(CP<sub>2</sub>).
</p><p>
What about the twistorial counterpart for the map of M<sup>4</sup>⊂ M<sup>8</sup>→ M<sup>4</sup>⊂ M<sup>8</sup>? One can consider several options.
<OL>
<LI> At the level of M<sup>8</sup>, M<sup>4</sup> is replaced by M<sup>6</sup> at least locally in the sense that one can use M<sup>6</sup> coordinates for the point of X<sup>6</sup>. Can one identify the M<sup>6</sup> image of this space as the projective space C<sup>4</sup>/C<sub>×</sub> obtained from C<sup>4</sup> by dividing with complex scalings? This would give the twistor space CP<sub>3</sub>= SU(4)/U(3) of M<sup>4</sup>. This is not obvious since one has (complexified) octonions rather than C<sup>4</sup> or its hypercomplex analog. This would be analogous to using several (4) coordinate charts glued together as in the case of sphere CP<sub>1</sub>.
<LI> If M<sup>8</sup>-H duality generalizes as such, the points of M<sup>6</sup> could be mapped to the 6-D analog of cd<sub>4</sub> such that the image point is defined as the intersection of a geodesic line with direction given by the 6-D momentum with the 5-D light-like boundary of 6-D counterpart cd<sub>6</sub> of cd? Does the slicing of M<sup>6</sup> by 5-D light-boundaries of cd<sub>6</sub> for various values of 6-D mass squared have interpretation as CP<sub>3</sub>? Note that the boundary of cd<sub>6</sub> does not contain origin and the same applies to CP<sub>3</sub>= C<sup>4</sup>/C<sub>×</sub>.
<LI> Or could one identify the octonionic analog of the projective space CP<sub>3</sub>=C<sup>4</sup>/C<sub>×</sub>? Could the octonionic M<sup>8</sup> momenta be scaled down by dividing with the momentum projection in the commutative normal space so that one obtains an analog of projective space? Could one use these as coordinates for M<sup>6</sup>?
</p><p>
The scaled 8-momenta would correspond to the points of the octonionic analog of CP<sub>3</sub>. The scaled down 8-D mass squared would have a constant value.
</p><p>
A possible problem is that one must divide either from left or right and results are different in the general case. Could one require that the physical states are invariant under the automorphisms generated o→ gog<sup>-1</sup>, where g is an element of the commutative subalgebra in question?
</OL>
What about the physical interpretation at the level of M<sup>8</sup><sub>c</sub>.
<OL>
<LI> The first thing to notice is that in the twistor Grassmann approach twistor space provides an elegant description of spin. Partial waves in the fiber S<sup>2</sup> of twistor space representation of spin as a partial wave. All spin values allow a unified treatment.
</p><p>
The problem is that this requires massless particles. In the TGD framework 4-D masslessness is replaced with its 8-D variant so that this difficulty is circumvented. This kind of description in terms of partial waves is expected to have a counterpart at the level of the twistor space T(M(<sup>4</sup>)× T(CP<sub>2</sub>). At level of M<sup>8</sup> the description is expected to be in terms of discrete points of M<sup>8</sup><sub>c</sub>.
<LI> Consider first the real part of X<sup>6</sup><sub>c</sub>⊂ M<sup>8</sup><sub>c</sub>. At the level of M<sup>8</sup> the points of X<sup>4</sup> correspond to points. The same must be true also at the level of X<sup>6</sup>. Single point in the fiber space S<sup>2</sup> would be selected. The interpretation could be in terms of the selection of the spin quantization axis.
</p><p>
Spin quantization axis corresponds to 2 diametrically opposite points of S<sup>2</sup>. Could the choice of the point also fix the spin direction? There would be two spin directions and in the general case of a massive particle they must correspond to the values S<sub>z</sub>= +/- 1/2 of fermion spin. For massless particles in the 4-D sense two helicities are possible and higher spins cannot be excluded. The allowance of only spin 1/2 particles conforms with the idea that all elementary particles are constructed from quarks and antiquarks. Fermionic statistics would mean that for fixed momentum one or both of the diametrically opposite points of S<sup>2</sup> defining the same and therefore unique spin quantization axis can be populated by quarks having opposite spins.
<LI> For the 6-D tangent space of X<sup>6</sup><sub>c</sub> or rather, its real projection, an analogous argument applies. The tangent space would be parametrized by a point of T(CP<sub>2</sub>) and mapped to this point. The selection of a point in the fiber S<sup>2</sup> of T(CP<sub>2</sub>) would correspond to the choice of the quantization axis of electroweak spin and diametrically opposite points would correspond to opposite values of electroweak spin 1/2 and unique quantization axis allows only single point or pair of diametrically opposite points to be populated.
</p><p>
Spin 1/2 property would hold true for both ordinary and electroweak spins and this conforms with the properties of M<sup>4</sup>× CP<sub>2</sub> spinors.
<LI> The points of X<sup>6</sup><sub>c</sub>⊂ M<sup>8</sup><sub>c</sub> would represent geometrically the modes of H-spinor fields with fixed momentum. What about the orbital degrees of freedom associated with CP<sub>2</sub>?
</p><p>
M<sup>4</sup> momenta represent orbital degrees of M<sup>4</sup> spinors so that E<sup>4</sup> parts of E<sup>8</sup> momenta should represent the CP<sub>2</sub> momenta. The eigenvalue of CP<sub>2</sub> Laplacian defining mass squared eigenvalue in H should correspond to the mass squared value in E<sup>4</sup> and to the square of the radius of sphere S<sup>3</sup> ⊂ E<sup>4</sup>.
</p><p>
This would be a concrete realization for the SO(4)=SU(2)<sub>L</sub>× SU(2)<sub>R</sub>↔ SU(3) duality between hadronic and quark descriptions of strong interaction physics. Proton as skyrmion would correspond to a map S<sup>3</sup> with radius identified as proton mass. The skyrmion picture would generalize to the level of quarks and also to the level of bound states of quarks allowed by the number theoretical hierarchy with Galois confinement. This also includes bosons as Galois confined many quark states.
<LI> The bound states with higher spin formed by Galois confinement should have the same quantization axis in order that one can say that the spin in the direction of the quantization axis is well-defined. This freezes the S<sup>2</sup> degrees of freedom for the quarks of the composite.
</OL>
<B>7-surfaces with real normal space and generalization of the notion of twistor space</B>
</p><p>
It would seem that twistorialization could correspond to the introduction
of 6-surfaces of M<sup>8</sup>, which have commutative normal space. The next step is to ask whether it makes sense to consider 7-surfaces with a real norma space allowing well-ordering? This would give a hierarchy of surfaces of M<sup>8</sup> with dimensions 7, 6, and 4. The 7-D space would have bundle projection to 6-D space having bundle projection to 4-D space.
</p><p>
What could be the physical interpretation of 7-D surfaces of M<sup>8</sup> with real normal space in the octonionic sense and of their H images?
<OL>
<LI> The first guess is that the images in H correspond to 7-D surfaces as generalizations of 6-D twistor space in the product of similar 7-D generalization of twistor spaces of M<sup>4</sup> and CP<sub>2</sub>. One would have a bundle projection to the twistor space and to the 4-D space-time.
<LI> SU(3)/U(1)× U(1) is the twistor space of CP<sub>2</sub>. SU(3)/SU(2)× U(1) is the twistor space of M<sup>4</sup>? Could 7-D SU(3)/U(1) <I> resp.</I> SU(4)/SU(3) correspond to a generalization of the twistor spaces of M<sup>4</sup> <I> resp.</I> CP<sub>2</sub>? What could be the interpretation of the fiber added to the twistor spaces of M<sup>4</sup>, CP<sub>2</sub> and X<sup>4</sup>? S<sup>3</sup> isomorphic to SU(2) and having SO(4) as isometries is the obvious candidate.
<LI> The analog of M<sup>8</sup>-H duality in Minkowskian sector in this case could be to use coordinates for M<sup>7</sup> obtained by dividing M<sup>8</sup> coordinates by the real part of the octonion. Is it possible to identify RP<sub>7</sub>= M<sup>8</sup>/R<sub>×</sub> with SU(4)/SU(3) or at least relate these spaces in a natural manner. It should be easy to answer these questions with some knowhow in practical topology.
</p><p>
A possible source of problems or of understanding is the presence of a commuting imaginary unit implying that complexification is involved in Minkowskian degrees of freedom whereas in CP<sub>2</sub> degrees of freedom it has no effect. RP<sub>7</sub> is complexified to CP<sub>7</sub> and the octonionic analog of CP<sub>3</sub> is replaced with its complexification.
</OL>
What could be the physical interpretation of the extended twistor space?
<OL>
<LI> Twistorialization takes care of spin and electroweak spin. The remaining standard model quantum numbers are Kähler magnetic charges for M<sup>4</sup> and CP<sub>2</sub> and quark number. Could the additional dimension allow their geometrization as partial waves in the 3-D fiber?
</p><p>
The first thing to notice is that it is not possible to speak about the choice of quantization axis for U(1) charge. It is however
possible to generalize the momentum space picture also to the 7-D branes X<sup>7</sup> of M<sup>8</sup> with real normal space and select only discrete points of cognitive representation carrying quarks. The coordinate of 7-D generalized momentum in the 1-D fiber would correspond to some charge interpreted as a U(1) momentum in the fiber of 7-D generalization of the twistor space.
<LI> One can start from the level of the 7-D surface with a real normal space. For both M<sup>4</sup> and CP<sub>2</sub>, a plausible guess for the identification of 3-D fiber space is as 3-sphere S<sup>3</sup> having Hopf fibration S<sup>3</sup>→ S<sup>2</sup> with U(1) as a fiber.
</p><p>
At H side one would have a wave exp(iQ φ/2π) in U(1) with charge Q and at M<sup>8</sup> side a point of X<sup>7</sup> representing Q as 7:th component of 7-D momentum.
</p><p>
Note that for X<sup>6</sup> as a counterpart of twistor space the 5:th and 6:th components of the generalized momentum would represent spin quantization axis and sign of quark spin as a point of S<sup>2</sup>. Even the length of angular momentum might allow this kind representation.
<LI> Since both M<sup>4</sup> and CP<sub>2</sub> allow induced Kähler field, a possible identification of Q would be as a Kähler magnetic charge. These charges are not conserved but in ZEO the non-conservation allows a description in terms of different values of the magnetic charge at opposite halfs of the light-cone of M<sup>8</sup> or CD.
</p><p>
Instanton number representing a change of magnetic charge would not be a charge in strict sense and drops from consideration.
</OL>
One expects that the action in the 7-D situation is analogous to Chern-Simons action associated with 8-D Kahler action, perhaps identifiable as a complexified 4-D Kähler action.
<OL>
<LI> At M<sup>4</sup> side, the 7-D bundle would be SU(4)/SU(3)→ SU(4)/SU(3)× U(1). At CP<sub>2</sub> side the bundle would be SU(3)/U(1)→ SU(3)/U(1)× U(1).
<LI> For the induced bundle as 7-D surface in the SU(4)/SU(3)× SU(3)/U(1), the two U(1):s are identified. This would correspond to an identification φ(M<sup>4</sup>)= φ(CP<sub>2</sub>) but also a more general correspondence φ(M<sup>4</sup>)= (n/m)φ(CP<sub>2</sub>) can be considered. m/n can be seen as a fractional U(1) winding number or as a pair of winding numbers characterizing a closed curve on torus.
<LI> At M<sup>8</sup> level, one would have Kähler magnetic charges Q<sub>K</sub>(M<sup>4</sup>), Q<sub>K</sub>(CP<sub>2</sub>) represented associated with U(1) waves at twistor space level and as points of X<sup>7</sup> at M<sup>8</sup> level involving quark. The same wave would represent both M<sup>4</sup> and CP<sub>2</sub> waves that would correlate the values of Kähler magnetic charges by Q<sub>K,m</sub>(M<sup>4</sup>)/Q<sub>K,m</sub>(CP<sub>2</sub>)= m/n if both are non-vanishing. The value of the ratio m/n affects the dynamics of the 4-surfaces in M<sup>8</sup> and via twistor lift the space-time surfaces in H.
</OL>
<B>How could the Grassmannians of standard twistor approach emerge number theoretically?</B>
</p><p>
One can identify the TGD counterparts for various Grassmann manifolds appearing in the standard twistor approach.
</p><p>
Consider first, the various Grassmannians involved with the standard twistor approach (<A HREF="https://cutt.ly/XE3vDKj">this</A>) can be regarded as flag-manifolds of 4-complex dimensional space T.
<OL>
<LI> Projective space is FP<sub>n-1</sub> the Grasmannian F<sub>1</sub>(F<sup>n</sup>) formed by the k-D planes of V<sup>n</sup> where F corresponds to the field of real, complex or quaternionic numbers, are the simplest spaces of this kind. The F-dimension is d<sub>F</sub>=n-1. In the complex case, this space can be identified as U(n)/U(n-1)× U(1)= CP<sub>n-1</sub>.
<LI> More general flag manifolds carry at each point a flag, which carries a flag which carries ... so that one has a hierarchy of flag dimensions d<sub>0</sub>=0<d<sub>1</sub><d<sub>2</sub>...d<sub>k</sub>=n. Defining integers n<sub>i</sub>= d<sub>i</sub>-d<sub>i-1</sub>, this space can in the complex case be expressed as U(n)/U(n<sub>1</sub>)×.....U(n<sub>k</sub>). The real dimension of this space is d<sub>R</sub>=n<sup>2</sup>-∑<sub>i</sub>n<sub>i</sub><sup>2</sup>.
<LI> For n=4 and F=C, one has the following important Grassmannians.
<OL>
<LI> The twistor space CP<sub>3</sub> is projective is of complex planes in T=C<sup>4</sup> and given by CP<sub>3</sub>=U(4)/U(3)× U(1) and has real dimension d<sub>R</sub>=6.
<LI> M=F<sub>2</sub> as the space of complex 2-flags corresponds to U(4)/U(2)× U(2) and has d<sub>R</sub>=16-8= 8. This space is identified as a complexified Minkowski space with D<sub>C</sub>= 4.
<LI> The space F<sub>1,2</sub> consisting of 2-D complex flags carrying 1-D complex flags has representation U(4)/U(2)× U(1)× U(1) and has dimension D<sub>R</sub>=10.
</p><p>
F<sub>1,2</sub> has natural projection ν to the twistor space CP<sub>3</sub> resulting from the symmetry breaking U(3)→ U(2)× U(1) when one assigns to 2-flag a 1-flag defining a preferred direction. F<sub>1,2</sub> also has a natural projection μ to the complexified and compactified Minkowski space M=F<sub>2</sub> resulting in the similar manner and is assignable to the symmetry breaking U(2)× U(2)→ U(1)× U(1) caused by the selection of 1-flag.
</p><p>
These projections give rise to two correspondences known as Penrose transform. The correspondence μ ∘ ν<sub>-1</sub> assigns to a point of twistor space CP<sub>3</sub> a point of complexified Minkowski space. The correspondence ν ∘ μ<sub>-1</sub> assigns to the point of complexified Minkowski space a point of twistor space CP<sub>3</sub>. These maps are obviously not unique without further conditions.
</OL>
</OL>
This picture generalizes to TGD and actually generalizes so that also the real Minkowski space is obtained naturally. Also the complexified Minkowski space has a natural interpretation in terms of extensions of rationals forcing complex algebraic integers as momenta. Galois confinement would guarantee that physical states as bound states have real momenta.
<OL>
<LI> The basic space is Q<sub>c</sub>=Q<sup>2</sup> identifiable as a complexified Minkowski space. The idea is that number theoretically preferred flags correspond to fields R,C,Q with real dimensions 1,2,4. One can interpret Q<sub>c</sub> as Q<sup>2</sup> and Q as C<sup>2</sup> corresponding to the decomposition of quaternion to 2 complex numbers. C in turn decomposes to R× R.
<LI> The interpretation C<sup>2</sup>= C<sup>4</sup> gives the above described standard spaces. Note that the complexified and compactified Minkowski space is not same as Q<sub>c</sub>=Q<sup>2</sup> and it seems that in TGD framework Q<sub>c</sub> is more natural and the quark momenta in M<sup>4</sup><sub>c</sub> indeed are complex numbers as algebraic integers of the extension.
</OL>
Number theoretic hierarchy R→ C→ Q brings in some new elements.
<OL>
<LI> It is natural to define also the quaternionic projective space Q<sub>c</sub>/Q=Q<sup>2</sup>/Q (see <A HREF="https://cutt.ly/LE3vM0G">this</A>), which corresponds to real Minkowski space. By non-commutativity this space has two variants corresponding to left and right division by quaternionic scales factor. A natural condition is that the physical states are invariant under automorphisms q→ hqh<sup>-1</sup> and depend only on the class of the group element. For the rotation group this space is characterized by the direction of the rotation axis and by the rotation angle around it and is therefore 2-D.
</p><p>
This space is projective space QP<sub>1</sub>, quaternionic analog of Riemann sphere CP<sub>1</sub> and also the quaternionic analog of twistor space CP<sub>3</sub> as projective space. Therefore the analog of real Minkowski space emerges naturally in this framework. More generally, quaternionic projective spaces Q<sup>n</sup> have dimension d=4n and are representable as coset spaces of symplectic groups defining the analogs of unitary/orthogonal groups for quaternions as Sp(n+1)/Sp(n)× Sp(1) as one can guess on basis of complex and real cases. M<sup>4</sup><sub>R</sub> would therefore correspond to Sp(2)/Sp(1)× SP(1).
</p><p>
QP<sub>1</sub> is homeomorphic to 4-sphere S<sup>4</sup> appearing in the construction of instanton solutions in E<sup>4</sup> effectively compactified to S<sup>4</sup> by the boundary conditions at infinity. An interesting question is whether the self-dual Kähler forms in E<sup>4</sup> could give rise to M<sup>4</sup> Kähler structure and could correspond to this kind of self-dual instantons and therefore what I have called Hamilton-Jacobi structures.
<LI> The complex flags can also contain real flags. For the counterparts of twistor spaces this means the replacement of U(1) with a trivial group in the decompositions.
</p><p>
The twistor space CP<sub>3</sub> would be replaced U(4)/U(3) and has real dimension d<sub>R</sub>=7. It has a natural projection to CP<sub>3</sub>. The space F<sub>1,2</sub> is replaced with representation U(4)/U(2) and has dimension D<sub>R</sub>=12.
</OL>
To sum up, the Grassmannians associated with M<sup>4</sup> as 6-D twistor space and its 7-D extension correspond to a complexification by a commutative imaginary unit i - that is "vertical direction". The Grassmannians associated with CP<sub>2</sub> correspond to "horizontal ", octonionic directions and to associative, commutative and well-ordered normal spaces of the space-time surface and its 6-D and 7-D extensions. Geometrization of the basic quantum states/numbers - not only momentum - representing them as points of these spaces is in question.
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGD2021.pdf">Summary of TGD as it is towards end of 2021</A> or the chapter <a HREF= "http://tgdtheory.fi/pdfpool/TGD2021.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-80999834645784320192021-10-26T03:19:00.002-07:002021-10-26T03:19:11.450-07:00Braids, anyons, and Galois groups
Braids and anyons in the TGD framework are discussed <A HREF="http://tgdtheory.fi/pdfpool/anyontgd.pdf">here</A>. Braid statistics has an interpretation in terms of rotations as homotopies at a 2-D plane of the space-time surfaces instead of rotations in M<sup>4</sup>. One can use M<sup>4</sup> coordinates for the M<sup>4</sup> projection of the space-time surface.
</p><p>
As a matter of fact, arbitrary isometry induced flows of H can be lifted to rotations as flows along the lifted curve at the space-time surface and for many-sheeted space-time the flows, which correspond to identity in H can lead to a different space-time sheet so that the braid groups structure emerges naturally (see <A HREF="http://tgdtheory.fi/public_html/articles/TGD2021.pdf">here</A>).
</p><p>
The representations of H isometries at the level of WCW act on the entire 3-surface identifiable as a generalized point-like particle and by holography on the entire space-time surface. The braid representations of isometries act inside the space-time surface. This suggests a generalization of the notions of gravitational and inertial masses so that they apply to all conserved charges. Generalization of Equivalence Principle would state that gravitational and inertial charges are identical.
</p><p>
The condition that the Dirac operator at the level of H has tangential part equivalent to the Dirac operator for induced spinors, implies that the conserved isometry currents of H are conserved along the flow lines of corresponding Killing vector fields and proportional to the Killing vectors lifted/projected to the space-time surface. This has an interpretation as a local hydrodynamics conservation law analogous to the conservation of ρ v<sup>2</sup>/2+p along a flow line.
</p><p>
One can ask whether the 2-dimensionality, which makes possible non-trivial and non-Abelian homotopy groups, is really necessary for the notion of the braid group in the TGD framework. As a matter of fact, the conditions are not expected to be possible for all conserved charges, and the intuitive guess that they hold true only for Cartan algebra representing maximal set of commuting observables would provide a space-time correlate of the Uncertainty Principle. If so, the space-time surface would depend on the choice of quantization axes. This conforms with quantum classical correspondence. For instance, the Cartan algebra of rotation group would act on a plane so that the effective 2-dimensionality of braid group and quantum group representations would hold true.
</p><p>
This view has some nice consequences.
<OL>
<LI> If the space-time surface is n-sheeted, the rotation of 2π can take the particle to a different space-time sheet, and only n fold-rotation brings it back to its original position. The formula for fractional Hall conductivity is the same as in the case of integer Hall effect except that the 1/ℏ-proportionality is replaced with 1/ℏ<sub>eff</sub>-proportionality in TGD framework (see <A HREF="http://tgdtheory.fi/pdfpool/anyontgd.pdf">this</A>).
<LI> Degeneracy of fermion states also makes non-Abelian braid statistics possible. Since the Galois group acts as a symmetry group, the degeneracy would be naturally associated with the representations of the Galois group. Galois singletness of the many-anyon states guarantees reduces braid statistics to ordinary statistics for these. Galois confinement is proposed to be a central element of quantum biology (see <A HREF="http://tgdtheory.fi/public_html/articles/darkchemi.pdf">this</A> and <A HREF="http://tgdtheory.fi/public_html/articles/Galoiscode.pdf">this</A>).
</OL>
Braid statistics could also relate to the problem created by Bose-Einstein and Fermi statistics.
<OL>
<LI> The problem is that many-boson and many-fermion states are maximally entangled so that state function reduction is in the QFT framework possible only for the entanglement between fermions and bosons.
</p><p>
In the TGD framework the situation is even more difficult since all elementary particles can be constructed from quarks. The replacement of point-like particles with 3-surfaces however forces us to re-consider the notion of particle identity. Number theoretic definition of identity applying to cognitive representations is attractive.
</p><p>
<LI> The intuitive proposal is that Galois representations can entangle and that the reduction of entanglement is possible. In particular, the decomposition of extension to a hierarchy of extensions with Galois groups forming a hierarchy of normal subgroups allows the notions of cognitive measurement cascade \cite{btart/SSFRGalois}.
<LI> A more rigorous basis for the intuition emerges from the TGD view about braiding. The Galois group can be always represented as a subgroup of a suitable symmetric group S<sub>n</sub>. S<sub>n</sub> allows braidings and therefore induces a braiding of the Galois group. The discrete subgroups of symmetry groups of TGD could allow representation as a Galois group of the space-time surface. They could also allow braiding defined by the lift of the continuous isometry flow to the space-time surface. This suggests that the notion of a quantum group could allow a geometric interpretation in terms of the braiding based on the many-sheeted sub-manifold geometry.
<LI> The Galois group is in general non-Abelian and the braided Galois group would define braid statistics allowing higher-D representations. This would also make possible a non-maximal entanglement and the reduction of entanglement for the tensor products would be possible..
</OL>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGDcondmatshort.pdf">TGD and condensed matter</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/TGDcondmatshort.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-77253147058775484832021-10-26T02:16:00.004-07:002021-10-26T02:16:46.402-07:00Adelic physics and quantum measurement theory
Adelic physics forces us to reconsider the notion of entanglement and what happens in state function reductions (SFRs). Let us leave the question whether the SFR can correspond to SSFR or BSFR or both open for a moment.
<OL>
<LI> The natural assumption is that entanglement is a number-theoretically universal concept and therefore makes sense in both real and various p-adic senses. This is guaranteed if the entanglement coefficients are in an extension E of rationals associated with the polynomial Q defining the space-time surface in M<sup>8</sup> and having rational coefficients.
</p><p>
In the general case, the diagonalized density matrix ρ produced in a state function reduction (SFR) has eigenvalues in an extension E<sub>1</sub> of E. E<sub>1</sub> is defined by the characteristic polynomial P of ρ.
<LI> Is the selection of one of the eigenstates in SFR possible if E<sub>1</sub> is non-trivial? If not, then one would have a number-theoretic entanglement protection.
<LI> On the other hand, if the SFR can occur, does it require a phase transition replacing E with its extension by E<sub>1</sub> required by the diagonalization?
</OL>
Let us consider the option in which E is replaced by an extension coding for the measured entanglement matrix so that something also happens to the space-time surface.
<OL>
<LI> Suppose that the observer and measured system correspond to 4-surfaces defined by the polynomials O and S somehow composed to define the composite system and reflecting the asymmetric relationship between O and S. The simplest option is Q=O∘ S but one can also consider as representations of the measurement action deformations of the polynomial O× P making it irreducible. Composition conforms with the properties of tensor product since the dimension of extension of rationals for the composite is a product of dimensions for factors.
<LI> The loss of correlations would suggest that a classical correlate for the outcome is a union of uncorrelated surfaces defined by O and S or equivalently by the reducible polynomial defined by the O× S (see <A HREF="http://tgdtheory.fi/public_html/articles/Galoiscode.pdf">this</A>). Information would be lost and the dimension for the resulting extension is the sum of dimensions for the composites. O however gains information and quantum classical correspondence (QCC) suggests that the polynomial O is replaced with a new one to realize this.
<LI> QCC suggests the replacement of the polynomial O the polynomial P∘ O, where P is the characteristic polynomial associated with the diagonalization of the density matrix ρ. The final state would be a union of surfaces represented by P∘ O and S: the information about the measured observable would correspond to the increase of complexity of the space-time surface associated with the observer. Information would be transferred from entangled Galois degrees of freedom including also fermionic ones to the geometric degrees of freedom P∘ O. The information about the outcome of the measurement would in turn be coded by the Galois groups and fermionic state.
<LI> This would give a direct quantum classical correspondence between entanglement matrices and polynomials defining space-time surfaces in M<sup>8</sup>. The space-time surface of O would store the measurement history as kinds of Akashic records. If the density matrix corresponds to a polynomial P which is a composite of polynomials, the measurement can add several new layers to the Galois hierarchy and gradually increase its height.
</p><p>
The sequence of SFRs could correspond to a sequence of extensions of extensions of..... This would lead to the space-time analog of chaos as the outcome of iteration if the density matrices associated with entanglement coefficients correspond to a hierarchy of powers P<sup>k</sup>.
</OL>
Does this information transfer take place for both BSFRs and SSFRs? Concerning BSFRs the situation is not quite clear. For SSFRs it would occur naturally and there would be a connection with SSFRs to which I have associated cognitive measurement cascades (see <A HREF="http://tgdtheory.fi/public_html/articles/SSFRGalois.pdf">this</A> and <A HREF="http://tgdtheory.fi/public_html/articles/GaloisTGD.pdf">this</A>).
<OL>
<LI> Consider an extension, which is a sequence of extensions E<sub>1</sub>→ ..E<sub>k</sub> → E<sub>k+1</sub>..→ E<sub>n</sub> defined by the composite polynomial P<sub>n</sub>∘ ....∘ P<sub>1</sub>. The lowest level corresponds to a simple Galois group having no non-trivial normal subgroups.
<LI> The state in the group algebra of Galois group G= G<sub>n</sub> having G<sub>n-1</sub> as a normal subgroup can be expressed as an entangled state associated with the factor groups G<sub>n</sub>/G<sub>n-1</sub> and subgroup G<sub>n-1</sub> and the first cognitive measurement in the cascade would reduce this entanglement. After that the process could but need not to continue down to G<sub>1</sub>. Cognitive measurements considerably generalize the usual view about the pair formed by the observer and measured system and it is not clear whether O-S pair can be always represented in this manner as assumed above: also small deformations of the polynomial O× S can be considered.
</p><p>
These considerations inspire the proposal the space-time surface assigned to the outcome of cognitive measurement G<sub>k</sub>,G<sub>k-1</sub> corresponds to polynomial the Q<sub>k,k-1</sub>∘ P<sub>n</sub>, where Q<sub>k,k-1</sub> is the characteristic polynomial of the entanglement matrix in question.
</OL>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGD2021.pdf">TGD Towards the End of 2021</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-9990180180358306082021-10-25T22:55:00.007-07:002021-10-25T23:06:22.891-07:00Uncertainty Principle and M8-H duality
<B>Uncertainty Principle and M<sup>8</sup>-H duality</B>
</p><p>
The detailed realization of M<sup>8</sup>-H duality (for what this duality means, see <a HREF= "http://tgdtheory.fi/public_html/articles/TGD2021.pdf">this</A>) involves still uncertainties. The quaternionic normal spaces containing fixed 2-space M<sup>2</sup> (or an integrable distribution of M<sup>2</sup>) are parametrized by points of CP<sub>2</sub>. One can map the normal space to a point of CP<sub>2</sub>.
</p><p>
The tough problem has been the precise correspondence between M<sup>4</sup> points in M<sup>4</sup>× E<sup>4</sup> and M<sup>4</sup>× CP<sub>2</sub> and the identification of the sizes of causal diamonds (CDs) in M<sup>8</sup> and H. The identification is naturally linear if M<sup>8</sup> is analog of space-time but if M<sup>8</sup> is interpreted as momentum space, the situation changes. The ealier proposal maps mass hyperboloids to light-cone proper time =constant hyperboloids and it has turned out that this correspondence does not correspond to the classical picture suggesting that a given momentum in M<sup>8</sup> corresponds in H to a geodesic line emanating from the tip of CD.
</p><p>
<B>M<sup>8</sup>-H duality in M<sup>4</sup> degrees of freedom</B>
</p><p>
The following proposal for M<sup>8</sup>-H duality in M<sup>4</sup> degrees of freedom relies on the intuition provided by UP and to the idea that a particle with momentum p<sup>k</sup> corresponds to a geodesic line with this direction emanating from the tip of CD.
<OL>
<LI> The first constraint comes from the requirement that the identification of the point p<sup>k</sup>∈ X<sup>4</sup>⊂ M<sup>8</sup> should classically correspond to a geodesic line m<sup>k</sup>= p<sup>k</sup>τ/m (p<sup>2</sup>=m<sup>2</sup>) in M<sup>8</sup> which in Big Bang analogy should go through the tip of the CD in H. This geodesic line intersects the opposite boundary of CD at a unique point.
</p><p>
Therefore the mass hyperboloid H<sup>3</sup> is mapped to the 3-D opposite boundary of cd⊂ M<sup>4</sup>⊂ H. This does not fix the size nor position of the CD (=cd× CP<sub>2</sub>) in H. If CD does not depend on m, the opposite light-cone boundary of CD would be covered an infinite number of times.
</p><p>
<LI> The condition that the map is 1-to-1 requires that the size of the CD in H is determined by the mass hyperboloid M<sup>8</sup>. Uncertainty Principle (UP) suggests that one should choose the distance T between the tips of the CD associated with m to be T= ℏ<sub>eff</sub>/m.
</p><p>
The image point m<sup>k</sup> of p<sup>k</sup> at the boundary of CD(m,h<sub>eff</sub>) is given as the intersection of the geodesic line m<sup>k</sup>= p<sup>k</sup>τ from the origin of CD(m,h<sub>eff</sub>) with the opposite boundary of CD(m,h<sub>eff</sub>):
</p><p>
m<sup>k</sup>=ℏ<sub>eff</sub>X× (p<sup>k</sup>/m<sup>2</sup>),</p><p>
X= 1/(1+ p<sub>3</sub>/p<sub>0</sub>) .
</p><p>
Here p<sub>3</sub> is the length of 3-momentum.
</p><p>
The map is non-linear. At the non-relativistic limit (X\rightarrow 1), one obtains a linear map for a given mass and also a consistency with the naive view about UP. m<sup>k</sup> is on the proper time constant mass shell so the analog of the Fermi ball in H<sup>3</sup> ⊂ M<sup>8</sup> is mapped to the light-like boundary of cd⊂ M<sup>4</sup>⊂ H.
<LI> What about massless particles? The duality map is well defined for an arbitrary size of CD. If one defines the size of the CD as the Compton length ℏ<sub>eff</sub>/m of the massless particle, the size of the CD is infinite. How to identify the CD? UP suggests a CD with temporal distance T= 2ℏ<sub>eff</sub>/p<sub>0</sub> between its tips so that the geometric definition gives p<sup>k</sup>= ℏ<sub>eff</sub>p<sup>k</sup>/p<sub>0</sub><sup>2</sup> as the point at the 2-sphere defining the corner of CD. p-Adic thermodynamics strongly suggests that also massless particles generate very small p-adic mass, which is however proportional to 1/p rather than 1/p<sup>1/2</sub>. The map is well defined also for massless states as a limit and takes massless momenta to the 3-ball at which upper and lower half-cones meet.
<LI> What about the position of the CD associated with the mass hyperboloid? It should be possible to map all momenta to geodesic lines going through the 3-ball dividing the largest CD involved with T determined by the smallest mass involved to two half-cones. This is because this 3-ball defines the geometric "Now" in TGD inspired theory of consciousness. Therefore all CDs in H should have a common center and have the same geometric "Now".
</p><p>
M<sup>8</sup>-H duality maps the slicing of momentum space with positive/negative energy to a Russian doll-like slicing of t≥0 by the boundaries of half-cones, where t has origin at the bottom of the double-cone. The height of the CD(m,h<sub>eff</sub>) is given by the Compton length L(m,h<sub>eff</sub>) = ℏ<sub>eff</sub>/m of quark. Each value of h<sub>eff</sub> corresponds its own scaled map and for h<sub>gr</sub>=GMm/v<sub>0</sub>, the size of CD(m,h<sub>eff</sub>)=GM/v<sub>0</sub> does not depend on m and is macroscopic for macroscopic systems such as Sun.
<LI> The points of cognitive representation at quark level must have momenta with components, which are algebraic integers for the extension of rationals considered. A natural momentum unit is m<sub>Pl</sub>=ℏ<sub>0</sub>/R, h<sub>0</sub> is the minimal value of h<sub>eff</sub>=h<sub>0</sub> and R is CP<sub>2</sub> radius. Only "active" points of X<sup>4</sup>⊂ M<sup>8</sup> containing quark are included in the cognitive representation. Active points give rise to active CD:s CD(m,h<sub>eff</sub>) with size L(m,h<sub>eff</sub>).
</p><p>
It is possible to assign CD(m,h<sub>eff</sub>) also to the composites of quarks with given mass. Galois confinement suggest a general mechanism for their formation: bound states as Galois singlets must have a rational total momentum. This gives a hierarchy of bound states of bound states of ..... realized as a hierarchy of CDs containing several CDs.
<LI> This picture fits nicely with the general properties of the space-time surfaces as associative "roots" of the octonionic continuation of a real polynomial. A second nice feature is that the notion of CD at the level H is forced by this correspondence. "Why CDs?" at the level of H has indeed been a longstanding puzzle. A further nice feature is that the size of the largest CD would be determined by the smallest momentum involved.
<LI> Positive and negative energy parts of zero energy states would correspond to opposite boundaries of CDs and at the level of M<sup>8</sup> they would correspond to mass hyperboloids with opposite energies.
<LI> What could be the meaning of the occupied points of M<sup>8</sup> containing fermion (quark)? Could the image of the mass hyperboloid containing occupied points correspond to sub-CD at the level of H containing corresponding points at its light-like boundary? If so, M<sup>8</sup>-H correspondence would also fix the hierarchy of CDs at the level of H.
</OL>
It is enough to realize the analogs of plane waves only for the actualized momenta corresponding to quarks of the zero energy state. One can assign to CD as total momentum and passive <I> resp.</I> active half-cones give total momenta P<sub>tot,P</sub> <I> resp.</I> P<sub>tot,A</sub>, which at the limit of infinite size for CD should have the same magnitude and opposite sign in ZEO.
</p><p>
The above description of M<sup>8</sup>-H duality maps quarks at points of X<sup>4</sup> ⊂ M<sup>8</sup> to states of induced spinor field localized at the 3-D boundaries of CD but necessarily delocalized into the interior of the space-time surface X<sup>4</sup> ⊂ H. This is analogous to a dispersion of a wave packet. One would obtain a wave picture in the interior.
</p><p>
<B>Does Uncertainty Principle require delocalization in H or in X<sup>4</sup>?</B>
</p><p>
One can argue that Uncertainty Principle (UP) requires more than the naive condition T=ℏ<sub>eff</sub>/m on the size of sub-CD. I have already mentioned two approaches to the problem: they could be called inertial and gravitational representations.
<OL>
<LI> The inertial representations assigns to the particle as a space-time surface (holography) an analog of plane wave as a superposition of space-time surfaces: this is natural at the level of WCW. This requires delocalization space-time surfaces and CD in H.
<LI> The gravitational representation relies on the analog of the braid representation of isometries in terms of the projections of their flows to the space-time surface. This does not require delocalization in H since it occurs in X<sup>4</sup>.
</OL>
Consider first the inertial representation. The intuitive idea that a single point in M<sup>8</sup> corresponds to a discretized plane wave in H in a spatial resolution defined by the total mass at the passive boundary of CD. UP requires that this plane wave should be realized at the level of H and also WCW as a superposition of shifted space-time surfaces defined by the above correspondence.
<OL>
<LI> The basic observation leading to TGD is that in the TGD framework a particle as a point is replaced with a particle as a 3-surface, which by holography corresponds to 4-surface.
</p><p>
Momentum eigenstate corresponds to a plane wave. Now planewave could correspond to a delocalized state of 3-surface - and by holography that of 4-surface - associated with a particle.
</p><p>
A generalized plane wave would be a quantum superposition of shifted space-time surfaces inside a larger CD with a phase factor determined by the 4-momentum. M<sup>8</sup>-H duality would map the point of M<sup>8</sup> containing an object with momentum p to a generalized plane wave in H. Periodic boundary conditions are natural and would force the quantization of momenta as multiples of momentum defined by the larger CD. Number theoretic vision requires that the superposition is discrete such that the values of the phase factor are roots of unity belonging to the extension of rationals associated with the space-time sheet. If momentum is conserved, the time evolutions for massive particles are scalings of CD between SSFRs are integer scalings. Also iterated integer scalings, say by 2 are possible.
<LI> This would also provide WCW description. Recent physics relies on the assumption about single background space-time: WCW is effectively replaced with M<sup>4</sup> since 3-surface is replaced with point and CP<sub>2</sub> is forgotten so that one must introduce gauge fields and metric as primary field variables.
</OL>
As already discussed, the gravitational representation would rely on the lift/projection of the flows defined by the isometry generators to the space-time surface and could be regarded as a "subjective" representation of the symmetries. The gravitational representation would generalize braid group and quantum group representations.
</p><p>
The condition that the "projection" of the Dirac operator in H is equal to the modified Dirac operator, implies a hydrodynamic picture. In particular, the projections of isometry generators are conserved along the lifted flow lines of isometries and are proportional to the projections of Killing vectors. QCC suggests that only Cartan algebra isometries allow this lift so that each choice of quantization axis would also select a space-time surface and would be a higher level quantum measurement.
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGD2021.pdf">TGD as it is towards the end of 2021</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/TGD2021.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-50743339600459417112021-10-25T03:43:00.004-07:002021-10-25T03:43:48.422-07:00Galois confinement
The notion of Galois confinement emerged in TGD inspired biology. Galois group for the extension of rationals determined by the polynomial defining the space-time surface X<sup>4</sup>⊂ M<sup>8</sup> acts as a number theoretical symmetry group and therefore also as a physical symmetry group.
<OL>
<LI> The idea that physical states are Galois singlets transforming trivially under the Galois group emerged first in quantum biology. TGD suggests that ordinary genetic code is accompanied by dark realizations at the level of magnetic body (MB) realized in terms of dark proton triplets at flux tubes parallel to DNA strands and as dark photon triplets ideal for communication and control. Galois confinement is analogous to color confinement and would guarantee that dark codons and even genes, and gene pairs of the DNA double strand behave as quantum coherent units.
<LI> The idea generalizes also to nuclear physics and suggests an interpretation for the findings claimed by Eric Reiter in terms of dark N-gamma rays analogous to BECs and forming Galois singlets. They would be emitted by N-nuclei - also Galois singlets - quantum coherently. Note that the findings of Reiter are not taken seriously because he makes certain unrealistic claims concerning quantum theory.
</OL>
</p><p>
<B>Galois confinement as a number theoretically universal manner to form bound states?</B>
</p><p>
It seems that Galois confinement might define a notion much more general than thought originally. To understand what is involved, it is best to proceed by making questions.
<OL>
<LI> Why not also hadrons could be Galois singlets so that the somewhat mysterious color confinement would reduce to Galois confinement? This would require the reduction of the color group to its discrete subgroup acting as Galois group in cognitive representations. Could also nuclei be regarded as Galois confined states? I have indeed proposed that the protons of dark proton triplets are connected by color bonds.
<LI> Could all bound states be Galois singlets? The formation of bound states is a poorly understood phenomenon in QFTs. Could number theoretical physics provide a universal mechanism for the formation of bound states. The elegance of this notion is that it makes the notion of bound state number theoretically universal, making sense also in the p-adic sectors of the adele.
<LI> Which symmetry groups could/should reduce to their discrete counterparts? TGD differs from standard in that Poincare symmetries and color symmetries are isometries of H and their action inside the space-time surface is not well-defined. At the level of M<sup>8</sup> octonionic automorphism group G<sub>2</sub> containing as its subgroup SU(3) and quaternionic automorphism group SO(3) acts in this way. Also super-symplectic transformations of δ M<sup>4</sup><sub>+/-</sub>× CP<sub>2</sub> act at the level of H.
In contrast to this, weak gauge transformations acting as holonomies act in the tangent space of H.
</p><p>
One can argue that the symmetries of H and even of WCW should/could have a reduction to a discrete subgroup acting at the level of X<sup>4</sup>. The natural guess is that the group in question is Galois group acting on cognitive representation consisting of points (momenta) of M<sup>8</sup><sub>c</sub> with coordinates, which are algebraic integers for the extension.
</p><p>
Momenta as points of M<sup>8</sup><sub>c</sub> would provide the fundamental representation of the Galois group. Galois singlet property would state that the sum of (in general complex) momenta is a rational integer invariant under Galois group. If it is a more general rational number, one would have fractionation of momentum and more generally charge fractionation. Hadrons, nuclei, atoms, molecules, Cooper pairs, etc.. would consist of particles with momenta, whose components are algebraic, possibly complex, integers.
Also other quantum numbers, in particular color, would correspond to representations of the Galois group. In the case of angular moment Galois confinement would allow algebraic half-integer valued angular momenta summing up to the usual half-odd integer valued spin.
<LI> Why Galois confinement would be needed? For particles in a box of size L the momenta are integer valued as multiples of the basic unit p<sub>0</sub>= ℏ n× 2π/L. Group transformations for the Cartan group are typically represented as exponential factors which must be roots of unity for discrete groups. For rational valued momenta this fixes the allowed values of group parameters. In the case of plane waves, momentum quantization is implied by periodic boundary conditions.
</p><p>
For algebraic integers the conditions satisfied by rational momenta in general fail. Galois confinement for the momenta would however guarantee that they are integer valued and boundary conditions can be satisfied for the bound states.
</OL>
Two further remarks are in order.
<OL>
<LI> Besides the simplest realization also a higher level realization is possible: Galois singlets are not realized in the space of momenta but in the space of wavefunctions of momenta. States of an electron in an atom serve as an analogy. Origin is invariant under the rotation group and electron at origin would be the classical analog of a rotationally invariant state. In quantum theory, this state is replaced with an s-wave invariant under rotations although its argument is not.
</p><p>
In the recent situation, one would have a wave function in the space of algebraic integers representing momenta, which are not Galois invariants but if one has Galois singlet, the average momentum as Galois invariant is ordinary integer. Also single-quark states could be Galois invariant in this sense.
<LI> The proposal inspired by TGD inspired quantum biology is that the polynomials defining 4-surface in M<sup>8</sup> vanish at origin: P(0)=0. One can form increasingly complex 4-surfaces in M<sup>8</sup> by forming composite polynomials P<sub>n</sub>∘ P<sub>n-1</sub>∘ ...∘ P<sub>1</sub> and these polynomials have roots of P<sub>1</sub>....and P<sub>n-1</sub> as their roots. These roots are like conserved genes: also the momentum spectra of Galois singlets are analogous to conserved genes. This construction applies to Galois singlets in both classical and quantal sense.
</p><p>
At the highest level one can construct states as singlets under the entire Galois group. One can use non-singlets of previous level as building bricks of these singlets.
</OL>
See the articles <A HREF="http://tgdtheory.fi/public_html/articles/TGD2010.pdf">TGD as it is towards the end of 2021</A> and
<A HREF="http://tgdtheory.fi/public_html/articles/TGDcondmatshort.pdf">TGD and condensed matter physics</A>
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-42613917900740431922021-10-25T01:19:00.004-07:002021-10-25T01:19:58.878-07:00Bird's Eye of View about the Topics of the Book "TGD and Condensed Matter"
This book tries to provide a view about the applications of TGD to condensed matter physics. Quantum TGD in its recent form. Quantum TGD relies on two different views about physics: physics as an infinite-dimensional spinor geometry based on the notion of "World of Classical Worlds" (WCW) and physics as a generalized number theory.
</p><p>
WCW picture generalizes Einstein's geometrization program to a geometrization of the entire quantum physics. Number theoretic vision states that so-called adelic physics provides a dual view about physics.
</p><p>
M<sup>8</sup>-H duality realizes these dual views in terms of space-time surfaces X<sup>4</sup>⊂ H and X<sup>4</sup>⊂ M<sup>8</sup> mapped to each other byM<sup>8</sup>-H duality. This duality turns out to be a generalization of momentum-position duality of wave mechanics. Also the duality of number theory and geometry suggested by Langlands correspondence pops up into mind.
</p><p>
<B>The view about physics at the level of H</B>
</p><p>
An important guiding principle in the development of TGD has been quantum classical correspondence (QCC), whose most profound implications follow almost trivially from the basic structure of the classical theory forming an exact part of quantum theory (here TGD differs from quantum field theories (QFTs)).
<OL>
<LI> 4-D General Coordinate Invariance (GCI) forces holography: the space-time surface associated with a given 3-surface is almost unique as an analog of Bohr orbit. X<sup>4</sup> is therefore a preferred extremal of an action principle. This realizes QCC at space-time level and leads to zero energy ontology (ZEO) generalizing the ontology of standard physics.
<LI> The new view about space-time as 4-surface X<sup>4</sup>⊂ H= M<sup>4</sup>× CP<sub>2</sub> is central for applications. One manner to formulate this is that X<sup>4</sup> is simultaneously minimal surface and extremal of Kähler action S<sub>K</sub> analogous to Maxwell action. The twistor lift of TGD forces the presence of both S<sub>K</sub> and the volume term in the action.
<LI> Especially important minimal surfaces are CP<sub>2</sub> type extremals representing building bricks of elementary particles, cosmic strings and magnetic flux tubes as their deformations so that their M<sup>4</sup> and CP<sub>2</sub> projections have dimension larger than 2, and so called massless extremals (MEs). Magnetic flux tubes appear as two variants depending on whether they carry monopole flux or not. Monopole flux tubes require no current to create the magnetic field and are not possible in Maxwellian theory. Both are in a crucial role also in condensed matter applications.
<LI> The new view about space-time differs dramatically from that of GRT. The space-time surface is topologically non-trivial in all scales and many-sheeted in the sense that CP<sub>2</sub> coordinates as function of M<sup>4</sup> coordinates and vice versa are many-valued. The space-time of GRT is obtained from the many-sheeted one in long length scale limit by replacing the sheets with a single region of M<sup>4</sup> and by deforming its metric. The gauge potentials are defined as sums of induced gauge potentials for sheets. The deviation of the metric is the sum of the deviations of the induced metric from the M<sup>4</sup> metric.
</p><p>
The new physics related to the many-sheetedness is not describable in terms of the QFT approach.
<LI> The classical field equations reduce to conservation laws for the conserved charges defined by the isometries of H. Therefore they are essentially hydrodynamical and this together with QCC is essential for TGD inspired quantum hydrodynamics (QHD). The conjecture that the extremals allow generalized Beltrami property, which implies the existence of a global coordinate varying along the flow lines of flow. For instance, Beltrami property provides purely classical geometric correlates for supra flows and supracurrents. Global coordinates allow identification of order parameters having interpretation in terms of quantum coherence.
<LI> The requirement that modified Dirac operator at the level of space-time surface is in a well-defined sense a projection of the Dirac operator of H implies that for preferred extremals the isometry currents are proportional to projections if the corresponding Killing vectors with proportionality factor constant along the projections of their flow lines.
This implies as generalization of the energy conservation along flow lines of hydrodynamical flow (ρ v<sup>2</sup>/2+p=constant).
</p><p>
This also leads to a braiding type representations for isometry flows of H in theirs of their projections to the space-time surface and it seems that quantum groups emerge from these representations. Physical intuition suggests that only the Cartan algebra corresponding to commuting observables allows this representation so that the selection of quantization axes would select also space-time surface as a higher level state function reduction.
</p><p>
One also ends up to a generalization of Equivalence Principle stating that the charges assignable to "inertial" or "objective" representations of H isometries in WCW affecting space-time surfaces as analogs of particles are identical with the charges of "gravitational" or subjective representations which act inside space-time surfaces. This has also implications for M<sup>8</sup>-H duality.
</OL>
</p><p>
<B>The view about physics at the level of M<sup>8</sup></B>
</p><p>
Over the years, the number theoretical vision has evolved to what I call adelic physics. M<sup>8</sup>-H duality as analog of momentum-position duality and of Complementarity Principle crystallizes number theoretical vision.
<OL>
<LI> Complexified octonions M<sup>8</sup><sub>c</sub> have interpretation as an analog of momentum space. There are 4-surfaces in both M<sup>8</sup> and H and they are related by M<sup>8</sup>-H duality. 4-surfaces X<sup>4</sup> ⊂ M<sup>8</sup> have associative normal spaces and are are defined as "roots" of an octonionic polynomial defined as an algebraic continuation of a real polynomial P with rational coefficients.
</p><p>
A real polynomial (its roots) therefore defines the entire 4-surface, which means holography taken to an extreme. This also motivates the preferred extremal property at the H side, where one has partial differential equations and variational principle instead of algebraic equations. The analog of Bohr orbit property is one manner to formulate the restrictions.
<LI> The notion of cognitive representation is motivated by adelic physics, and corresponds to a subset of points of X<sup>4</sup> ⊂ M<sup>8</sup> in M<sup>8</sup><sub>c</sub> such that the coordinates of the points are in the extension of rationals defined by P. They define a unique discretization of the 4-surface. Cognitive representation contains common points of the real 4-surface and its p-adic variants and makes possible the number-theoretical universality of adelic physics. It is not completely clear whether the cognitive representations are needed only on the M<sup>8</sup> side or at both sides of the duality.
<LI> The interpretation of the points of M<sup>8</sup> as momenta leads to the proposal that the points of cognitive representation are algebraic integers. Contrary to the naive expectations, all algebraic numbers of extension belong to the cognitive representation for the roots of octonionic polynomials. A natural restriction is that 4-momenta correspond to algebraic integers. A further restriction is that the "active" points of the cognitive representation are occupied by quarks (in TGD leptons can correspond to bound states of quarks). Therefore the 4-surface in H is analogous to the Fermi ball containing discrete quark momenta as an analog of cognitive representation. Condensed matter physics relies strongly on the use of momentum space so that M<sup>8</sup> picture is central for the applications in condensed matter.
<LI> Galois group of the polynomial defining X<sup>4</sup> ⊂ M<sup>8</sup> acts as symmetries for the roots of the polynomials. The order of Galois group of P is identified as effective Planck constant h<sub>eff</sub>/h<sub>0</sub>=n, where the ordinary Planck constant is a multiple of h<sub>0</sub>. n is in general not the same as the dimension of extension defined by the degree of P. The original identification was as the dimension of extension counting the number of roots. One must keep an open mind here.
</p><p>
The identification of as h<sub>eff</sub> as the order of the Galois group (rather than dimension of extension) finds support from the following. The order gives the number of regions at the orbit of Galois group (in the case that the isotropy group of the point is trivial!) and at the level of H, the action is sum of identical contributions over these regions so that Planck constant would be nh<sub>0</sub>.
</p><p>
The phases of matter with different values of h<sub>eff</sub> behave like dark matter relative to each other (this does not of course imply that galactic dark matter and energy correspond to h<sub>eff</sub>>h phases). Various quantum scales, typically proportional to h<sub>eff</sub>, can be arbitrarily long and quantum coherence is possible in all scales and assigned with the magnetic body (MB).
</p><p>
Galois confinement generalizes the notion of periodic boundary conditions and could serve as a universal mechanism for the formation of bound states. One implication is that the total momentum of a bound state formed by quarks has total momentum, which is a rational integer. This serves as an extremely powerful constraint.
</p><p>
The evolutionary hierarchies formed by the extensions of rationals defined by functional composition of polynomials are characterized by root inheritance if the condition P(0)=0 is satisfied. This gives rise to an evolutionary hierarchy of bound states such that the new level contains all bound states of the previous level (conserved genes serve as an analogy). What is nice is that the states of Galois representations which are not Galois singlets can serve as composites of Galois singlets at the next level.
</p><p>
For these reasons Galois confinement is expected to play a fundamental role in TGD and in the TGD based view about condensed matter.
<LI> Nottale introduced the notion of gravitational Planck constant ℏ<sub>eff</sub>=ℏ<sub>gr</sub> = GMm/v<sub>0</sub>, which allows us to understand the planetary planets as Bohr orbits. The form of ℏ<sub>gr</sub> is dictated by the Equivalence Principle. In the TGD framework, the hierarchy of Planck constants, the infinite range of gravitational interaction, and the absence of screening motivate this notion.
</p><p>
The gravitational Compton length for a particle with mass m is Λ<sub>gr</sub> = GM/v<sub>0</sub>=r<sub>s</sub>c/2v<sub>0</sub> and thus expressible in terms of Schwarzschild radius and has r<sub>s</sub>/2 as lower bound. For Earth of order TGD view about living matter involves ℏ<sub>gr</sub> in an essential manner.
</p><p>
Amazingly, there are indications that Λ<sub>gr</sub> for Earth could play a key role in super-conductivity, superfluidity, and in quantum analogies of hydrodynamical systems. Also the role of Λ<sub>gr</sub> for Sun is suggestive. Note that the original motivation for the large value of h<sub>eff</sub> was the TGD based model for effects of em radiation at ELF frequencies on the vertebrate brain.
</OL>
</p><p>
<B>M<sup>8</sup>-H duality</B>
</p><p>
The proposal is that the description of physics in terms of geometry and number theory are dual to each other. There are several observations motivating M<sup>8</sup>-H duality.
<OL>
<LI> There are four classical number fields: reals, complex numbers, quaternions, and octonions with dimensions 1,2,4,8. The dimension of the embedding space is D(H)= 8, the dimension of octonions. Spacetime surface has dimension D(X<sup>4</sup>)=4 of quaternions. String world sheet and partonic 2-surface have dimension D(X<sup>2</sup>) =2 of: complex numbers. The dimension D(string)=1 of string is that of reals.
<LI> Isometry group of octonions is a subgroup of automorphism group G<sub>2</sub> of octonions containing SU(3) as a subgroup. CP<sub>2</sub>=SU(3)/U(2) parametrizes quaternionic 4-surfaces containing a fixed complex plane.
</OL>
Could M<sup>8</sup> and H= M<sup>4</sup>× CP<sub>2</sub> provide alternative dual descriptions of physics?
<OL>
<LI> Actually a complexification M<sup>8</sup><sub>c</sub>== E<sup>8</sup><sub>c</sub> by adding an imaginary unit i commuting with octonion units is needed in order to obtain sub-spaces with real number theoretic norm squared. M<sup>8</sup><sub>c</sub> fails to be a field since 1/o does not exist if the complex valued octonionic norm squared ∑ o<sub>i</sub><sup>2</sup> vanishes.
<LI> The four-surfaces X<sup>4</sup> ⊂ M<sup>8</sup> are identified as "real" parts of 8-D complexified 4-surfaces X<sup>4</sup><sub>c</sub> by requiring that M<sup>4</sup>⊂ M<sup>8</sup> coordinates are either imaginary or real so that the number theoretic metric defined by octonionic norm is real. Note that the imaginary unit defining the complexification commutes with octonionic imaginary units and number theoretical norm squared is given by ∑<sub>i</sub> z<sub>i</sub><sup>2</sup> which in the general case is complex.
<LI> The space H would provide a geometric description, classical physics based on Riemann metric, differential geometric structures and partial differential equations deduced from an action principle. M<sup>8</sup><sub>c</sub> would provide a number theoretic description: no partial differential equations, no Riemannian metric, no connections...
</p><p>
M<sup>8</sup><sub>c</sub> has only the number theoretic norm squared and bilinear form, which are real only if M<sup>8</sup><sub>c</sub> coordinates are real or imaginary. This would define "physicality". One open question is whether all signatures for the number theoretic metric of X<sup>4</sup> should be allowed? Similar problem is encountered in the twistor Grassmannian approach.
<LI> The basic objection is that the number of algebraic surfaces is very small and they are extremely simple as compared to extremals of action principle. Second problem is that there are no coupling constants at the level of M<sup>8</sup> defined by action.
</p><p>
Preferred extremal property realizes quantum criticality with universal dynamics with no dependence on coupling constants. This conforms with the disappearance of the coupling constants from the field equations for preferred extremals in H except at singularities, with the Bohr orbitology, holography and ZEO. X<sup>4</sup>⊂ H is analogous to a soap film spanned by frame representing singularities and implying a failure of complete universality.
<LI> In M<sup>8</sup>, the dynamics determined by an action principle is replaced with the condition that the <I> normal</I> space of X<sup>4</sup> in M<sup>8</sup> is associative/quaternionic. The distribution of normal spaces is always integrable to a 4-surface.
</p><p>
One cannot exclude the possibility that the normal space is complex 2-space, this would give a 6-D surface. Also this kind of surfaces are obtained and even 7-D with a real normal space. They are interpreted as analogs of branes and are in central role in TGD inspired biology.
</p><p>
Could the twistor space of the space-time surface at the level of H have this kind of 6-surface as M<sup>8</sup> counterpart? Could M<sup>8</sup>-H duality relate these spaces in 16-D M<sup>8</sup><sub>c</sub> to the twistor spaces of the space-time surface as 6-surfaces in 12-D T(M<sup>4</sup>)× T(CP<sub>2</sub>)?
<LI> Symmetries in M<sup>8</sup> number theoretic: octonionic automorphism group G<sub>2</sub> which is complexified and contains SO(1,3). G<sub>2</sub> contains SU(3) as M<sup>8</sup> counterpart of color SU(3) in H. Contains also SO(3) as automorphisms of quaternionic subspaces. Could this group appear as an (approximate) dynamical gauge group?
</p><p>
M<sup>8</sup>=M<sup>4</sup>× E<sup>4</sup> as SO(4) as a subgroup. It is not an automorphism group of octonions but leaves the octonion norm squared invariant. Could it be analogous to the holonomy group U(2) of CP<sub>2</sub>, which is not an isometry group and indeed is a spontaneously broken symmetry.
</p><p>
A connection with hadron physics is highly suggestive. SO(4)=SU(2)<sub>L</sub>× SU(2)<sub>R</sub> acts as the symmetry group of skyrmions identified as maps from a ball of M<sup>4</sup> to the sphere S<sup>3</sup>⊂ E<sup>4</sup>. Could hadron physics ↔ quark physics duality correspond to M<sup>8</sup>-H duality. The radius of S<sup>3</sup> is proton mass: this would suggest that M<sup>8</sup> has an interpretation as an analog of momentum space.
</p><p>
One implication of M<sup>8</sup>-H duality is that the image of a generic point of X<sup>4</sup>⊂ M<sup>8</sup> is a single point of CP<sub>2</sub>. There is complete localization in color degrees of freedom in Einsteinian regions of space-time having 4-D M<sup>4</sup> projection and one cannot even speak of color. This would solve in a trivial manner the problem of color confinement.
</p><p>
CP<sub>2</sub> type extremals however correspond to singularities for which a single point of line has ill-defined normal space and normal spaces correspond to a 3-D surface in CP<sub>2</sub>. In this case, one can assign representations of color group SU(3) to the image of the line which is essentially CP<sub>2</sub>. Color therefore makes sense inside the Euclidean wormhole contacts.
</p><p>
For the string-like entities, the singularity at a given point of string world sheets corresponds to a 2-D surface of CP<sub>2</sub>, which is a complex manifold or Lagrangian manifold. For the two geodesic spheres the color group reduces to U(2) or SO(3), and one can speak about spontaneous breaking of the color symmetry. Also S<sup>1</sup> singularity is possible for objects with 3-D M<sup>4</sup> projection. In this case the color symmetry reduces to U(1).
<LI> What is the interpretation of M<sup>8</sup>? Massless Dirac equation in M<sup>8</sup> for the octonionic spinors must be algebraic. This would be analogous to the momentum space Dirac equation. Solutions would be discrete points having interpretation as quark momenta! Quarks pick up discrete points of X<sup>4</sup>⊂ M<sup>8</sup>. </p><p>
States turn out to be massive in the M<sup>4</sup> sense: this solves the basic problem of 4-D twistor approach (it works y for massless states only). Fermi ball is replaced with a region of a mass shell (hyperbolic space H<sup>3</sup>).
</p><p>
M<sup>8</sup> duality would generalize the momentum-position duality of the wave mechanics. QFT does not generalize this duality since momenta and position are not anymore operators.
</OL>
See the book <a HREF= "http://www.tgdtheory.fi/tgdhtml/TGDcondmat.html">TGD and Condensed Matter</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-91269669297335217832021-10-23T03:11:00.006-07:002021-10-23T03:11:59.725-07:00Condensate of electron quadruplets as a new phase of condensed matter
The formation of <A HREF="https://cutt.ly/TRcxQtz">fermion quadruplet condensates</A> is a new exotic condensed matter phenomenon discovered by Prof. Egor Babaev almost 20 years ago and 8 years after publishing a paper predicting it. Recently Babaev and collaborators presented in Nature Physics evidence of fermion quadrupling in a series of experimental measurements on the iron-based material, Ba<sub>1-x</sub>K<sub>x</sub>Fe<sub>2</sub>As<sub>2</sub>.
</p><p>
The abstract of the article summarizes the finding.
</p><p>
<I> The most well-known example of an ordered quantum state superconductivity is caused by the formation and condensation of pairs of electrons. Fundamentally, what distinguishes a superconducting state from a normal state is a spontaneously broken symmetry corresponding to the long-range coherence of pairs of electrons, leading to zero resistivity and diamagnetism.
</p><p>
Here we report a set of experimental observations in hole-doped Ba<sub>1-x</sub>K<sub>x</sub>Fe<sub>2</sub>As<sub>2</sub>. Our specific-heat measurements indicate the formation of fermionic bound states when the temperature is lowered from the normal state. However, when the doping level is x ∼ 0.8, instead of the characteristic onset of diamagnetic screening and zero resistance expected below the superconducting phase transition, we observe the opposite effect: the generation of self-induced magnetic fields in the resistive state, measured by spontaneous Nernst effect and muon spin rotation experiments. This combined evidence indicates the existence of a bosonic metal state in which Cooper pairs of electrons lack coherence, but the system spontaneously breaks time-reversal symmetry. The observations are consistent with the theory of a state with fermionic quadrupling, in which long-range order exists not between Cooper pairs but only between pairs of pairs.</I>
</p><p>
Fermion quadruplets are proposed to be formed as pairs of Cooper pairs are formed somewhat above the critical temperature T<sub>c</sub> for a transition to superconductivity. Breaking of the time reversal symmetry T is involved.
</p><p>
The question is why quadruplets are stable against thermal noise above the critical temperature. Superconductivity is thought to be lost by the thermal noise making the bound states of electrons in Cooper pair unstable. Is the binding energy for quadruplets larger than for Cooper pairs so that quadruplet condensate is possible below higher critical temperature. What is the mechanism of binding?
</p><p>
The discovery is highly interesting from the TGD point of view.
<OL>
<LI> TGD leads to a model of super-conductivity involving new physics predicted by TGD.
<LI> Adelic physics number theoretic view about dark matter as h<sub>eff</sub> >h phases h<sub>eff</sub> proportional to the order of the Galois group. This leads to the notion of Galois confinement. Galois confinement could serve as a universal mechanism for the formation of bound states including also Cooper pairs and even quadruplets. In quantum biology triplets of protons representing genetic codons and even their sequences representing genes would be formed by Galois confinement.
<LI> The finding also allows to develop more preices view of TGD view concerning discrete symmetries and their violation.
</OL>
</p><p>
<B>Time reversal symmetry in TGD</B>
</p><p>
What do time reversal symmetry and its violation mean in TGD.
<OL>
<LI> The presence of magnetic field causes violation of T in condensed matter systems.
<LI> Second, not necessarily independent, manner to violate T in TGD framework is analogous to that in strong CP breaking but different from it many crucial aspects. Vacuum functional is exponent of Kähler function but exponent can contain also an instanton term I, which is equal to a divergence of topological instant current which is axial. so that non-vanishing I suggests parity violation. The fact that exponent of I is imaginary while exponent of Kähler action is real, means C violation. If instanton current is proportional to conserved Kähler current its divergence is vanishing and M<sup>4</sup> projection is less than 4-D.
</p><p>
I is non-vanishing only if the space-time sheet in X<sup>4</sup>\subset M<sup>4</sup>\times CP<sub>2</sub> has 4-D CP<sub>2</sub> or M<sup>4</sup> projection. The first case corresponds to CP<sub>2</sub> instanton term I(CP<sub>2</sub>) and second case to I(M<sup>4</sup>) present since twistor lift forces also M<sup>4</sup> to have an analog of Kähler structure. The two Kähler currents are separately conserved.
<LI> These two mechanisms of T violation might be actually equivalent if the T violation is caused by the M<sup>4</sup> part of Kähler action. Consider a space-time surface with 2-D string world sheet as M<sup>4</sup> projection carrying Kähler electric field but necessarily vanishing Kähler magnetic field B<sub>K</sub>. If it is deformed to make M<sup>4</sup> projection 4-D, B<sub>K</sub> is generated and T is violated. Therefore generation of B<sub>K</sub> in M<sup>4</sup> can lead to a T violation.
</OL>
</p><p>
<B>Generalized Beltrami currents</B>
</p><p>
Generalized Beltrami currents are nother key notion in TGD based view about superconductivity (see <A HREF="http://tgdtheory.fi/public_html/articles/SCBerryTGD.pdf">this</A>).
<OL>
<LI> The existence of a generalized Beltrami current j= Ψ dΦ implies the existence of global coordinate Φ varying along the flow lines of the current. Also the condition dj∧ j=0 follows. The 4-D generalization states that Lorentz force and electric force vanish.
In effectively 3-D situation, j could correspond to magnetic field B and dj to current as its rotor and the Beltrami condition fof B implies that Lorentz force vanishes.
<LI> The proposal is that for the preferred extremals CP<sub>2</sub> <I> resp.</I> M<sup>4</sup> Kähler current is proportional to instanton current I(CP<sub>2</sub>) <I> resp.</I> I(M<sup>4</sup>) and therefore topological for D(CP<sub>2</sub>)=3 <I> resp.</I> D(M<sup>4</sup>)=3. For D=2 the contribution to instanton current vanishes. In this case the Lorentz force vanishes so that the divergence of the energy momentum tensor is proportional to I and vanishes so that dissipation is absent. One can verify this result using the effective 3-dimensionality of the projection and using 3-D notations: in this formulation the vanishing of Lorentz force reduces to Beltrami property for B as 3-D vector.
With this assumption, dissipation for the preferred extremals of Kähler action just as it is absent in Maxwell's theory. An open question is
whether this situation is true always so that dissipation and the observed loss of quantum coherence would be due to the finite size of space-time sheet of the system considered.
<LI> Beltrami property would serve as a classical space-time correlate for the absence of dissipation and presence of quantum coherence. Beltrami property allows defining of a supra current like quantity in terms of Ψ and Φ. Usually the superconducting order parameter Ψ is actually not an order parameter for a coherent state as a superposition of states with a varying number of Cooper pairs. Now the geometry of the space-time sheets (magnetic flux tube carrying dark Cooper pairs) allows the identification of this order parameter below the quantum coherence scale. The TGD interpretation is that the coherent state is an approximation, which does not take into account the fact that the system is not closed. There is exchange of electron pairs between ordinary and dark space-time sheets with h<sub>eff</sub>>h (see <A HREF="http://tgdtheory.fi/public_html/articles/SCBerryTGD.pdf">this</A>). Dark Cooper pairs would form bound states by Galois confinement.
<LI> In the superconducting state space-time regions would have at most 3-D M<sup>4</sup> projection at fundamental level and T would not be violated. There is no dissipation and pairs are possible below critical temperature.
</p><p>
One can also understand the Meissner effect. According to the TGD view, the monopole flux tubes generate the analog of the field H perhaps serving as an approximate average description for the field of monopole flux tubes. This field induces the analog of magnetization M involving non-monopole flux tubes. Also M would be an average field. For superconductors in the diamagnetic phase, the sum would be zero: B= H+M=0. If the Cooper pairs have spin, the supracurrents of Cooper pairs at monopole flux tubes could generate the compensating magnetization.
</OL>
</p><p>
<B>TGD view about quadruplet condensate</B>
</p><p>
How could one understand quadruplet condensate in the TGD framework?
<OL>
<LI> T violation could be accompanied by the presence of Kähler instanton term I(M<sup>4</sup>) or I(CP<sub>2</sub>) requiring 4-D M<sup>4</sup> or CP<sub>2</sub> projection: this would also generate M<sup>4</sup> magnetic fields. The M<sup>4</sup> option would bring in new physics for which also the Magnus effect of hydrodynamics suggesting Lorentz force serves as an indication <A HREF="http://tgdtheory.fi/public_html/articles/TGDhydro.pdf">this</A>).
</p><p>
For 4-D M<sup>4</sup> projection, the divergence of the axial instanton current would be non-vanishing and the proportionality of Kähler current and instanton current implying a vanishing classical dissipation would be impossible. The instanton number can be expressed as instanton flux over 3-D surfaces, which would be "holes".
<LI> For the quadruplet condensate M<sup>4</sup> projection is 4-D and T is violated. Kähler magnetic fields originating from M<sup>4</sup> part of Kähler action would be present as also dissipation. For quadruplet condensate M would not compensate for H so that net magnetic fields B would be generated and correspond to space-time sheets with 4-D M<sup>4</sup> projection.
<LI> Dark matter as phases with h<sub>eff</sub>>h would however be present and quadruplets would correspond to bound states of 4 electrons formed by Galois confinement (see <A HREF="http://tgdtheory.fi/public_html/articles/TGD2021.pdf">this</A> and <A HREF="http://tgdtheory.fi/public_html/articles/TGDcondmatshort.pdf">this</A>) stating that the total momentum of the bound state as sum of momenta, which are algebraic - possibly complex - integers, is a rational integer in accordance with the periodic boundary conditions.
<LI> What prevents the formation of Cooper pairs? Above T<sub>c</sub> thermal energy exceeds the gap energy so that Cooper pairs are thermally stable. If the binding energy for quadruplets is larger, they are stable.
<LI> In what sense the quadruplets could be regarded as bound states of Cooper pairs? Since the ordinary Cooper pairs are Galois singlets, bound state formation does not look plausible since Cooper pairs themselves are unstable. A more plausible option is that Cooper pairs involved are "off-mass-shell" in that they have momenta, which are non-trivial algebraic integers and that the sum of these momenta is a rational integer in the bound state.
</OL>
<I> Remark</I>: Four-momenta as algebraic integers are in general complex. Usual charge conjugation involves complex conjugation in CP<sub>2</sub> degrees of freedom. Is it accompanied by conjugation of the complex 4-momenta. Kähler currents of M<sup>4</sup> and CP<sub>2</sub> are separately conserved: should one regard complex conjugations in M<sup>4</sup> and CP<sub>2</sub> as independent charge conjugation like symmetries. C(M<sup>4</sup>) would however leav Galois singlets invariant.
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGDcondmatshort.pdf">TGD and condensed matter physics</A> and the book <A HREF="http://www.tgdtheory.fi/tgdhtml/TGDcondmat.html">TGD and Condensed Matter"</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-20809892615857774422021-09-28T03:55:00.005-07:002021-10-26T01:12:17.686-07:00TGD and Quantum Hydrodynamics
This work is devoted to the question of what quantum hydrodynamics could mean in the TGD framework. In the standard picture quantum hydrodynamics (see <A HREF="https://cutt.ly/JEAumRZ">this</A>) is obtained from the hydrodynamic interpretation of the Schrödinger equation. Bohm theory involves this interpretation.
<OL>
<LI> Quantum hydrodynamics appears in TGD as an <I> exact</I> classical correlate of quantum theory. Modified Dirac equation forces as a consistency condition classical field equations for X<sup>4</sup>. Actually, a TGD variant of the supersymmetry, which is very different from the standard SUSY, is in question.
<LI> TGD itself has the structure of hydrodynamics. Field equations for a single space-time sheet are conservation laws. Minimal surfaces as counterparts of massless fields emerge as solutions satisfying simultaneously analogs of Maxwell equations. Beltrami flow for classical Kähler field defines an integrable flow. There is no dissipation classically and this can be interpreted as a correlate for a quantum coherent phase.
<LI> Induced Kähler form J is the fundamental field variable. Classical em and Z<sup>0</sup> fields have it as a part. For S<sup>3</sup>⊂ CP<sub>2</sub> em and Z<sup>0</sup> fields are proportional to J: which suggests large parity breaking effects. Hydrodynamic flow would naturally correspond to a generalized Beltrami flow and flow lines would integrate to a hydrodynamic flow.
<LI> The condition that Kähler magnetic field defines an integrable flow demands that one can define a coordinate along the flow line. This would suggest non-dissipating generalized Beltrami flows as a solution to the field equations and justifies the expectation that Einstein's equations are obtained at QFT limit.
<LI> If one assumes that a given conserved current defines an integrable flow, the current is a gradient. The strongest condition is that this is true for all conserved currents. The non-triviality of the first homotopy group could allow gradient flows at the fundamental level. The situation changes at the QFT limit.
<LI> Beltrami conditions make sense also for fermionic conserved currents as purely algebraic linear conditions stating that fermionic current is a gradient of some function bilear in oscillator operators. Whether they are actually implied by the classical Beltrami conditions, is an interesting question.
<LI> The requirement that modified Dirac operator at the level of space-time surface is in a well-defined sense a projection of the Dirac operator of H implies that for preferred extremals the isometry currents are proportional to projections if the corresponding Killing vectors with proportionality factor constant along the projections of their flow lines.
This implies as generalization of the energy conservation along flow lines of hydrodynamical flow (ρ v<sup>2</sup>/2+p=constant).
</p><p>
This also leads to a braiding type representations for isometry flows of H in theirs of their projections to the space-time surface and it seems that quantum groups emerge from these representations. Physical intuition suggests that only the Cartan algebra corresponding to commuting observables allows this representation so that the selection of quantization axes would select also space-time surface as a higher level state function reduction.
</p><p>
One also ends up to a generalization of Equivalence Principle stating that the charges assignable to "inertial" or "objective" representations of H isometries in WCW affecting space-time surfaces as analogs of particles are identical with the charges of "gravitational" or subjective representations which act inside space-time surfaces. This has also implications for M<sup>8</sup>-H duality.
<LI> Minimal surfaces as analogs of solutions of massless field equations and their additional property of being extremals of Kähler action gives a very concrete connection with Maxwell's theory öcite{btart/minimal}.
</OL>
In the sequel some key challenges of hydrodynamics are considered from TGD point of view.
<OL>
<LI> The generation of turbulence is one of the main problems of classical hydrodynamics and TGD inspired quantum hydrodynamics suggests a solution to this problem. Not only "classical" is replaced with "quantum" but also quantum theory is generalized.
</p><p>
The key notion is magnetic body (MB): MB carries dark matter as h<sub>eff</sub>=nh<sub>0</sub> phases and controls the flow at the level of ordinary matter. Magnetic flux tubes would be associated with the vortices. The proposal inspired by super-fluidity is that velocity field is proportional to Kähler gauge potential and that the cores of vortices corresponds to monopole flux tubes whereas their exteriors would correspond to Lagrangian flux tubes with a vanishing Kähler field so that velocity field is gradient. Vorticity field would correspond to the Z<sup>0</sup> magnetic field so that a very close analogy with superconductivity emerges.
</p><p>
The model is applied to several situations. The generation of turbulence and its decay in a flow near boundaries is discussed. ZEO suggests that the generation of turbulence could correspond to temporary time reversal associated with a macroscopic "big" (ordinary) state function reduction (BSFR).
</p><p>
Also the connection with magnetohydrodynamics (MHD) is considered. The reconnection of the field lines is replaced with the reconnection of flux tubes. The fact that monopole flux tubes require no current to generate the magnetic field provides a new insight to the problem of how magnetic fields in astrophysical scales are generated.
</p><p>
The topological picture based on flux tubes can be applied to the collisions of circular vortices. Also the violations of the circulation theorem of Kelvin is discussed.
<LI> Second section is devoted to hydrodynamic quantum analogs studied by Bush et al. These intriguing phenomena, in particular Couder walker bounces along a Faraday wave that it generates. Also surfing mode is possible. The energy feed comes from shaking the water pool and plays a role of metabolic energy feed leading to self-organization. This phenomenon allows in the TGD framework a modelling based on quantum gravitational hydrodynamics. MB serves as a "boss" and therefore takes the role of the pilot wave proposed by Bush. The key prediction that the Faraday wave length analogous to Compton wavelength equals to the gravitational Compton length Λ<sub>gr</sub>= GM/v<sub>0</sub> is correct.
<LI> Also the electromagnetic and Z<sup>0</sup> analogs of ℏ<sub>gr</sub> make sense and it asked whether in these scales the gravitational, Z<sup>0</sup> and electromagnetic Compton lengths are identical at gravitational flux tubes and that particles are at flux tubes with length of order this wavelength.
The twistor lift predicts that also M<sup>4</sup> has Kähler structure and M<sup>4</sup> Kähler form could give contribution to electromagnetic and Z<sup>0</sup> fields. Kähler currents for M<sup>4</sup> and CP<sub>2</sub> parts are separately conserved and this leads to ask whether Magnus forces resembling Lorentz force could reflect the presence of classical Z<sup>0</sup> force or M<sup>4</sup> contribution to the Kähler force.
<LI> One section is devoted to the attempt to understand the origin of viscosity and interpret critical Reynolds numbers in the TGD framework. In TGD quantum gravitation involves quantum coherence in astrophysical scales so that it is not totally surprising that the critical Reynolds numbers associated with the turbulence in pipe flow and flow past a plate relate directly to the gravitational Compton lengths of Earth and Sun: In the case of Sun ℏ<sub>gr</sub> involves two values of the velocity parameter β<sub>0</sub> appearing in the Nottale formula. Also a model for the ordinary viscosity and its increase with a decreasing temperature is discussed.
<LI> Also nuclear and hadron physics suggests applications for QHD. The basic vision about what happens in high energy nuclear and hadron collisions is that two BSFRs ("big" state function reductions changing the arrow of time) take place. The first BSFR creates the intermediate state with h<sub>eff</sub>>h: the entire system formed by colliding systems need not be in this state. In nuclear physics this state corresponds to a dark nucleus which decays in the next BSFR to ordinary nuclei. The basic notions are the notion of dark matter at MB and ZEO, in particular the change of the arrow of time in BSFR.
</p><p>
See the article
<a HREF= "http://tgdtheory.fi/public_html/articles/TGDhydro.pdf">TGD and Quantum Hydrodynamics</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/TGDhydro.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com2tag:blogger.com,1999:blog-10614348.post-32560516679429925972021-09-22T22:39:00.002-07:002021-09-22T22:39:55.531-07:00Breakthrough prediction for quantum hydrodynamics and quantum gravity according to TGD
The hydrodynamic quantum analogs are highly interesting from TGD point of view and Wikipedia article gives a nice summary about them (see <A HREF="https://cutt.ly/xEk5Api">this</A>). The quantum-like aspects are associated with a hydrodynamical system consisting of a liquid layer and liquid drop. Liquid surface in a periodic accelerated motion due to shaking: this means energy feed. The fluid bath is just below the criticality for a generation of standing Faraday wave and the bouncind particle indeed generates this kind of wave.
</p><p>
Depending on the values of the parameters, the liquid drop is surfing, bouncing at a fixed position, or "walking" along the surface wave. The surface wave is created by the interaction of particle with the surface. These findings suggest that macrosopic quantum coherence could be involved and quantum phenomena have also classical description.
There is energy feed to the systems.
</p><p>
The findings of the group led by Bush and describe in his Youtube lecture (see <A HREF="https://cutt.ly/xEk5Api">this</A>) give a nice overall view about the quantum analogs. Bush also suggests a generalization of theory of Vigier involving two pilot waves, which correspond to those associated with wave function and to classical system and theory of Bohm involving single pilot wave assigned to wave function.
</p><p>
The article of Bush et al (see <A HREF="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.01100">this</A>) describes the findings about the analog of quantum corral. The latter involves electrons inside a circular corral defined by negative ions.
</p><p>
"<I> Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field . The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical
behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.</I></I>
</p><p>
The key questions are following.
<OL>
<LI> Could quantum classical correspondence (QCC) be more than an approximation (stationary phase approximation). Note that in TGD QCC is in a well-defined sense exact.
<LI> Can a macroscopic system can exhibit quantal looking behavior and is there a genuine quantum behavior behind it? In the TGD framework, the hierarchy of effective Planck constants h<sub>eff</sub>=nh<sub>0</sub> labelling phases of ordinary matter located at magnetic body (MB). MB has a hierarchical structure and defines a master slave hierarchy.
</p><p>
A given level of the hierarchy controls the physics at the lower levels. h<sub>eff</sub> hierarchy makes quantum coherence possible in arbitrarily long scales at MB and this induces coherence at the level of ordinary matter and makes possible self-organization. The increase of h<sub>eff</sub> requires however the analogy of metabolic energy feed quite generally.
</p><p>
There is indeed energy feed to the studied system at frequency of f=50 Hz of the vibrating cylindrical shaker. The standing wave resonance occurs at Faraday frequency f<sub>F</sub>= f/2. The Faraday frequency has slow time variation with the frequency f and slightly below f<sub>F</sub>.
</p><p>
The system system should be near criticality for the generation of h<sub>eff</sub> phases. These phases at MB would induce long range correlations of ordinary matter near criticality. The system studied is indeed near criticality for the generation of standing Faraday waves.
<LI> What could the value of h<sub>eff</sub> be? The Faraday wave length λ<sub>F</sub>= 2\pi\sqrt{2ν/μ</sub> should be equal to the analog of Compton wavelength λ<sub>c</sub> =ℏ<sub>eff</sub>/m, m the mass of the water droplet. λ<sub>F</sub> does not however depend on the mass of the droplet and in the model of the Faraday waves hydrodynamical is determined in the model considered by the properties of the fluid that is friction and kinematic viscosity.
</p><p>
The only possibility is that one has ℏ<sub>eff</sub>= ℏ<sub>gr</sub> = GMm/v<sub>0</sub>, where ℏ<sub>gr</sub> is the gravitational Planck constant introduced by Nottale and also appearing in the TGD based model of superconductivity. This would give λ<sub>F</sub>= λ<sub>>gr</sub>= GM/v<sub>0</sub>= r<sub>s</sub>(M)/2v<sub>0</sub>, where r<sub>s</sub>(M) is Schartschild radius. M is naturally the mass of Earth. The minimum value of λ<sub>gr</sub> corresponds to v<sub>0</sub>/c=1 and is λ<sub>gr</sub>= r<sub>s</sub>/2. Earth's Scwartschild radius is 8.7 mm so that one would have λ<sub>F</sub>= 4.35 mm.
</p><p>
The value of λ<sub>F</sub> for the system studied in the analog of quantum corral by Bush et al is 4.75 mm \cite{bcond/qcorral} and about 10 per cent larger than the minimal value suggesting that β<sub>0</sub>=v<sub>0</sub>/c\simeq .92!
</p><p>
If this single testable prediction is not a nasty coincidence, it would mean an instantaneous breakthrough for the TGD view about quantum gravitation as macroscopic and even astrophysical phenomenon. The only parameter that can be varied in the prediction is β<sub>0</sub>. One could measure λ<sub>F</sub>=2\pi (2ν/μ)<sup>1/2</sup> for different liquids to see whether v<sub>0</sub> codes for the properties of the liquid or whether λ<sub>F</sub> is independent of the liquid so that the classical model for Faraday waves could be wrong.
<LI> The system has a memory in the sense that the induced Faraday wave interpreted as an analog of pilot wave is affected by the bouncing particle and in turns determines particle behavior but not quite completely: an analog of non-deterministic "zitterbewegung" seems to be present for strong enough acccelerations. The observations about the double slit experiment and also about approach to chaotic behavior indeed suggests that the system is not completely deterministic. The findings also suggest that the statistical description of this non-determinism is analogous that in quantum systems.
</p><p>
In ZEO quantum state as time= constant snapshot is replaced with a space-time surface as preferred extremal (PE) analogous to Bohr orbit. What comes in mind, is that the bouncing corresponds to "small" SFRs (SSFRs). The determinism of PEs is not quite exact that it would serve as correlate for what I call cognitive measurements as SSFRs. In the TGD inspired theory of consciousness, the loci of non-determism for space-time surfaces as analogs of soap filmds would serve as the seats of mental images quite universally and also represent conscious memories.
<LI> In this talk Bush interprets the Faraday wave induced by the motion of the droplet along the surface as a kind of pilot wave. In the TGD framework the counterpart of the pilot wave would be the magnetic body (MB) carrying h<sub>eff</sub>=nh<sub>0</sub> phases quantum controlling the behavior of ordinary matter. The magnetic flux tubes assignable to the exteriors of vortex cores are proposed to be present in microscopic scale also below turbulence and to serve as correlates for the vorticity caused by the boundary conditions at the boundary of flowing liquid. Now these boundaries correspond to the boundary between air and liquid bath and air and liquid droplet and could explain how the gravitational magnetic body characterized by ℏ<sub>gr</sub> enters into the physics of the moving water droplet.
</OL>
The results discussed in the talk of Bush and the article provide a benchmark test for the general picture provided by TGD and allows to sharpen the TGD view about QCC in quantum hydrodynamics (QHD).
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGDcondmat.pdf">TGD and condensed matter</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-78436673520828894462021-09-11T00:32:00.003-07:002021-09-11T00:32:52.673-07:00TGD as it is towards end of 2021
Writing a summary about Topological Geometrodynamics (TGD) as it is now led to considerable progress in several aspects of TGD.
<OL>
<LI> The mutual entanglement of fermions (bosons) as elementary particles is always maximal so that only fermionic and bosonic degrees can entangle in QFTs. The replacement of point-like particles with 3-surfaces forces us to reconsider the notion of identical particles from the category theoretical point of view. The number theoretic definition of particle identity seems to be the most natural and implies that the new degrees of freedom make possible geometric entanglement.
</p><p>
Also the notion particle generalizes: also many-particle states can be regarded as particles with the constraint that the operators creating and annihilating them satisfy commutation/anticommutation relations. This leads to a close analogy with the notion of infinite prime.
<LI> The understanding of the details of the M<sup>8</sup>-H duality forces us to modify the earlier view. The notion of causal diamond (CD) central to zero energy ontology (ZEO) emerges as a prediction at the level of H. The pre-image of CD at the level of M<sup>8</sup> is a region bounded by two mass shells rather than CD. M<sup>8</sup>-H duality maps the points of cognitive representations as momenta of quarks with fixed mass in M<sup>8</sup> to either boundary of CD in H. Mass shell (its positive and negative energy parts) is mapped to a light-like boundary of CD with size T= h<sub>eff</sub>/m, m the mass associated with momentum.
<LI> Galois confinement at the level of M<sup>8</sup> is understood at the level of momentum space and is found to be necessary. Galois confinement implies that quark momenta in suitable units are algebraic integers but integers for Galois singlet just as in ordinary quantization for a particle in a box replaced by CD. Galois confinement could provide a universal mechanism for the formation of all bound states.
<LI> There is considerable progress in the understanding of the quantum measurement theory based on ZEO. From the point of view of cognition BSFRs would be like heureka moments and the sequence of SSFRs would correspond to an analysis having as a correlate the decay of 3-surface to smaller 3-surfaces.
</OL>
See the the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGD2021.pdf">TGD as it is towards end of 2021</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/TGD2021.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-17598832877132905512021-09-08T01:21:00.003-07:002021-09-08T01:21:51.284-07:00TGD and Condensed Matter
The writing of an article about possible condensed matter applications of TGD led to a considerable progress in TGD itself and in the following I shall briefly summarize also the new perspectives.
</p><p>
It is perhaps good to explain what TGD is not and what it is or hoped to be.
</p><p>
<OL>
<LI> "Geometro-" refers to the idea about the geometrization of physics. The geometrization program of Einstein is extended to gauge fields allowing realization in terms of the geometry of surfaces so that Einsteinian space-time as abstract Riemann geometry is replaced with sub-manifold geometry. The basic motivation is the loss of classical conservation laws in General Relativity Theory (GRT). Also the interpretation as a generalization of string models by replacing string with 3-D surface is natural.
</p><p>
Standard model symmetries uniquely fix the choice of 8-D space in which space-time surfaces live to H=M<sup>4</sup>× CP<sub>2</sub>. Also the notion of twistor is geometrized in terms of surface geometry and the existence of twistor lift fixes the choice of H completely so that TGD is unique. The geometrization applies even to the quantum theory itself and the space of space-time surfaces - "world of classical worlds" (WCW) - becomes the basic object endowed with Kähler geometry. General Coordinate Invariance (GCI) for space-time surfaces has dramatic implications. Given 3-surface fixes the space-time surface almost completely as analog of Bohr orbit (preferred extremal).This implies holography and leads to zero energy ontology (ZEO) in which quantum states are superpositions of space-time surfaces.
</p><p>
<LI> Consider next the attribute "Topological". In condensed matter physical topological physics has become a standard topic. Typically one has fields having values in compact spaces, which are topologically non-trivial. In the TGD framework space-time topology itself is non-trivial as also the topology of H=M<sup>4</sup>× CP<sub>2</sub>.
</p><p>
The space-time as 4-surface X<sup>4</sup> ⊂ H has a non-trivial topology in all scales and this together with the notion of many-sheeted space-time brings in something completely new. Topologically trivial Einsteinian space-time emerges only at the QFT limit in which all information about topology is lost.
</p><p>
Practically any GCI action has the same universal basic extremals: CP<sub>2</sub> type extremals serving basic building bricks of elementary particles, cosmic strings and their thickenings to flux tubes defining a fractal hierarchy of structure extending from CP<sub>2</sub> scale to cosmic scales, and massless extremals (MEs) define space-time correletes for massless particles. World as a set or particles is replaced with a network having particles as nodes and flux tubes as bonds between them serving as correlates of quantum entanglement.
</p><p>
"Topological" could refer also to p-adic number fields obeying p-adic local topology differing radically from the real topology.
</p><p>
<LI> Adelic physics fusing real and various p-adic physics are part of the number theoretic vision, which provides a kind of dual description for the description based on space-time geometry and the geometry of "world of classical" orders. Adelic physics predicts two fractal length scale hierarchies: p-adic length scale hierarchy and the hierarchy of dark length scales labelled by h<sub>eff</sub>=nh<sub>0</sub>, where n is the dimension of extension of rational. The interpretation of the latter hierarchy is as phases of ordinary matter behaving like dark matter. Quantum coherence is possible in all scales.
</p><p>
The concrete realization of the number theoretic vision is based on M<sup>8</sup>-H duality. The physics in the complexification of M<sup>8</sup> is algebraic - field equations as partial differential equations are replaced with algebraic equations associating to a polynomial with rational coefficients a X<sup>4</sup> mapped to H by M<sup>8</sup>-H duality. The dark matter hierarchy corresponds to a hierarchy of algebraic extensions of rationals inducing that for adeles and has interpretation as an evolutionary hierarchy.
</p><p>
M<sup>8</sup>-H duality provides two complementary visions about physics, and can be seen as a generalization of the q-p duality of wave mechanics, which fails to generalize to quantum field theories (QFTs).
</p><p>
<LI> In Zero energy ontology (ZEO), the superpositions of space-time surfaces inside causal diamond (CD) having their ends at the opposite light-like boundaries of CD, define quantum states. CDs form a scale hierarchy.
</p><p>
Quantum jumps occur between these and the basic problem of standard quantum measurement theory disappears. Ordinary state function reductions (SFRs) correspond to "big" SFRs (BSFRs) in which the arrow of time changes. This has profound thermodynamic implications and the question about the scale in which the transition from classical to quantum takes place becomes obsolete. BSFRs can occur in all scales but from the point of view of an observer with an opposite arrow of time they look like smooth time evolutions.
</p><p>
In "small" SFRs (SSFRs) as counterparts of "weak measurements" the arrow of time does not change and the passive boundary of CD and states at it remain unchanged (Zeno effect).
</OL>
The writing of the article summarizing TGD and its possible condensed matter applications led to considerable progress in several aspects of TGD and also forced to challenge some aspects of the earlier picture.
<OL>
<LI> The mutual entanglement of fermions (bosons) as elementary particles is always maximal so that only fermionic and bosonic degrees can entangle in QFTs. The replacement of point-like particles with 3-surfaces forces us to reconsider the notion of identical particles from the category theoretical point of view. The number theoretic definition of particle identity seems to be the most natural and implies that the new degrees of freedom make possible geometric entanglement.
</p><p>
Also the notion particle generalizes: also many-particle states can be regarded as particles with the constraint that the operators creating and annihilating them satisfy commutation/anticommutation relations. This leads to a close analogy with the notion of infinite prime.
<LI> The understanding of the details of the M<sup>8</sup>-H duality forces us to modify the earlier view. The notion of causal diamond (CD) central to zero energy ontology (ZEO) emerges as a prediction at the level of H. The pre-image of CD at the level of M<sup>8</sup> is a region bounded by two mass shells rather than CD. M<sup>8</sup>-H duality maps the points of cognitive representations as momenta of quarks with fixed mass in M<sup>8</sup> to either boundary of CD in H.
<LI> Galois confinement at the level of M<sup>8</sup> is understood at the level of momentum space and is found to be necessary. Galois confinement implies that quark momenta in suitable units are algebraic integers but integers for Galois singlet just as in ordinary quantization for a particle in a box replaced by CD. Galois confinement could provide a universal mechanism for the formation of all bound states.
<LI> There is considerable progress in the understanding of the quantum measurement theory based on ZEO. From the point of view of cognition BSFRs would be like heureka moments and the sequence of SSFRs would correspond to an analysis having as a correlate the decay of 3-surface to smaller 3-surfaces.
</OL>
The improved vision allows us to develop the TGD interpretation for various condensed matter notions.
<OL>
<LI> TGD is analogous to hydrodynamics in the sense that field equations at the level of H reduce to conservation laws for isometry charges. The preferred extremal property meaning that space-time surfaces are simultaneous extremals of volume action and Kähler action allows interpretation in terms of induced gauge fields. The generalized Beltrami property implies the existence of an integrable flow serving as a correlate for quantum coherence. Conserved Beltrami flows currents correspond to gradient flows. At the QFT limit this simplicity would be lost.
<LI> The fields H, M, B and D, P, E needed in the applications of Maxwell's theory could emerge at the fundamental level in the TGD framework and reflect the deviation between Maxwellian and the TGD based view about gauge fields due to CP<sub>2</sub> topology.
<LI> The understanding of macroscopic quantum phases improves. The role of the magnetic body carrying dark matter is central. The understanding of the role of WCW degrees of freedom improves considerably in the case of Bose-Einstein condensates of bosonic particles such as polaritons. M<sup>8</sup> picture allows us to understand the notion of skyrmion. The formation of Cooper pairs and analogous states with higher energy would correspond to a formation of Galois singlets liberating energy used to increase h<sub>eff</sub>. What is new is that energy feed makes possible supra-phases and their analogs above the critical temperature.
<LI> Fermi surface emerges as a fundamental notion at the level of M<sup>8</sup> but has a counterpart also at the level of H. Galois groups would be crucial for understanding braids, anyons and fractional Quantum Hall effect. Space-time surface could be seen as a curved quasicrystal associated with the lattice of M<sup>8</sup> defined by algebraic integers in an extension of rationals. Also the TGD analogs of condensed matter Majorana fermions emerge.</OL>
</OL>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/TGDcondmat.pdf">TGD and Condensed Matter</A> or a <a HREF= "http://tgdtheory.fi/pdfpool/TGDcondmat.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-44658636798601194032021-08-18T04:05:00.035-07:002021-09-06T05:49:08.703-07:00Could the notion of a polynomial of infinite degree make sense?
TGD provides motivations for the question whether the notion polynomial of infinite degree could make sense. In the following I consider this question from the point of view of physicist and start from the vision about physics as generalized number theory.
</p><p>
<B>1. Background and motivations for the idea</B>
</p><p>
M<sup>8</sup>-H-duality (H= M<sup>4</sup>× CP<sub>2</sub>) states that space-time surfaces defined as 4-D roots of complexified octonionic polynomials so that they have quaternionic normal space, can be mapped to 4-surfaces in H (see <a HREF= "http://tgdtheory.fi/public_html/articles/M8H1.pdf">this</A>, <a HREF= "http://tgdtheory.fi/public_html/articles/M8H2.pdf">this</A>, and <a HREF= "http://tgdtheory.fi/public_html/articles/M8Hperiodic.pdf">this</A>).
</p><p>
The octonionic polynomials are obtained by algebraic continuation of ordinary real polynomials with rational coefficients although one can also consider algebraic coefficients.
</p><p>
This construction makes sense also for analytic functions with rational (or algebraic) coefficients. For the twistor lift of TGD, cosmological constant Λ emerges via the coefficient of a volume term of the action containing also Kähler action (see <a HREF= "http://tgdtheory.fi/public_html/articles/smatrix.pdf">this</A>). This leads to an action consisting of Kähler action with both CP<sub>2</sub> and M<sup>4</sup> terms having very interesting and physically attractive properties, such as spin glass degeneracy. Λ=0 would correspond to an infinite volume limit making the QFT description possible as an approximate description. Also the thermodynamic limit could correspond to this limit (see <a HREF= "http://tgdtheory.fi/public_html/articles/GaloisTGD.pdf">this</A>).
</p><p>
Irreducible polynomials of rational coefficients give rise to algebraic extensions characterized by the Galois group and these notions are central in adelic vision (see <a HREF= "http://tgdtheory.fi/public_html/articles/adelephysics.pdf">this</A>).
</p><p>
I do not know of any deep reason preventing analytic functions with rational Taylor coefficients. These would make possible transcendental extensions. For instance, the product ∏<sub>p</sub>(e<sup>x</sup>-p) for some set of primes p would give as roots transcendental numbers log(p). The Galois group would be however trivial although the extension is infinite. Second example is provided by trigonometric functions sin(x) and cos(x) with roots coming as multiples of nπ and (2n+1)π/2. This might be necessary in order to have Fourier analysis. The translations by a multiple of π for x act permuted roots but do not leave rational numbers rational so that the interpretation as a Galois group is not possible so that also now Galois group would be trivial.
</p><p>
A long standing question has been whether these analytic functions could be regarded as polynomials of infinite order by posing some conditions to the Taylor coefficients. If so, one might hope that the notion of Galois group could make sense also now, and one would obtain a unified view about transcendental extensions of rationals.
<OL>
<LI> For polynomials as roots of octonionic polynomials space-time surfaces are finite and located inside finite-sized causal diamond (CD).
</p><p>
In the TGD Universe cosmological constant Λ depends on the p-adic length scale and approaches zero at infinite length scale. At the Λ=0 limit, which corresponds also to QFT and thermodynamical limits, space-time surfaces would have infinite size. Only Kähler action with M<sup>4</sup> and CP<sub>2</sub> parts and having ground state degeneracy analogous to spin glass degeneracy would be present.
<LI> The octonionic algebraic continuations of analytic functions with rational coefficients and subject to restrictions guaranteeing that the notion of prime function makes sense, would define space-time surfaces as their roots.
<LI> Prime analytic functions defining space-time surfaces would in some sense be polynomials of infinite degree and could be even characterized by the Galois group. For real polynomials complex conjugations for the roots is certainly this kind of symmetry.
</p><p>
These functions should have Taylor series at origin, which is a special point for octonionic polynomials with rational (or perhaps even algebraic) coefficients. The selection of origin as a preferred point relates directly to the condition eliminating possible problems due to the loss of associativity and commutativity.
</p><p>
The prime property is possible only if the set of these polynomials fails to have a field property (so that the inverse of any element would be well-defined) since for fields one does not have the notion of prime. The field property is lost if the allowed functions vanish at origin so that one cannot have a Taylor series at origin and the inverse diverges at origin.
</p><p>
The vanishing at origin guarantees that the functional composite f○g of f and g has the roots of g. Roots are inherited as algebraical complexity as a kind of evolution increases. In TGD inspired biology, the roots of polynomials are analogous to genes and the conservation of roots in the function composition would be analogous to the conservation of genes.
</OL>
<B>2. Intuitive view about the situation</B>
</p><p>
Could one make anything concrete about this idea? What kind of functions f could serve as analogs of polynomials of infinite degree with transcendental roots. Could any analytic function with rational coefficients vanishing at origin have a possibly unique decomposition to prime analytic functions?
<OL>
<LI> Suppose that the analytic prime decomposes to a product over monomials x-x<sub>i</sub> with transcendental roots x<sub>i</sub> such that the Taylor series has rational coefficients. This requires an infinite Taylor series.
<LI> One obtains an infinite number of conditions. Each power x<sup>n</sup> in f has a rational coefficient f<sub>n</sub> equal to the sum over all possible products ∏<sub>k=1</sub><sup>n</sup> x<sub>i<sub>k</sub></sub> of n transcendental roots x<sub>i<sub>k</sub></sub>. This gives an infinite number of conditions and each condition involves an infinite number of roots. If the number N of transcendental roots is finite as it is for polynomials, each term involves a finite number of products and the conditions imply that the roots are algebraic. The number of transcendental roots must therefore be infinite. At least formally, these conditions make sense.
<LI> The sums of products are generalized symmetric functions of transcendental roots and should have rational values equal to x<sub>n</sub>. This generalizes the corresponding condition for ordinary polynomials. Symmetric functions for S<sub>n</sub> have S<sub>n</sub> as a group of symmetries. For a Galois extension of a polynomial of order n, the Galois group is a subgroup of S<sub>n</sub>. This suggests that the Galois group is a subgroup of S<sub>∞</sub>. S<sub>∞</sub> has the simple A<sub>∞</sub> of even permutations as a subgroup . The simple groups are analogs of primes for finite groups and one can hope that this is true for infinite and discrete groups (see <a HREF= "http://tgdtheory.fi/public_html/articles/GaloisTGD.pdf">this</A>).
</OL>
There are infinitely many ways to represent an algebraic extension in terms of a polynomial and the same is true for transcendental extensions with the rationality condition.
<OL>
<LI> Consider a general decomposition of the polynomial of an infinite order to a product of monomials with roots spanning the possibly transcendental extension. Could a suitable representation of extension as an infinite polynomial allow rational coefficients f<sub>n</sub> for the function ∑ f<sub>n</sub> x<sup>n</sup> defined by the infinite product?
<LI> f<sub>n</sub> is the sum over all possible products of roots obtained by dropping n different roots from the product of all roots which should be finite and equal to one for the generalization of monic polynomials. Therefore there is an infinite sum of terms, which are inverses of finite products and therefore transcendental but one can hope that the infinite number of the summands allows the rationality condition to be satisfied.
</OL>
</p><p>
<B>3. Profinite groups and Galois extensions as inverse limits</B>
</p><p>
Infinite groups indeed appear as Galois groups of infinite extensions. Absolute Galois groups, say Galois groups of algebraic numbers, provide the basic example.
<OL>
<LI> There exists a natural topology, known as Krull topology, which turns Galois group to a profinite group (totally disconnected, Hausdorff topological group) (see <A HREF="https://en.wikipedia.org/wiki/Profinite_group">this</A>), which is also Stone space (see <A HREF="https://en.wikipedia.org/wiki/Stone_space">this</A>).
<LI> Profinite groups are not countably infinite but are effectively finite just as hyper-finite factors of type II<sub>1</sub> are finite-dimensional: they appear naturally in the TGD framework. Profinite groups have Haar measure giving them a finite volume. Profinite groups behave in many respects like finite groups (compact groups also behave in this manner as far representations are considered). Profiniteness is possessed by products, closed subgroups, and the coset groups associated with the closed normal subgroups.
<LI> Every profinite infinite group is a Galois group for an infinite extension for some field K but one cannot control which field K is realized for a given profinite group. Additive p-adics groups and their products appear as Galois groups of an infinite extension for some field K. The Galois theory of infinite field extensions involves profinite groups obtained as Galois groups for the inverse limits of finite field extensions ..F<sub>n</sub>→ F<sub>n+1</sub>→ .
<LI> This kind of iterated extensions are of special interest in the TGD framework and an infinite extension would be obtained at the limit (see <A HREF="http://tgdtheory.fi/public:_htmla/articles/SSFRGalois.pdf">this</A>). The naive expectation is that the polynomial of infinite degree is a limit of a composite ...P<sub>n</sub>○ P<sub>n-1</sub>..○ P<sub>1</sub> of rational polynomials. The number of infinite extensions obtained in this manner would be infinite.
</p><p>
An interesting question is under what conditions the limiting infinite polynomial exists as an analytic function and whether the Taylor coefficients are rational or in some extension of rationals. The naive intuition is that the inverse limit preserves rationality.
<LI> The identification as the iterate ...P<sub>n</sub>○ P<sub>n-1</sub>○ P<sub>1</sub> is indeed suggestive. Infinite cyclic extension defined at the limit by the polynomial x<sup>N</sup>, N=∞, to be discussed below, has this kind of interpretation. The Galois group of this kind of extension is however not simple.
</p><p>
<B> Remark</B>: The polynomials in question satisfying P(0)=0 are not irreducible: the composite of N polynomials has x<sup>N</sup> as a factor
and has 0 as N-fold root. The origin of octonionic M<sup>8</sup> appears as an isolated root.
<LI> Is the infinite-D extension obtained as an inverse limit transcendental or algebraic? In the TGD framework the condition that the polynomial P<sub>1</sub>○ P<sub>2</sub> has the roots of P<sub>1</sub> as roots implies the loss of the field property of analytic functions making the notion of analytic prime possible. The roots of the infinite polynomial contain all roots of finite polynomials appearing in the sequences. This would suggest that the extension is not transcendental. Giving up the property P<sub>i</sub>(0)=0 also leads to a loss of root inheritance.
</OL>
For finite-dimensional Galois extensions, there exists an infinite number of polynomials generating the extension and one can consider families of extensions parametrized by a set of rational parameters. The Galois group does not change under small variations of parameters (see <a HREF= "http://tgdtheory.fi/public_html/articles/GaloisTGD.pdf">this</A>). If the inverse limit based on an infinite composite of polynomials makes sense, the situation could be the same for possibly existing rational polynomials of infinite order? The study of infinite Galois groups could provide insights on the problem.
</p><p>
<B>4. Could infinite extensions of rationals with a simple Galois group exist?</B>
</p><p>
Simple Galois groups have no normal subgroups and are of special interest as the building bricks of extensions by functional composition of polynomials. The infinite Galois groups obtained as inverse limit have however an infinite hierarchy of normal subgroups and simple argument suggests that the extensions are algebraic. Could infinite-D transcendental extensions defined by an analytic function with rational coefficients and with a simple infinite Galois group, exist?
</p><p>
A<sub>∞</sub> is simple and could not be seen as an inverse limit. Also the groups PSL<sub>n</sub>(K) are infinite discrete groups for K=Z or Q. A further example is provided by Tarski monster groups (see <A HREF="https://en.wikipedia.org/wiki/Tarski_monster_group">this</A>) having only cyclic groups Z<sub>p</sub> for a fixed p as subgroups and existing or p>10<sup>75</sup>. For these primes, there is a continuum of these monsters.
</p><p>
If the inverse limit is essential for profiniteness for infinite groups, then simple infinite groups are excluded as Galois groups. Indeed, the topology of an infinite simple group G cannot be profinite. The Krull topology has as a basis for open sets all cosets of normal subgroups H of finite index (the number of cosets gH is finite). Simple group has no normal subgroups except a trivial group consisting of a unit element and the group itself. The only open sets would be the empty set and G itself.
</p><p>
In fact, there is also a theorem stating that every Galois group is profinite (see <A HREF="https://cutt.ly/wQ2W1Of">this</A>). All finite groups are profinite in discrete topology. This theorem however excludes infinite simple Galois groups. If one allows only polynomials with P(0)=0, the conservation of algebraic roots suggests that infinite polynomials with transcendental roots are not possible.
</p><p>
The condition for the failure of the field property however leaves the iterates of polynomials for which only the highest polynomial in the infinite sequence of functional compositions vanishes at origin. These infinite polynomials could have transcendental roots.
</p><p>
</p><p>
<B>Two examples</B>
</p><p>
In the following two examples are consider to test whether the notion of a polynomial of infinite order might work.
</p><p>
<B>5.1 Cyclic extension as an example</B>
</p><p>
The natural question is whether the transcendental roots be regarded as limits of roots for a polynomial with rational coefficients at the limit when the degree N approaches infinity. The above arguments suggest that the limits involve an infinite function composition.
</p><p>
Consider as an example cyclic extension defined by a polynomial X<sup>N</sup>, which can be regarded as a composite of polynomials x<sup>p<sub>i</sub></sup> for ∏ p<sub>i</sub>=N. This is perhaps the simplest possible extension than one can imagine.
<OL>
<LI> The roots are now powers of roots of unity. The notion of the root of unity as e<sup>i2π/N</sup> does not make sense at the limit N→ ∞. One can however consider the roots e<sup>i2π M/N</sup> and its powers such that the limit M/N → α is irrational. The powers of e<sup>i nα</sup> give a dense subset of the circle S<sup>1</sup> consisting of irrational points. Note that one obtains an infinite number of extensions labelled by irrational values of α.
<LI> The polynomial should correspond to the limit P<sub>N</sub>(x)=x<sup>N</sup>-1, N→ ∞. For each finite value of N, one has P<sub>N</sub>(x)= ∏<sub>n=1</sub><sup>N</sup>(x-U<sup>n</sup>)-1, U= e<sup>i2π/N</sup> . The reduction to P=x<sup>N</sup>-1 follows from the vanishing of all terms involving lower powers of x than x<sup>N</sup>.
<LI> If these conditions hold true at the limit N→ ∞, one obtains the same result. The coefficient of x<sup>N</sup> equals to 1 trivially. The coefficient of x<sup>N-1</sup> is the sum over all roots and should vanish. This is also assumed in Fourier analysis ∑<sub>n</sub> e<sup>iα n</sup>=0 for irrational α. For α=0 the sum equals to N=∞ identified as Dirac delta function. The lower terms give conditions expected to reduce to this condition. This can be explicitly checked for the coefficient f<sub>1</sub>.
<LI> The Galois group is in this case the cyclic group U<sub>∞,α</sub> defined by the powers of U<sub>α</sub>.
</OL>
<B>5.2 Infinite iteration yield contimuum or roots</B>
</p><p>
The iterations of polynomials define an N→ ∞ limit, which can be handled mathematically whereas for an arbitrary sequence of polynomials in the functional composition it is difficult to say anything about the possible emergence of transcendental roots. Note however that the Lim<sub>N→ ∞</sub> (1+1/N)<sup>N</sup>=e shows that transcendentals can appear as limits of rationals. I have considered iterations of polynomials and approach to chaos from the point of view of M<sup>8</sup>-H duality in (see <a HREF= "http://tgdtheory.fi/public_html/articles/tgdchaos.pdf">this</A>).
</p><p>
Consider polynomials P<sub>N</sub>= Q<sub>N</sub>○ R, where R with Q(0)=0 is fixed polynomial and Q<sub>N</sub>=Q<sup>○N</sup> is the N:th iterate of some irreducible polynomial Q with Q(0)=0 and dQ/dz(0)=0. Origin is a fixed critical point of Q and the attractor towards which the points in the attractor basin of origin end up in the iteration and become roots of P<sub>∞</sub> and are roots at this limit. For the real points in the intersection of the positive real axis and attractor basin are roots at this limit so that one has a continuum of roots. The set of roots consists of a continuous segment [0,T) and a discrete set coming from the Julia set defining the boundary of the attractor basin.
</p><p>
Profiniteness suggests an interpretation of this set in terms of p-adic topology or a product of a subset p-adic topologies somehow determined by the number theoretic properties of Q. p-Adic number fields are indeed profinite and as additive groups can act as infinite Galois groups permuting the zeros. The action of p-adic translations could indeed leave the basin of attraction invariant.
</p><p>
In the TGD framework these roots correspond to values of M<sup>4</sup> time (or actually energy!) in M<sup>8</sup> mapped to actual time values in H by M<sup>8</sup>-H duality. I have referred to them as "very special moments in the life of self" with a motivation coming from TGD inspired theory of consciousness (see <a HREF= "http://tgdtheory.fi/public_html/articles/M8H1.pdf">this</A>, <a HREF= "http://tgdtheory.fi/public_html/articles/M8H2.pdf">this</A>). One might perhaps say that at this limit subjective time consisting of these moments becomes continuous in the interval [0,T].
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/trascendGalois.pdf">Does the notion of polynomial of infinite order make sense?</A>
or the chapter <a HREF= "http://tgdtheory.fi/pdfpool/GaloisTGD.pdf">About the role of Galois groups in TGD framework</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com1tag:blogger.com,1999:blog-10614348.post-58102227233705185092021-08-13T01:49:00.005-07:002021-08-13T01:51:36.069-07:00Empirical support for the Expanding Earth Model and TGD view about classical gauge fields
I learned about some new-to-me empirical facts providing further support for the Expanding Earth Model (EEM). The first strange finding is the large fluctuations of oxygen levels during the Cambrian Explosion. The general form of EEM applies to all astrophysical objects and could explain the strange lack of craters and volcanic activity in Venus suggesting a global resurfacing for 750 million years ago.
</p><p>
Contrary to expectations, the magnetic field of Venus vanishes. The TGD based view about gauge fields differs from the standard view in that it allows the notion of monopole flux. The monopole part field would be analogous to the external magnetic field H inducing magnetization M as the non-monopole part of B. Venus would be a perfect diamagnet and even a superconductor whereas Earth would be a paramagnet. In the TGD framework, superconductivity driven by the thermal energy feed from the interior of Venus would be possible. The interior of Venus could be a living system but in a very different sense than Earth.
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/expearthnewest.pdf">Empirical support for the Expanding Earth Model and TGD view about classical gauge fields</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf"> Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-87133098765201437202021-08-12T00:59:00.007-07:002021-08-12T03:37:45.252-07:00Expanding Venus modelNews about unexpected findings relating to the physics of astrophysical objects emerge on an almost daily basis. The most recent <A HREF="https://astronomy.com/magazine/greatest-mysteries/2019/07/33-why-did-venus-turn-itself-inside-out">news </A> relates to <A HREF="https://en.wikipedia.org/wiki/Venus">Venus</A>.
</p><p>
<B> Has Venus turned itself inside-out?</B>
</p><p>
The surface of Venus was expected to have craters, just like the surface of Earth, Moon, and Mars but the number of craters is very small. The surface of Venus also has weird features and many volcanoes. Also trace signs of erosion and tectonic shifts were found. The impression is that the surface of Venus had been turned inside out in a catastrophic event that occurred about 750 million years ago.
</p><p>
Since Venus is our sister planet with almost the same mass and radius, it is interesting to notice that the biology of Earth experienced Cambrian explosion 541 million years ago.
<OL>
<LI> The TGD explanation for Cambrian Explosion relies on Expanding Earth model (see <A HREF="http://tgdtheory.fi/pdfpool/expearth.pdf">this</A>, <A HREF="http://tgdtheory.fi/public_html/articles/expearthnew.pdf">this</A>,
<A HREF="http://tgdtheory.fi/public_html/articles/platoplate.pdf">this</A>, and <A HREF ="http://tgdtheory.fi/public_html/articles/expearth2021.pdf">this</A>).
</p><p>
There was a relatively fast increase of the Earth's radius by factor, which led to the burst of underground oceans to the surface of the Earth and led to the formation of oceans. Standard cosmology predicts a continuous smooth expansion of astrophysical objects. Contrary to this prediction, astrophysical objects do not seem to expand smoothly. In the TGD Universe, the smooth expansion is replaced by rapid jerks and the Cambrian Explosion would be associated with this kind of phase transitions.
<LI> In this expansion the multicellular photosynthesizing life burst to the surface. This explains the sudden emergence of highly evolved life forms during the Cambrian Explosion that Darwin realized to be a heavy objection against his theory.
</p><p>
<LI> There are many objections to be circumvented. For instance, how photosynthesis could evolve in the underground ocean. Here TGD
views dark matter as h<sub>eff</sub>=nh<sub>0</sub> phases of ordinary matter, which are relatively dark with respect to each other, come in rescue. Dark water blobs could leak into the interior of Earth and the solar light possessing dark portion could do the same so that photosynthesis became possible (see <A HREF="http://tgdtheory.fi/public_html/articles/expearth2021.pdf">this</A>).
<LI> Did Venus experience a similar rapid expansion 200 million years earlier, about 750 million years ago (or maybe roughly at the same time). Venus does not have water at its surface. This can be understood in terms of heat from solar radiation forcing the evaporation of water and subsequent loss. This also prevented the leakage of the water to the interior of Venus. If there were no water reservoirs inside Venus, no oceans were formed. The cracks of the crust created expanding areas of magma, which were like the bottoms of the oceans at Earth. Also at Earth a fraction about 2/3 of the Earth's surface is sea bottom.
</OL>
</p><p>
<B> Why does Venus not possess a magnetic field?</B>
</p><p>
Venus offers also a second puzzle. Venus does not have an appreciable magnetic field although it has been speculated that it has had it (see <A HREF="https://astrobiology.nasa.gov/news/in-search-of-an-ancient-global-magnetic-field-on-venus/">this</A>). The solar dynamo mechanism would suggest its presence.
<OL>
<LI> TGD predicts that there are two kinds of flux tubes carrying Earth's magnetic field B<sub>E</sub> with a nominal value of .5 Gauss. This applies quite generally. The flux tubes have a closed cross section - this is possible only in TGD Universe, where the space-time is 4-surface in M<sup>4</sup>× CP<sub>2</sub>. The flux tubes can have a vanishing Kähler magnetic flux or non-vanishing quantized monopole flux: this has no counterpart in Maxwellian electrodynamics.
For Earth, the monopole part would correspond to about .2 Gauss - 2/5 of the full strength of B<sub>E</sub>.
<LI> Monopole part needs no currents to maintain it and this makes it possible to understand how the Earth's magnetic field has not disappeared a long time ago. This also explains the existence of magnetic fields in cosmological scales.
</p><p>
The orientation of the Earth's magnetic field is varying. In the TGD based model the monopole part plays the role of master. When the non-monopole part becomes too weak, the magnetic body defined by the monopole part changes its orientation. This induced currents refresh the non-monopole part (<A HREF= "http://tgdtheory.fi/public_html/articles/Bmaintenance.pdf">this</A>).
The standard dynamo model is part of this model.
<LI> There is an interesting (perhaps more than) analogy with the standard phenomenological description of magnetism in condensed matter. One has B= H+M. H field is analogous to the monopole part and the non-monopole part is analogous to the magnetization M induced by H. B= H+M would represent the total field. If this description corresponds to the presence of two kinds of flux tubes, the TGD view about magnetic fields would have been part of electromagnetism from the beginning!
</p><p>
Flux tubes can also carry electric fields and also for them this kind of decomposition makes sense. Could also the fields D and H have a similar interpretation?
</p><p>
In the linear model of magnetism, one has M= χH and B=μH= (1+χ)H. For diamagnets one has χ<0 and for paramagnets χ>0. Earth would be paramagnet with χ ≈ 3/2 if the linear model works. χ is a tensor in the general case so that B and H can have different directions.
<LI> All stars and planets, also Venus, correspond to flux tube tangles formed from monopole flux tubes. This leaves only one possibility. Venus behaves like a super-conductor and is an ideal diamagnet with χ=-1 so that B vanishes. The monopole part would be present however.
</p><p>
This could provide a totally new insight to the Meissner effect and loss of superconductivity. In TGD the based model (see <A HREF= "http://tgdtheory.fi/public_html/articles/SCBerryTGD.pdf">this</A>), monopole flux tubes carry supracurrent. The BSC model however requires the absence of a magnetic field. Could the induced non-monopole field cancel the monopole part. Venus would indeed be a superconductor!
<LI> The tilt of the rotation axis relative to the plane of rotation around the Sun is very small for Venus, about 3 degrees and much smaller than for the Earth. This implies that the surface temperature of Venus is roughly constant. At Earth plate tectonics makes possible the heat transfer from the interior to the surface and its leakage to the outer space. For Venus this is not possible.
</p><p>
Could this relate to the different magnetization properties of Earth and Venus? The TGD based model also predicts superconductivity driven by external energy feed. This would be possible also above critical temperature. The energy feed would increase the value of h<sub>eff</sub> and below the critical temperature it would be provided by the energy liberated in the formation of Cooper pairs which need not actually be the current carriers since dark electrons can carry the current without dissipation. In TGD inspired biology and quite universally, the basic role of metabolic energy feed is to prevent the reductions of the values of h<sub>eff</sub>.
</p><p>
Could the superconductivity be forced by the thermal energy feed from the interior of Venus? Superconductivity means in the TGD framework large h<sub>eff</sub> and therefore complexity, intelligence, and long quantum coherence length (see <a HREF= "http://tgdtheory.fi/public_html/articles/darkchemi.pdf">this</A>). Could Venus be alive but in a very different sense than Earth? The same question can be of course made in the case of Sun.
</p><p>
The possibility that life actually appears in cosmic scales and is associated with quantum coherent flux tube networks associated with active galactic nuclei usually identified as super-massive blackholes containing stellar and planetary systems as tangles is discussed <a HREF= "http://tgdtheory.fi/public_html/articles/galjets.pdf">here</A>.
</p><p>
Also Mars lacks the global magnetic field although it has auroras assigned with local fields. Could also Mars be alive in the same same sense as Venus? Note that the recent radius of Mars is about 1/2 of Earth's radius. If Venus expanded by factor 2, all these 3 planets would have had roughly the same radius for about 750 million years ago. Mars would be waiting for the moment of expansion.
</OL>
See the article <A HREF="http://tgdtheory.fi/public_html/articles/expearth2021.pdf"> Updated version of Expanding Earth model</A> or the chapter <A HREF="http://tgdtheory.fi/pdfpool/expearth.pdf">Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.</p><p><A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-6485415929669086732021-08-10T00:14:00.001-07:002021-08-10T00:16:28.051-07:00The conjectured duality between number theory and geometry from TGD point of viewThere was a Quanta Magazine article about the link between geometry and numbers (see <A HREF="https://www.quantamagazine.org/with-a-new-shape-mathematicians-link-geometry-and-numbers-20210719/">this</A>). In the following I consider this idea from the TGD view point.
</p><p>
<B> 1. M<sup>8</sup>-M<sup>4</sup>× CP<sub>2</sub> duality</B>
</p><p>
What makes the proposed connection so interesting from the TGD point of view is that in TGD M<sup>8</sup>-M<sup>4</sup>× CP<sub>2</sub> duality
(see for instance <A HREF="http://tgdtheory.fi/public_html/articles/M8H1.pdf">this</A>, <A HREF="http://tgdtheory.fi/public_html/articles/M8H2.pdf">this</A> and <A HREF="http://tgdtheory.fi/public_html/articles/M8Hperiodic.pdf">this</A>)
states number theoretic and geometric descriptions of physics are dual and this duality is the generalization of wave-mechanical momentum-position duality having no generalization in quantum field theory since position is not an observable in quantum field theory but mere coordinate of space-time.
<OL>
<LI> M<sup>8</sup> picture about space-time surface provides a number theoretic description of physics based on the identification of space-time surfaces as algebraic surfaces. Dynamics is coded by the condition that the normal space of the space-time surface is associative.
<LI> H= M<sup>4</sup>× CP<sub>2</sub> provides a geometric description of space-time surfaces based on differential geometry, partial differential equations, and action principle. The existence of twistor lift of TGD fixes the choice of H uniquely (<A HREF="http://tgdtheory.fi/public_html/articles/smatrix.pdf">this</A>).
</p><p>
The solutions of field equations reduce to minimal surfaces as counterparts for solutions of massless field equations and the simultaneous extremal of Kähler action implies a close connection with Maxwell's theory. Space-time surfaces would be analogous to soap films spanned by dynamically generated frames (<A HREF="http://tgdtheory.fi/public_html/articles/minimal.pdf">this</A>).
</p><p>
Beltrami field property implies that dissipation is absent at the space-time level and gives support for the conjecture that the QFT limit gives Einstein-YM field equations in good approximation. The absence of dissipation is also a correlate for quantum coherence implying absence of dissipation (<A HREF="http://tgdtheory.fi/public_html/articles/SCBerryTGD.pdf">this</A>).
</OL>
It would be very nice if this duality between number theory and geometry would be present at the level of mathematics itself.
</p><p>
<B> 2. Adelic physics as unified description of sensory experience and cognition</B>
</p><p>
Adelic physics involves both real and p-adic number fields (see for instance <A HREF ="http://tgdtheory.fi/public_html/articles/adelephysics.pdf">this</A>). p-Adic variants of the space-time surface are an essential piece and give rise to mathematical correlates of cognition. Cognitive representations are discretizations, which consist of points of space-time surface a, whose imbedding space coordinates are in an extension of rationals characterizing a given adele are common to real and various p-adic variants of space-time, define the intersection of cognitive and sensory realities.
</p><p>
What is so nice from the physics point of view, is that these discretizations are unique for a given adele and adeles form an infinite evolutionary cognitive hierarchy . The p-adic geometries proposed by Scholze would be very interesting from this point of view and I wonder whether there might be something common between TGD and the work done by Scholze. Unfortunately, I do not have the needed knowledge about technicalities.
</p><p>
<B> 3. Langlands corresponds and TGD</B>
</p><p>
Also Langlands correspondence, which I have tried to understand several times with my tiny physicist's brain, is involved.
<OL>
<LI> Global Langlands correspondence (GLC) states that there is a connection between representations of continuous groups and Galois groups of extensions of rationals.
<LI> Local LC states (LLC) states this in the case of p-adics.
</OL>
There is a nice interpretation for the two LCs in terms of sensory experience and cognition in the TGD inspired theory of consciousness.
<OL>
<LI> In adelic physics real numbers and p-adic number fields define the adele. Sensory experience corresponds to reals and cognition to p-adics. Cognitive representations are in their discrete intersection and for extensions of rationals belonging to the intersection.
<LI> Sensory world, "real" world corresponds to representation of continuous groups/Galois groups of rationals: this would be GLC.
<LI> "p-Adic" worlds correspond to cognition and representations of p-adic variants of continuous groups and Galois groups over p-adics: this would be LLC.
<LI> One could perhaps talk also about Adelic LC (ALC) in the TGD framework. Adelic representations would combine real and p-adic representations for all primes and give as complete a view about reality as possible.
</OL>
</p><p>
<B> 4. Galois groups, physics and cognition</B>
</p><p>
TGD provides a geometrization for the action of Galois groups (see <A HREF="http://tgdtheory.fi/public_html/articles/GaloisTGD.pdf">this</A> and
<A HREF="http://tgdtheory.fi/public_html/articles/Galoiscode.pdf">this</A>).
<OL>
<LI> Galois groups are symmetry groups of TGD since space-time surfaces are determined by polynomials with rational (possibly also algebraic) coefficients continued to octonionic polynomials Galois groups relate to each other sheets of space-time time and a very nice physical picture emerges. Physical states correspond to the representations of Galois groups and are crucial in the dark matter sector, especially important in quantum biology. Space-time surface provides them and also the fermionic Fock states realize them.
<LI> The order n of the Galois group over rationals corresponds to an effective Planck constant h<sub>eff</sub>= nh<sub>0</sub> so that there is a direct connection to a generalization of quantum physics (see for instance <A HREF="http://tgdtheory.fi/public_html/articles/ccheff.pdf">this</A>). The phases of ordinary matter with h<sub>eff</sub>=nh<sub>0</sub> behave like dark matter. n measures the algebraic complexity of space-time surfaces and serves also as a kind of IQ. Evolution means an increase of n and therefore increase of IQ.
</p><p>
The representation of real continuous groups assignable to the real numbers as a piece of adele would be related to the representations of Galois groups in GLC.
</p><p>
Also p-adic representations of groups are needed to describe cognition and these p-adic group representations and representations of p-adic Galois groups would be related by LLC.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p><A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com1tag:blogger.com,1999:blog-10614348.post-81404782993323639122021-08-07T22:47:00.003-07:002021-08-07T22:47:52.024-07:00Do we really understand how galaxies are formed?The continual feed of unexpected observations is forcing a critical re-evaluation of what we really know about galaxies and their formation thought to be due a condensation of matter under gravitational attraction. Even the Milky Way yields one surprise after another. It is amusing to witness how empirical findings are gradually leaving TGD as the only viable option.
</p><p>
Today's surprise was from Science alert
(see <A HREF="https://www.sciencealert.com/this-new-structure-in-the-milky-way-is-so-big-we-aren-t-sure-it-isn-t-a-spiral-arm">this</A>). It tells at layman level about the findings reported in an article accepted to The Astrophysical Journal Letters (see <A HREF="https://arxiv.org/abs/2108.01905">this</A>).
</p><p>
Cattail is a gigantic structure with a length which can be as much as 16,300 light-years, discovered in the Milky Way. It is a filament which does not seem to be analogous to a spiral arm since it does not follow the warping of the galactic disk which is thought to be an outcome of some ancient collision. In the TGD framework this structure would be associated with a cosmic string, which has in some places thickened to a flux tube and generated ordinary matter in this process.
</p><p>
Also the spiral arms might be accompanied by cosmic strings. In any case, there would be a long cosmic string orthogonal to the galactic plane (jets are parallel to it quite generally) having galaxies along it and generated by the thickening of the cosmic string generating blackhole-like entities as active galactic nuclei.
</p><p>
Just yesterday I learned that the Milky Way also offers other surprises
(see <A HREF="https://www.quantamagazine.org/the-new-history-of-the-milky-way-20201215/">this</A>).
</p><p>
One of them is that the galactic disk contains old stars that should not be there but in the outskirts of the galaxy which is the place for oldies whereas younger stars live active life in the galactic disk. This if one assumes that the usual view about the formation of galaxies is correct. This applies also to the weird filaments mentioned above.
</p><p>
In the TGD Universe, galaxies are not formed by a condensation of gas but by a process replacing inflation with a process in which cosmic strings thicken and their string tension - energy density - is reduced. The liberated energy forms the ordinary matter giving rise to the galaxy. This solves the dark matter problem: strings define dark matter and energy and no halo is needed to produce a flat velocity spectrum of distant stars. The collisions of cosmic strings are unavoidable in 3-D space and could have induced the thickening process creating the active galactic nuclei (quasars).
</p><p>
This process would be opposite to what is believed to occur in the standard model. What comes to mind is that the oldies in the disk are formed from a cosmic string portion in the galactic plane. The tangle of the cosmic string can indeed extend in the galactic plane over long distances and there can also be cosmic strings (associated with galactic spiral arms?) in the galactic plane, which would have almost intersected a cosmic string orthogonal to the plane inducing the formation of the Milky Way.
</p><p>
For the TGD views see for instance <A HREF="http://tgdtheory.fi/public_html/articles/meco.pdf">this</A>, <A HREF="http://tgdtheory.fi/public_html/articles/galaxystars.pdf">this</A>) and
<A HREF="http://tgdtheory.fi/public_html/articles/galjets.pdf">this</A>).
</p<p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-34523359373417052672021-08-07T21:36:00.005-07:002021-08-07T21:41:12.976-07:00Fractons and TGDIn Quanta Magazine there was a highly interesting article about entities known as fractons (see <A HREF=
"https://www.quantamagazine.org/fractons-the-weirdest-matter-could-yield-quantum-clues-20210726/">this</A>).
</p><p>
There seems to be two different views about fractons as one learns by going to Wikipedia. Fracton can be regarded as a as self-similar particle-like entity (see <A HREF=
"https://en.wikipedia.org/wiki/Fracton">this</A> or as "sub-dimensional" particle unable to move in isolation (see <A HREF="https://en.wikipedia.org/wiki/Fracton_subdimensional_particle">this</A>). I do not understand the motivation for "sub-dimensional". It is also unclear whether the two notions are related. The popular article assigns to the fractons both the fractal character and the inability to move in isolation.
</p><p>
The basic idea is however that discrete translational symmetry is replaced with a discrete scaling invariance. The analog of lattice which is invariant under discrete translations is fractal invariant under discrete scalings.
</p><p>
One can also consider the possibility that the time evolution operator would act as scaling rather than translation. This is something totally new from quantum field theory (QFT) point of view. In QFTs energy corresponds to time translational symmetry and Hamiltonian generates infinitesimal translations. In string models the analog of stringy Hamiltonian is the infinitesimal scaling operator, Virasoro generator L<sub>0</sub>.
</p><p>
In TGD the extension of physics to adelic physics provides number theoretic and geometric descriptions as dual descriptions of physics
(see for instance <A HREF=
"https://tgdtheory.fi/public_html/articles/adelephysics.pdf">this</A>, <A HREF="https://tgdtheory.fi/public_html/articles/M8H1">this</A>, and <A HREF=
"https://tgdtheory.fi/public_html/articles/M8H2">this</A>).
This approach also provides insights about fractons as scale invariant entities and.
<OL>
<LI> In TGD the analog of time evolution between "small" state function reductions is the exponent of the infinitesimal scaling operator, Virasoro generator L<sub>0</sub>. One could imagine fractals as states invariant under discrete scalings defined by the exponential of L<sub>0</sub>. They would be counterparts of lattices but realized at the level of space-time surfaces having quite concrete fractal structure.
<LI> In p-adic mass calculations the p-adic analog of thermodynamics for L<sub>0</sub> proportional to mass squared operator M<sup>2</sup> replaces energy. This approach is the counterpart of the Higgs mechanism which allows only to reproduce masses but does not predict them. I carried out the calculations already around 1995 and the predictions were amazingly successful and eventually led to what I call adelic physics fusing real and various p-adic physics (see <A HREF=
"https://tgdtheory.fi/public_html/bookpdf/padphys.pdf">this</A>).
<LI> Long range coherence and absence of thermal equilibrium are also mentioned as properties of fractons (at least those of the first kind). Long range coherence could be due to the predicted hierarchy of Planck constants h<sub>eff</sub>=n×h<sub>0</sub> assigned with dark matter and predicting quantum coherence in arbitrarily long scales and associated with what I called magnetic bodies.
</p><p>
If translations are replaced by discrete scalings, the analogs of thermodynam equilibria would be possible for L<sub>0</sub> rather than energy. Fractals would be the analogs of thermodynamic equilibria. In p-adic thermodynamic elementary particles are thermodynamic equilibria for L<sub>0</sub> but it is not clear whether the analogy with fractal analog of a plane wave in lattice makes sense.
</OL>
Fractons are also reported to be able to move only in combinations. This need not relate to the scaling invariance. What this actually means, remained unclear to me from the explanation. What comes to mind is color confinement: free quarks are not possible. Quarks are unable to exist as isolated entities, not only to move as in isolated entities.
</p><p>
In TGD number theoretical vision leads to the notion of Galois confinement analogous to color confinement. The Galois group of a given extension of rationals indeed acts as a symmetry at space-time level. In TGD inspired biology Galois groups would play a fundamental role. For instance, dark analogs of genetic codons, codon pairs, and genes would be singlets (invariant) under an appropriate Galois group and therefore behave as a single quantum coherent dynamical and informational unit. See (see <A HREF=
"http://tgdtheory.fi/public_html/articles/darkchemi.pdf">this</A> and <A HREF=
"http://tgdtheory.fi/public_html/articles/TIH.pdf">this</A>) .
</p><p>
Suppose that one has a system - say a fractal analog of a lattice consisting of Galois singlets. Could fracton be identified as a state which is analogous to quark or gluon and therefore not invariant under the Galois group. The physical states could be formed from these as Galois singlets and are like hadrons.
</p><p>For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-1534089303600824262021-08-06T00:32:00.006-07:002021-08-07T19:53:06.264-07:00Hyperon problem of stellar cores in TGD frameworkHyperon problem is a mystery related to the physics of neutron stars (see
<A HREF="https://www.secretsofuniverse.in/hyperon-puzzle-neutron-stars">this</A>).
In neutron star neutrons temperature is zero in good approximation and Fermi statistics implies that the all states characterized by momentum and spin are filled up to maximum energy, known as Fermi energy E<sub>F</sub> identifiable as a chemical potential determined by the number density of fermions.
</p><p>
The increase of density inside a neutron star increases the total Fermi energy. Above a critical Fermi temperature possible in the core of the neutron star, the transformation of neutrons to hyperons which are baryons with some strange quarks becomes possible. λ hyperon with mass about 10 percent higher than neutron mass becomes possible. In a thermo-dynamical equilibrium the chemical potentials of hyperons and neutrons are identical. Note that chemical potentials are in a good approximation Fermi energies at zero temperature.
</p><p>
If part of neutrons transform to hyperons, the total energy decreases since the Fermi energy scales like 1/mass. One therefore expects the presence of hyperons in the cores of neutron stars, where the density and therefore also Fermi energy is high enough. The problem is that the maximal mass for known stars is above the maximal mass expected if hyperon fraction is present. Hyperon cores seem to be absent.
</p><p>
If further neutrons are added part of them transforms to hyperons and eventually all particles transform to neutrons and one can even think of the doomsday option that all matter transforms to hyperon stars.
</p><p>
Can one imagine any manner to prevent the formation of the hyperon core? Could the Fermi energy in the core remain below the needed critical Fermi energy by some new physics mechanism.
<OL>
<LI> Apart from numerical constants, the Fermi energy for effectively n-D system is given by E<sub>F</sub>= ℏ<sup>2</sup> k<sub>F</sub><sup>2</sup>/2, where k<sub>F</sub> is some power of number density (N/V<sub>n</sub>)<sup>2/n</sup>, where V<sub>n</sub> refers to volume, area, or length for n=3, 2, 1. Since zero temperature approximation is good, Fermi energy depends only on the density.
<LI> Could one think that part of neutrons transforms to dark neutrons in the transformation h<sub>eff</sub>→ kh<sub>eff</sub> such that neither mass, energy, and Fermi energy are not affected but that wavelength is scaled up as also the volume. For an effectively 3-D system, dark neutrons would occupy a volume which is scaled up by factor k<sup>3</sup>.
<LI> The Fermi energies as chemical potentials for both ordinary neutrons and their dark variants could remain the same in thermal equilibrium and remain below the critical value so that the transformation to hyperons would not take place? The condition that Fermi energies are the same implies that the numbers of ordinary and dark neutrons are the same. This would reduce individual Fermi energies by a factor 1/2<sup>2/3</sup> but is only a temporary solution.
</p><p>
One can however introduce phases with k different values of h<sub>eff</sub> and in this case the reduction of Fermi energies is 1/k<sup>2/3</sup>.
<LI> Fermi statistics might however pose a problem. The second quantization of the induced spinor fields at the space-time surface is induced by the second quantization of free spinor fields in the embedding space M<sup>4</sup>×CP<sub>2</sub>. Could the CP<sub>2</sub> degrees of freedom give additional degrees of freedom realized as many-sheeted structures allowing to avoid the problems with Fermi statistics?
</OL>
See the article <A HREF="http://tgdtheory.fi/public_html/articles/darkcore.pdf"> Solar Metallicity Problem from TGD Perspective</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-13522200941515802772021-08-05T21:00:00.004-07:002021-08-07T19:53:53.748-07:00Time crystals in TGD framework
Google has reported about a realization of a time crystal as a spin system.
A rather hypish layman article at <A HREF="https://thenextweb.com/news/google-may-have-achieved-breakthrough-time-crystals">here</A> creates the impression that perpetuum mobile has been discovered. Also the Quanta Magazine <AS HREF="https://www.quantamagazine.org/first-time-crystal-built-using-googles-quantum-computer-20210730/">article</A> creates this impression. The original research article can be found in
<A HREF="https://arxiv.org/abs/2107.13571">arXiv.org</A>.
</p><p>
It is interesting to look at the situation in the TGD framework. From the abstract of the article also from the Wikipedia article about time crystals one learns that the system has periodic energy feed and is therefore not closed so that the finding is not in conflict with the second law and perpetuum mobile is not in question.
</p><p>
<B>1. What is time crystal?</B>
</p><p>
The notion of time crystal (<A HREF="https://cutt.ly/2n65xOk">this</A>) is a temporal analog of ordinary crystals in the sense that there is temporal periodicity, was proposed by Frank Wilczeck in 2012. Experimental realization was demonstrated in 2016-2017 but not in the way theorized by Wilczek. Soon also a no-go theorem against the original form of the time crystal emerged and motivated generalizations of Wilzeck's proposal.
</p><p>
The findings reported by Google are however extremely interesting. Very concisely, researchers study a spin system, which has two directions of magnetization and the external laser beam induces the system to oscillate between the two magnetization directions with a period, which is a multiple of the period of the laser beam. It is interesting to consider the system in TGD framework and I have actually discussed time crystals briefly in a recent <A HREF="http://tgdtheory.fi/public_html/articles/minimal.pdf">article</A>.
</p><p>
<B> 2. Space-time surfaces as periodic minimal surfaces as counterparts of time crystals</B>
</p><p>
In TGD, classical physics is an exact part of quantum theory and quantum classical correspondence holds true. Hence it is interesting to consider first the situation at the classical space-time level. In TGD time crystals have as classical correlates space-time surfaces which are periodic minimal surfaces.
</p><p>
It is possible to have analogs of time-crystals and also more general structures built as piles of lego like basic pieces in time direction bringing in mind sentences of language and DNA, which is quasi-periodic structure and more general than crystal.
</p><p>
<B>3. What about thermodynamics of time crystals?</B>
</p><p>
Could the time crystal be possible also in thermodynamic sense and even for thermodynamically closed systems? In TGD Negentropy Maximization Principle (NMP) (see <A HREF="http://www.tgdtheory.fi/articles/nmpsecondlaw.html">this</A>) and zero energy ontology (ZEO) (see <A HREF="http://www.tgdtheory.fi/articles/zeoquestions.html">this</A> and <A HREF="http://www.tgdtheory.fi/articles/zeonew.html">this</A>) forces to generalize thermodynamics to allow both time arrows. ZEO is forced by TGD inspired theory of consciousness and solves the basic paradox of quantum measurement theory. The arrow of time would change in ordinary ("big") state function reduction (BSFR) and would remain unaffected in "small" SFR (SSFR). Second law holds true at the level of real physics but in the cognitive sector information increases and NMP holds true.
</p><p>
<B> 4. Is new quantum theory making possible quantum coherence in long scales needed?</B>
</p><p>
Also new quantum theory might be needed to explain why the period is multiple of the driving period. The first possibly needed new element is hierarchy of effective Planck constants h<sub>eff</sub>= n×h_0 having number theoretical interpretation. h<sub>eff</sub> measures the scale of quantum coherence and has also interpretation as the order of Galois group for a polynomial defining the space-time surface in M<sup>8</sup> mapped to M<sup>4</sup>×CP<sub>2</sub> by M<sup>8</sup>-H duality (see <A HREF="http://www.tgdtheory.fi/articles/M8H1.html">this</A> and <A HREF="http://www.tgdtheory.fi/articles/M8H2.html">this</A>).
</p><p>
The replacement of h with h<sub>eff</sub> scales the periods by n and keeps energies unchanged. In TGD inspired biology h<sub>eff</sub> hierarchy is in a crucial role and its levels behave relative to each other like dark matter.
</p><p>
In the recent case, the magnetic body (MB)of the spin system controlling its behavior would have h<sub>eff</sub>=nh. Each period would be initiated by BSFR at the level of MB and change the arrow of time and induce effective change of it also at the level of the ordinary matter.
</p><p>
In ZEO, time crystal-like entities, which live in cycle by extracting back part of the energy that they have dissipated in a time reversed mode, are in principle possible. System "breathes". Various bio-rhythms could correspond to time crystals. The biological analogy is obvious and we know that life requires a metabolic energy feed: in TGD Universe it prevents the decrease of h<sub>eff</sub> (see <A HREF="http://www.tgdtheory.fi/articles/darkchemi.html">this</A>).
</p><p>
<B> 5. Perpetuum mobile?: almost but not quite!</B>
</p><p>
For an thermodynamically open system, part of the dissipated energy leaks into the external world during each half cycle. Same happens in the time reversed mode and would mean that the system apparently receives positive energy also from the external world. Could this energy feed compensate for the energy loss to the external world by dissipation so that no external energy feed would be needed? Perhaps this might be the case in the ideal situation.
</p><p>
One would have almost a perpetuum mobile! Periodic driving feeding energy to the system would be needed to take care that h<sub>eff</sub> is not reduced.
</p><p>
See the chapter <A HREF="http://tgdtheory.fi/pdfpool/qcritdark2.pdf">Quantum Criticality and Dark Matter: part II</A>.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-56706474450819843362021-08-05T20:02:00.004-07:002021-08-07T19:54:47.823-07:00Huge fluctuations in oxygen concentration during Cambrian Explosion and Expanding Earth model
I encountered two interesting articles related to the Great Oxidation Event that started long before the Cambrian Explosion (CE) and reached its climax during CE (about 541 million years ago) leading to the oxygen based multicellular life in a very rapid time scale.
</p><p>
The standard view is that oceans before CE had very low oxygen content. The emergence of photosynthesizing cyanobacteria producing oxygen as a side product led to the oxygenation of the atmosphere and to mysteriously rapid evolution of life. How this is possible at all is not understood.
</p><p>
The first article
(see <A HREF="https://www.sciencealert.com/earth-s-spin-is-slowing-down-and-that-might-be-why-it-has-enough-oxygen-for-life">this</A>)
proposes that the slowing down of the spinning of Earth was somehow related to this.
</p><p>
The second article in Quanta Magazine (see <A HREF="https://www.quantamagazine.org/rapid-oxygen-changes-fueled-an-explosion-in-ancient-animal-diversity-20190509/">this</A>) tells about finding that during the Cambrian Explosion (see <A HREF="https://en.wikipedia.org/wiki/Cambrian_explosion">this</A>") the oxygen content of the studied shallow ocean show fluctuations with with about 4-5 peaks. The reduction/increase of the oxygen content was even 40 per cent, which is a huge number. The reduction of oxygen content caused extinctions and its increase was accompanied by the emergence of new species. The mystery is how this could happen so fast and which caused the fluctuations.
</p><p>
<B>1. Expanding Earth hypothesis</B>
</p><p>
Expanding Earth theory hypothesis is not originally TGD based but TGD provides its realization. The proposal is that the Cambrian Explosion was caused by a rapid increase of the radius of Earth by factor 2 (see <A HREF="http://tgdtheory.fi/public_html/articles/expearth.pdf">this</A>
and <A HREF ="http://tgdtheory.fi/public_html/articles/expearth2021.pdf">this</A>).
</p><p>
This hypothesis also solves one of the basic mysteries of cosmology. Astrophysical objects participate in cosmological expansion by comoving with it but do not expand themselves. Why? The prediction that the expansion of the astrophysical objects did not occur smoothly but as rapid phase transitions and the expansion was very slow in the intermediate states. Cambrian Explosion would correspond to one particular jerk of this kind in which the radius of Earth grew by a factor 2 (p-adic length scale hypothesis). The length of the day increased by factor 4 from conservation of angular momentum. This might relate to the conjecture of the first article.
</p><p>
The rapid expansion led to the breakage of the Earth crust and to the birth of plate tectonics. It also led to the burst of underground oceans to the surface of the Earth. The photosynthesizing multicellular life had developed in these oceans and emerged almost instantaneously and led to a rapid oxygenation of the atmosphere. One can say that life evolved in the womb of Mother Gaia shielded from meteorites and cosmic rays. No superfast evolution was needed. Already Charles Darwin realized that the sudden appearance of trilobites was a heavy objection against the theory of natural selection.
</p><p>
Possible scenarios for the phase transition are discussed <a HREF= "http://tgdtheory.fi/public_html/articles/expearth2021.pdf">here</A>. The thickening of magnetic flux tubes for water blobs at the surface of Earth led to the increase of the volume of water blob and induced the increase of h<sub>eff</sub> a factor 2 for valence electrons but not for the inner electrons. Since valence electrons are responsible for chemistry, atoms became effectively dark and the water blobs could leak to the interior of Earth. By their darkness they could have much lower temperature and pressure than the matter around them and the life could evolve.
</p><p>
<B> 2. How photosynthesis was possible underground?</B>
</p><p>
What made photosynthesis possible in the underground oceans? One possible explanation is that the photons from the Sun propagated along flux tubes of the "endogenous" part of the Earth's magnetic field as dark photons with h<sub>eff</sub>=nh_0>h. Endogenous part would be the part of Earth's magnetic field with a strength about 2/5 of the Earth's magnetic field for which flux tubes carry monopole flux: this is possible in TGD but not in Maxwell's theory.
</p><p>
Since these photons behave like dark matter with respect to the ordinary matter, they were not absorbed considerably and reached the water blobs (or actually their magnetic bodies consisting of flux tubes) in underground oceans having a portion with the same value of h<sub>eff</sub>>h. Of course, several values of h<sub>eff</sub> were possible since this is the case in quantum critical system (large values of h<sub>eff</sub> characterize the quantum scales of long range fluctuations). One can also consider other variants of the model. The ordinary matter in Earth's crust had h<sub>eff</sub> =h/2 and photons with h<sub>eff</sub>=h propagated to the interior and reached the water blobs with h<sub>eff</sub>=h.
</p><p>
<B> 3. The sudden emergence of multicellulars and oxygen fluctuations</B>
</p><p>
Before the expansion period was much like the surface of Mars now and contained no oceans, perhaps some ponds allowing primitive monocellular lifeforms. As the ground of Earth broke here and there during the rapid expansion period, lakes and oceans were formed at the surface of Earth. The multicellulars bursted to these oceans and oxygenation of the atmosphere started locally.
</p><p>
Since the oxygen rich water was mixed with the water in the shallow oceans, the local oxygen content of the burst water was reduced and this led to an eventual extinction of many multicellulars in the burst. Burgess Shale fauna contained entire classes, which suffered extinction. In the average sense the oxygen concentration increased and led to the apparent very rapid evolution of multicellulars, which had actually already occurred underground. Of course, also evolution at the surface of Earth took place.
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/expearth2021.pdf">Updated version of Expanding Earth model</A> or the chapter <A HREF="http://tgdtheory.fi/pdfpool/expearth.pdf">Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life</A> .
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-80568562561518002652021-08-03T20:20:00.003-07:002021-08-03T20:20:34.534-07:00Gentral engine of galactic nuclei as a time reversed blackhole-like object
The work to be summarize was inspired by a Quanta Magazine article "Physicists Identify the Engine Powering blackhole Energy Beams" (see <A HREF="https://cutt.ly/dQtDK7Q">this</A>) telling about an empirical support for the model of Blandford and Znajek (BZ model in the sequel) for the central engine providing energy for jets from active galactic nuclei (AGNs). In the BZ model (see <A HREF="https://cutt.ly/9QtD0kK">this</A>) AGN is identified as a blackhole and the Penrose process would provide the energy of the jets emerging from the blackhole. The energy would come basically from the blackhole mass. The empirical support is found by studying the supermassive blackhole associated with a galaxy known as Messier 87 (M87).
</p><p>
The basic problem is the identification of the central engine of the active galactic nuclei (AGNs) (see <A HREF="https://cutt.ly/eQtFyib">this</A>) providing the huge energy feed to the the jets.
</p><p>
<B>1. Typical properties of active galactic nuclei</B>
</p><p>
The power emitted by active galactic nuclei (AGNs) is typically of the order of 10<sup>38</sup> W corresponding to a transformation of a mass of 10<sup>22</sup> kg per second to energy. The typical radius of the AGN is R ≈ 2 AU for the active region.
</p><p>
One must distinguish between magnetic fields associated with the interior of the central objects, the region near its surface, and the jet region with the scale of visible jets about 10<sup>5</sup> ly. According to the estimate of (see <A HREF="https://cutt.ly/DQtFzD3">this</A>), the magnetic field is about 10 Gauss in the jet region. About 10<sup>6</sup>-10<sup>7</sup> Gauss near horizon. Also near-horizon magnetic fields in the range 10<sup>8</sup>-10<sup>11</sup> Tesla have been proposed for some AGNs.
</p><p>Quasars are examples of AGNs and also M87 central region idenfied as a blackhole is such. In this case the mass is 6.5× 10<sup>9</sup>M<sub>Sun</sub> and Scwartschild radius is about 2× 10<sup>10</sup> km = 1.3× 10<sup>3</sup> AU. For M87 central object the magnetic field in the jet region is that of refrigerator magnet and about 100 Gauss. For Sagittarius A has a in the center of the Milky Way the radius of the central object .4 AU.
</p><p>
One can pose some general conditions on the central engine serving as the energy source for the jets. The time scale Δt for the luminosity fluctuations in the power should satisfy Δt< R. For M87 one has Δt ≤ 10<sup>4</sup> s. The gravitational force is assumed to be balanced by the radiation pressure of the outward radiation.
</p><p>
Consider now the observations about the M87 blackhole-like entity (see <A HREF="https://cutt.ly/dQtDK7Q">this</A>).
</p><p>
<OL>
<LI> The mass of the M87 black-hole-like entity is about 6.5× 10<sup>9</sup>M<sub>Sun</sub>.
<LI> There are two 5,000 ly long white hot plasma jets travelling in opposite time directions with emitted power of 3× 10<sup>36</sup> W. They have blobs at their ends. Synchrotron radiation is emitted at radio wavelengths in the magnetic field, which according to the popular article has the strength of a refrigerator magnet, so that one would have B ≈ 100 Gauss. Both the intensity of B and the size of the emitting region contribute to the intensity of the energy flow.
<LI> There are two alternatives for the BZ process that have been developed and explored in hundreds of computer simulations in recent decades. They have acronyms MAD and SANE.
</p><p>
For the SANE option B is weak: charged matter dominates over B. For the MAD option B is strong, has a spiral structure and acts as a "boss" of the matter. The tight spiral structure forms a sleeve around the jet preventing charges from entering the central object. This inspires a critical question: doesn't the object look more like a whitehole.
</p><p>
The strongly polarized light in the Event Horizon Telescope's new photo suggests strong magnetic fields, and supports the MAD version. B has a strength of about 100 Gauss, that is 200 times the strength of the Earth's magnetic field with the nominal value B<sub>E</sub>≈ .5 Gauss. The polarization pattern for the radio waves is found to be stripy and the polarization in a plane locally: this allows us to conclude that the magnetic field is indeed helical and non-random.
</p><p>
</OL>
</p><p>
<B>2. Central engine as a Penrose process?</B>
</p><p>
In the BZ model, the central object is assumed to be a blackhole and Penrose process would provide the energy feed to the jets of length about 5000 ly. Note that the Milky Way is about 1,0000 ly thick.
</p><p>
<OL>
<LI> The blackhole is surrounded by an accretion disk from which the matter ends down to the BH.
<LI> Kerr solution of Einstein-Maxwell field equations (<A HREF="https://cutt.ly/sQuKXth">this</A>) involving magnetic field is the starting point. Matter falling into the Kerr blackhole rotates and the magnetic field lines are twisted to helical shape. By Faraday law, an electric field along field lines is generated by the rotation of the flux lines. Electrons and positrons created in the annihilation photons emitted as the particles fall to the region near the blachole, start to flow along the field lines of the electric field in opposite directions and generate the jets.
<LI> The model assumes that the electromagnetic field is force free so that it does not dissipate and Lorentz force vanishes. At a single particle level this implies the condition E+qv× B=0. Vanishing dissipation requires v· E=0. This helical structure would be in the direction of the jet.
<LI> The basic question has been whether it is accretion disk or magnetic field that controls the dynamics. The first option, known as SANE, corresponds to weak and incoherent magnetic fields. The second option, known as MAD, corresponds to strong and coherent magnetic fields.
<LI> MAD is favoured by the recent observations. Magnetic field would form a sleeve around the jet and the synchrotron radiation pressure would prevent matter from falling into the blackhole. Matter can only occasionally leak to blackhole.
</p><p>
One can however wonder whether it makes sense to talk about blackhole anymore! Doesn't this look more like white hole as a time reversal of blackhole feeding energy and matter to the environment?
</p><p>
</OL>
</p><p>
<B>3. TGD inspired view of the central engine</B>
</p><p>
In the TGD framework the model of the central engine as a Penrose process is replaced by the following picture. The key concepts are following:
</p><p>
<OL>
</p><p><LI> Space-time is identified as a 4-D minimal surface in H=M<sup>4</sup>× CP<sub>2</sub> or as an algebraic surface in complexified M<sup>8</sup> having octonionic interpretation. These descriptions are related by M<sup>8</sup>-H duality analogous to momentum-position duality, which does not generalize from wave mechanics to quantum field theory (QFT). Therefore the points or M<sup>8</sup> are 8-momenta.
</p><p>
The classical dissipation is absent for the generalized Beltrami fields and the proposal is that minimal surfaces (apart form singularities defining dynamically generated frame for space-time surfaces as analog of a soap film) define locally generalized Beltrami fields.
<LI> Zero energy ontology (ZEO) predicts that time reversal occurs in the TGD counterparts of ordinary state function reductions ("big" SFRs) but not in "small" SFRs (SSFRs).
<LI> The hierarchy of effective Planck constants predicts a hierarchy of phases of ordinary matter labelled by the values of effective Planck constant h<sub>eff</sub>= nh<sub>0</sub>. The phases with different values of h<sub>eff</sub> behave in many respects like dark matter with respect to each other. The findings of Randell Mills suggest ℏ/ℏ<sub>0</sub>=6 but also larger values for this ratio can be considered. I have proposed that the ℏ<sub>0</sub>/ℏ is equal to the ratio l<sub>P</sub>/R of Planck length l<sub>P</sub> to CP<sub>2</sub> radius R.
</p><p>
As a special case, one obtains gravitational Planck constant satisfying h<sub>eff</sub>= h<sub>gr</sub>= GMm/β<sub>0</sub>, where β<sub>0</sub>=v<sub>0</sub>/c and β<sub>0</sub><c has dimensions of velocity, as a generalization of Nottale's hypothesis. The gravitational Compton length λ<sub>gr</sub>=ℏ<sub>gr</sub>/m=GM/β<sub>0</sub> does not depend on m and is equal to Schwartschild radius r<sub>s</sub> for β<sub>0</sub>= 1/2. Also the cyclotron energy spectrum E<sub>c</sub>=n GMqB/β<sub>0</sub> is independent of the mass of the charged particle.
</p><p>
The hierarchy of Planck constants, the notion of ℏ<sub>gr</sub>, and coupling constant evolution are discussed in detail <A HREF="http://tgdtheory.fi/public_html/articles/ccheff.pdf">here</A>.
</OL>
Consider next the key elements of the model.
<OL>
<LI> TGD leads to a general model for the formation of galaxies, stars, planets,... in terms of cosmic strings thickening to flux tubes. The energy of the flux tube, which consists of a volume energy and Kähler magnetic energy, is transformed to ordinary matter as the string tension is reduced in a sequence of phase transitions reducing the length scale dependent cosmological constant λ.
</p><p>
This process is analogous to the decay of an inflaton field to matter. The model (there are actually several basic variants of it) explains the flat velocity spectra associated with the spiral galaxies. For the first option, a long cosmic string normal to the galactic plane causes the gravitational field explaining the flat velocity spectrum of spiral galaxies. For galaxies formed around closed flux loops the velocity spectrum is not flat. There is no dark matter halo although it is possible that the galactic plane contains cosmic strings parallel to the plane.
<LI> Zero energy ontology (ZEO), which predicts that the TGD counterparts of ordinary state function reductions (SFRs) involve time reversal, is involved in an essential manner. TGD predicts both blackhole-like objects (BH) and whitehole-like objects (WH) as the time reversals of BHs. The seed of the galaxy, active galactic nucleus (AGN), involves WH. Quasars are cases of AGNs as WHs.
<LI> In the TGD framework, the Kerr blackhole is replaced with a whitehole-like object (WH). Kerr blackhole indeed has an opposite arrow of time reversal as the distant environment. The WH is time reversal of BH and feeds matter and energy to the environment. This serves as an analog of the Penrose process in the TGD based model.
<LI> The TGD analog for the rotation of spacetime and the twisting of the magnetic field lines near the Kerr blackhole is very concrete. Space-time is a 4-surface and the flux tubes carrying monopole flux are pieces of 3-space as a 3-surface. They quite concretely rotate and get twisted in the process. Analogous process occurs in the Sun with a period of 11 years ending as reconnections untwist the flux tubes.
<LI> WH would correspond to a tangle of a long cosmic string in the direction of the jet thickened to a flux tube but still carrying an extremely strong magnetic field. The helical magnetic field in the exterior of the jet would not represent return flux of this field as one might first think. There is a current ring associated with the equator of Earth, which carries a parallel magnetic field analogous to the helical magnetic field.
</p><p>
The magnetic field in the exterior of WH is associated with a space-time surface, which is many-sheeted with respect to CP<sub>2</sub> rather than M<sup>4</sup> so that either CP<sub>2</sub> or cosmic string world sheet M<sup>2</sup>× S<sup>2</sup>⊂ M<sup>4</sup>× CP<sub>2</sub>) would serve as the arena of physics rather than M<sup>4</sup>, which is quantum coherent flux tube bundle analogous to BE-condensate. M<sup>4</sup> coordinates as functions of CP<sub>2</sub> or M<sup>2</sup>× CP<sub>2</sub> coordinates would be many-valued rather than vice versa. This picture is very natural if one accepts M<sup>8</sup>-H duality.
</p><p>
Cosmic strings dominate during the primordial cosmology in TGD Universe, and the analog of the inflationary period corresponds to the transition to a phase in which the Einsteinian space-time with M<sup>4</sup> as the arena of physics is a good approximation. Hence the M<sup>2</sup>× CP<sub>2</sub> option looks more plausible.
<LI> The force-free em fields appearing in the BZ model correspond to space-time surfaces as minimal surfaces realizing a 4-D generalization of 3-D Beltrami fields, which do not not dissipate classically. The interpretation of the non-dissipating Kähler currents is as classical correlates for supracurrents. The prediction is that charged particles flow without dissipation that is as supra currents: not only Cooper pairs but also charged fermions. Also the analogs of laser beams of dark photons are expected.
<LI> The hierarchy of Planck constants is an important piece of the picture emerging from adelic physics. From ℏ<sub>gr</sub>=GMm/β<sub>0</sub> realizing Equivalence Principle, the gravitational Compton length λ<sub>gr</sub>= r<sub>s</sub>/2β<sub>0</sub> is universal and equals to r<sub>S</sub> for β<sub>0</sub>== β<sub>0</sub>=1/2.
</p><p>
All astrophysical objects are predicted to be quantum coherent in the scale of λ<sub>gr</sub>= r<sub>s</sub>/2β<sub>0</sub> at least. The quantum coherence would be at the level of magnetic body (MB). WH/BH as a thickened flux tube tangle would not have large h<sub>eff</sub> but would be accompanied by a large scale quantum object.
</p><p>
The astroscopic quantum coherence would be associated with the helical magnetic field surrounding the long cosmic string having BH or WH as a tangle.
<LI> Also the cyclotron energy spectrum is universal and does not depend on the mass of the charged particles so that all charged particles rather than only electrons are expected to form supracurrents. Dark matter would flow along flux tubes and form the dark core of the jet, perhaps extending over cosmic distances to other galaxies identified as tangles of the one and the same cosmic string.
</p><p>
Stars and even planets would be parts of this fractal network. Dark cyclotron states have huge energies for h<sub>eff</sub>=h<sub>gr</sub> serving also as a measure for algebraic complexity and, in the TGD inspired theory of consciousness, also for intelligence and scale of quantum coherence. The analogy with a cosmic nervous system is obvious.
<LI> The decay of quantum coherent states to ordinary states takes place by the loss of quantum coherence in which h<sub>eff</sub>= h<sub>gr</sub> is reduced. This would create the visible jets and blobs at their ends. For M87, which is elliptical for which the velocity spectrum is not flat, the flux tubes would be closed in a relatively short scale. Their length scale could be that of the jets in the case of ellipticals. The thickening of the cosmic string at the core leads to the reduction of mass of WH and gives rise to the flow of mass and energy to the environment. One could see this process as a time reversal for the generation of BH and perhaps also as an analogy for the evaporation of BH.
<LI> M<sup>8</sup>-H duality and adelic physics help to understand the decoherence process geometrically. The reduction of h<sub>eff</sub> and thus of the length scale of quantum coherence, allows a number theoretic description at the level of M<sup>8</sup>. An irreducible polynomial, which depends on parameters, reduces to a product of polynomials for some critical values of the parameters. This gives rise to a set of disjoint space-time surfaces, which are not correlated. This means decoherence. This includes as a special case the description of catastrophic changes in catastrophe theory of Thom. The maximal decoherence produces a product of first order polynomials with rational roots.
</p><p>
At the level of H =M<sup>4</sup>×CP<sub>2</sub> this corresponds to a decay of coherent flux tube bundle to disjoint uncorrelated flux tubes.
</OL>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/galjets.pdf">TGD view of the engine powering jets from active galactic nuclei</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/galjets.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-49236593619341603952021-08-03T02:46:00.011-07:002021-10-05T22:04:56.561-07:00Evolution of Kähler coupling strength
The evolution of Kähler coupling strength α<sub>K</sub>= g<sub>K</sub><sup>2</sup>/2h<sub>eff</sub> gives the evolution of α<sub>K</sub> as a function of dimension n of EQ: α<sub>K</sub>= g<sub>K</sub><sup>2</sup>/2nh<sub>0</sub>. If g<sub>K</sub><sup>2</sup> corresponds to electroweak U(1) coupling, it is expected to evolve also with respect to PLS so that the evolutions would factorize.
</p><p>
Note that the original proposal that g<sub>K</sub><sup>2</sup> is renormalization group invariant was later replaced with a piecewise constancy: α<sub>K</sub> has indeed interpretation as piecewise constant critical temperature
<OL>
<LI> In the TGD framework, coupling constant as a continuous function of the continuous length scale is replaced with a function of PLS so that coupling constant is a piecewise constant function of the continuous length scale.
</p><p>
PLSs correspond to p-adic primes p, and a hitherto unanswered question is whether the extension determines p and whether p-adic primes possible for a given extension could correspond to ramified primes of the extension appearing as factors of the moduli square for the differences of the roots defining the space-time surface.
</p><p>
In the M<sup>8</sup> picture the moduli squared for differences r<sub>i</sub>-r<sub>j</sub> of the roots of the real polynomial with rational coefficients associated with the space-time surfaces correspond to energy squared and mass squared. This is the case of p-adic prime corresponds to the size scale of the CD.
</p><p>
The scaling of the roots by constant factor however leaves the number theoretic properties of the extension unaffected, which suggests that PLS evolution and dark evolution factorize in the sense that PLS reduces to the evolution of a power of a scaling factor multiplying all roots.
</p><p>
<LI> If the exponent Δ K/log(p) appearing in p<sup>Δ K/log(p))</sup>=exp(Δ K) is an integer, exp(Δ K) reduces to an integer power of p and exists p-adically. If Δ K corresponds to a deviation from the Kähler function of WCW for a particular path in the tree inside CD, p is fixed and exp(Δ K) is integer. This would provide the long-sought-for identification of the preferred p-adic prime. Note that p must be same for all paths of the tree. p need not be a ramified prime so that the trouble-some correlation between n and ramified prime defining padic prime p is not required.
</p><p>
<LI> This picture makes it possible to understand also PLS evolution if Δ K is identified as a deviation from the Kähler function. p<sup>Δ K/log(p))</sup>=exp(Δ K) implies that Δ K is proportional to log(p). Since Δ K as 6-D Kähler action is proportional to 1/α<sub>K</sub>, log(p)-proportionality of Δ K could be interpreted as a logarithmic renormalization factor of α<sub>K</sub>∝ 1/log(p).
</p><p>
<LI> The universal CCE for α<sub>K</sub> inside CDs would induce other CCEs, perhaps according to the scenario based on M"obius transformations.
</OL>
<B>Dark and p-adic length scale evolutions of Kähler coupling strength</B>
</p><p>
The original hypothesis for dark CCE was that h<sub>eff</sub>=nh is satisfied. Here n would be the dimension of EQ defined by the polynomial defining the space-time surface X<sup>4</sup>subset M<sup>8</sup><sub>c</sub> mapped to H by M<sup>8</sup>-H correspondence. n would also define the order of the Galois group and in general larger than the degree of the irreducible polynomial.
</p><p>
<B> Remark</B>: The number of roots of the extension is in general smaller and equal to n for cyclic extensions only. Therefore the number of sheets of the complexified space-time surface in M<sup>8</sup><sub>c</sub> as the number of roots identifiable as the degree d of the irreducible polynomial would in general be smaller than n. n would be equal to the number of roots only for cyclic extensions (unfortunately, some former articles contain the obviously wrong statement d=n).
</p><p>
Later the findings of Randell Mills, suggesting that h is not a minimal value of h<sub>eff</sub>, forced to consider the formula h<sub>eff</sub>=nh<sub>0</sub>, h<sub>0</sub>=h/6, as the simplest formula consistent with the findings of Mills. h<sub>0</sub> could however be a multiple of even smaller value of h<sub>eff</sub>, call if h<sub>0</sub> and the formula h<sub>0</sub>=h/6 could be replaced by an approximate formula.
</p><p>
The value of h<sub>eff</sub>=nh<sub>0</sub> can be understood by noticing that Galois symmetry permutes "fundamental regions" of the space-time surface so that action is n times the action for this kind of region. Effectively this means the replacement of α<sub>K</sub> with α<sub>K</sub>/n and implies the convergence of the perturbation theory. This was actually one of the basic physical motivations for the hierarchy of Planck constants. In the previous section, it was argued that h<sub>0</sub> is given by the square of the ratio l<sub>P</sub>/R of Planck length and CP<sub>2</sub> length scale identified as dark scale and equals to n<sub>0</sub>=(7!)<sup>2</sup>.
</p><p>
The basic challenge is to understand p-adic length scale evolutions of the basic gauge couplings. The coupling strengths should have a roughly logarithmic dependence on the p-adic length scale p≈ 2<sup>k/2</sup> and this provides a strong number theoretic constraint in the adelic physics framework.
</p><p>
Since Kähler coupling strength α<sub>K</sub> induces the other CCEs it is enough to consider the evolution of α<sub>K</sub>.
</p><p>
<B>p-Adic CCE of α from its value at atomic length scale?</B>
</p><p>
If one combines the observation that fine structure constant is rather near to the inverse of the prime p=137 with PLS, one ends up with a number theoretic idea leading to a formula for α<sub>K</sub> as a function of p-adic length scale.
<OL>
<LI> The fine structure constant in atomic length scale L(k=137) is given α (k)=e<sup>2</sup>/2h ≈ 1/137. This finding has created a lot of speculative numerology.
<LI> The PLS L(k)= 2<sup>k/2</sup>R(CP<sub>2</sub>) assignable to atomic length scale p≈ 2<sup>k</sup> corresponds to k=137 and in this scale α is rather near to 1/137. The notion of fine structure constant emerged in atomic physics. Is this just an accident, cosmic joke, or does this tell something very deep about CCE?
</p><p>
Could the formula
</p><p>
α(k)= e<sup>2</sup>(k)/2h= 1/k
</p><p>
hold true?
</OL>
There are obvious objections against the proposal.
<OL>
<LI> α is length scale dependent and the formula in the electron length scale is only approximate. In the weak boson scale one has α≈ 1/127 rather than α= 1/89.
<LI> There are also other interactions and one can assign to them coupling constant strengths. Why electromagnetic interactions in electron Compton scale or atomic length scales would be so special?
</OL>
The idea is however plausible since beta functions satisfy first order differential equation with respect to the scale parameter so that single value of coupling strength determines the entire evolution.
</p><p>
<B>p-Adic CCE from the condition α<sub>K</sub>(k=137)= 1/137</B>
</p><p>
In the TGD framework, Kähler coupling strength α<sub>K</sub> serves as the fundamental coupling strength. All other coupling strengths are expressible in terms of α<sub>K</sub>, and I have proposed that M"obius transformations relate other coupling strengths to α<sub>K</sub>. If α<sub>K</sub> is identified as electroweak U(1) coupling strength, its value in atomic scale L(k=137) cannot be far from 1/137.
</p><p>
The factorization of dark and p-adic CCEs means that the effective Planck constant h<sub>eff</sub>(n,h,p) satisfies
</p><p>
h<sub>eff</sub>(n,h,p)=h<sub>eff</sub>(n,h) = nh .
</p><p>
and is independent of the p-adic length scale. Here n would be the dimension of the extension of rationals involved. h<sub>eff</sub>(1,h,p) corresponding to trivial extension would correspond to the p-adic CCE as the TGD counterpart of the ordinary evolution.
</p><p>
The value of h need not be the minimal one as already the findings of Randel Mills suggest so that one would have h=n<sub>0</sub>h<sub>0</sub>.
</p><p>
h<sub>eff</sub>= nn<sub>0</sub>h ,
</p><p>
α<sub>K,0</sub>= g<sub>K,max</sub><sup>2</sup>/2h<sub>0</sub> =n<sub>0</sub> .
</p><p>
This would mean that the ordinary coupling constant would be associated with the non-trivial extension of rationals.
</p><p>
Consider now this picture in more detail.
<OL>
<LI> Since dark and p-adic length scale evolutions factorize, one has
</p><p>
α<sub>K</sub> (n)= g<sub>K</sub><sup>2</sup>(k)/2h<sub>eff</sub></sub> ,
</p><p>
h<sub>eff</sub>= nh<sub>0</sub> .
</p><p>
U(1) coupling indeed evolves with the p-adic length scale, and if one assumes that g<sub>K</sub><sup>2</sup>(k,n<sub>0</sub>) (h=n<sub>0</sub>h<sub>0</sub>) is inversely proportional to the logarithm of p-adic length scale, one obtains
</p><p>
g<sub>K</sub><sup>2</sup>(k,n<sub>0</sub>) =g<sub>K</sub><sup>2</sup>(max)/k ,
</p><p>
α<sub>K</sub> = g<sub>K</sub><sup>2</sup>(max)/2kh<sub>eff</sub> .
<LI> Since k=137 is prime (here number theoretical physics shows its power!), the condition α<sub>K</sub> (k=137,h<sub>0</sub>)=1/137 gives
</p><p>
g<sub>K</sub><sup>2</sup>(max)/2h<sub>0</sub>}= α<sub>K</sub>(max) =(7!)<sup>2</sup> .
</p><p>
The number theoretical miracle would fix the value of α<sub>K</sub>(max) to the ratio of Planck mass and CP<sub>2</sub> mass n<sub>0</sub>= M<sup>2</sup><sub>P</sub>/M<sup>2</sup>(CP<sub>2</sub>)= (7!)<sup>2</sup> if one takes the argument of the previous section
seriously.
</p><p>
The convergence of perturbation theory could be possible also for h<sub>eff</sub>=h<sub>0</sub> if the p-adic length scale L(k) is long enough to make α<sub>K</sub>= n<sub>0</sub>/k small enough.
<LI> The outcome is a very simple formula for α<sub>K</sub>
</p><p>
α<sub>K</sub>(n,k) = n<sub>0</sub>/kn ,
</p><p>
which is a testable prediction if one assumes that it corresponds to electroweak U(1) coupling strength at QFT limit of TGD. This formula would give a practically vanishing value of α<sub>K</sub> for very large values of n associated with h<sub>gr</sub>. Here one must have n>n<sub>0</sub>.
</p><p>
For h<sub>eff</sub>=nn<sub>0</sub>h characterizing extensions of extension with h<sub>eff</sub>=h one can write
</p><p>
α<sub>K</sub>(nn<sub>0</sub>,k) = 1/kn .
<LI> The almost vanishing of α<sub>K</sub> for the very large values of n associated with ℏ<sub>gr</sub> would practically eliminate the gauge interactions of the dark matter at gravitational flux tubes but leave gravitational interactions, whose coupling strength would be beta<sub>0</sub>/4pi. The dark matter at gravitational flux tubes would be highly analogous to ordinary dark matter.
</OL>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/ccheff.pdf">Questions about coupling constant evolution</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/ccheff.pdf">chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-50498348541971017302021-08-03T02:45:00.005-07:002021-08-03T21:15:43.148-07:00Minimal value of heff from the ratio of Planck mass and CP2 mass?
<B>Minimal value of h<sub>eff</sub> from the ratio of Planck mass and CP<sub>2</sub> mass?</B>
</p><p>
Could one understand and perhaps even predict the minimal value h<sub>0</sub>of h<sub>eff</sub>? Here number theory and the notion of n-particle Planck constant h<sub>eff</sub>(n) suggested by Yangian symmetry could serve as a guidelines.
<OL>
<LI> Hitherto I have found no convincing empirical argument fixing the value of r=ℏ/ℏ<sub>0</sub>: this is true for both single particle and 2-particle case.
</p><p>
The value h<sub>0</sub>=h/6 as a maximal value of h<sub>0</sub> is suggested by the findings of Randell Mills and by the idea that spin and color must be representable as Galois symmetries so that the Galois group must contain Z<sub>6</sub>=Z<sub>2</sub>× Z<sub>3</sub>. Smaller values of h<sub>0</sub> cannot be however excluded.
<LI> A possible manner to understand the value r geometrically would be following. It has been assumed that CP<sub>2</sub> radius R defines a fundamental length scale in TGD and Planck length squared l<sub>P</sub><sup>2</sup>= ℏ G =x<sup>-2</sup> × 10<sup>-6</sup>R<sup>2</sup> defines a secondary length scale. For Planck mass squared one has m<sub>Pl</sub><sup>2</sup>= m(CP<sub>2</sub>,ℏ)<sup>2</sup>× 10<sup>6</sup>x<sup>2</sup> , m(CP<sub>2</sub>,ℏ)<sup>2</sup>= ℏ/R<sup>2</sup>. The estimate for x from p-adic mass calculations gives x≈ 4.2. It is assumed that CP<sub>2</sub> length is fundamental and Planck length is a derived quantity.
</p><p>
But what if one assumes that Planck length identifiable as CP<sub>2</sub> radius is fundamental and CP<sub>2</sub> mass corresponds the minimal value h<sub>0</sub> of h<sub>eff</sub>(2)? That the mass formula is quadratic and mass is assignable to wormhole contact connecting two space-time sheets suggests in the Yangian framework that h<sub>eff</sub>(2) is the correct Planck constant to consider.
</OL>
One can indeed imagine an alternative interpretation. CP<sub>2</sub> length scale is deduced indirectly from p-adic mass calculation for electron mass assuming h<sub>eff</sub>=h and using Uncertainty Principle. This obviously leaves the possibility that R= l<sub>P</sub> apart from a numerical constant near unity, if the value of h<sub>eff</sub> to be used in the mass calculations is actually h<sub>0</sub>= (l<sub>P</sub>/R)<sup>2</sup>ℏ. This would fix the value of ℏ<sub>0</sub> uniquely.
</p><p>
The earlier interpretation makes sense if R(CP<sub>2</sub>) is interpreted as a dark length scale obtained scaling up l<sub>P</sub> by ℏ/ℏ<sub>0</sub>. Also the ordinary particles would be dark.
</p><p>
h<sub>0</sub> would be very small and α<sub>K</sub>(ℏ<sub>0</sub>)= (ℏ/ℏ<sub>0</sub>)α<sub>K</sub> would be very large so that the perturbation theory for it would not converge. This would be the reason for why ℏ and in some cases some smaller values of h<sub>eff</sub> such as ℏ/2 and ℏ/4 seem to be realized.
</p><p>
For R=l<sub>P</sub> Nottale formula remains unchanged for the identification M<sup>2</sup><sub>P</sub>= ℏ/R<sup>2</sup> (note that one could consider also ℏ<sub>0</sub>/R<sup>2</sup> used in p-adic mass calculations).
</p><p>
<B>Various options</B>
</p><p>
Number theoretical arguments allow to deduce precise value for the ratio ℏ/ℏ<sub>0</sub>. Accepting the Yangian inspired picture, one can consider two options for what one means with ℏ.
<OL>
<LI> ℏ refers to the single particle Planck constant ℏ<sub>eff</sub>(1) natural for point-like particles.
<LI> ℏ refers to h<sub>eff</sub>(2). This option is suggested by the proportionality M<sup>2</sup>∝ ℏ in string models due to the proportionality M<sup>2</sup>∝ℏ/G in string models. At a deeper level, one has M<sup>2</sup> ∝ L<sub>0</sub>, where L<sub>0</sub> is a scaling generator and its spectrum has scale given by ℏ.</p><p>Since M<sup>2</sup> is a p-adic thermal expectation of L<sub>0</sub> in the TGD framework, the situation is the same. This also due the fact that one has In TGD framework, the basic building bricks of particles are indeed pairs of wormhole throats.
</OL>
One can consider two options for what happens in the scaling h<sub>eff</sub>→ kh<sub>eff</sub>. </p><p><B> Option 1</B>: Masses are scaled by k and Compton lengths are unaffected.
</p><p>
<B> Option 2</B>: Compton lengths are scaled by k and masses are unaffected.
</p><p>
The interpretation of M<sub>P</sub><sup>2</sup>= (ℏ/ℏ<sub>0</sub>) M<sup>2</sup>(CP<sub>2</sub>) assumes Option 1
whereas the new proposal would correspond to Option 2 actually assumed in various applications.
</p><p>
The interpretation of M<sub>P</sub><sup>2</sup>= (ℏ/ℏ<sub>0</sub>) M<sup>2</sup>(CP<sub>2</sub>) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.
</p><p>
For Option 1 m<sub>Pl</sub><sup>2</sup>= (ℏ<sub>eff</sub>/ℏ) M<sup>2</sup>(CP<sub>2</sub>). The value of M<sup>2</sup>(CP<sub>2</sub>)= ℏ/R<sup>2</sup> is deduced from the p-adic mass calculation for electron mass. One would have R<sup>2</sup> ≈ (ℏ<sub>eff</sub>/ℏ) l<sub>P</sub><sup>2</sup> with ℏ<sub>eff</sub>/ℏ = 2.54× 10<sup>7</sup>. One could say that the real Planck length corresponds to R.
</p><p>
<B>Quantum-classical correspondence favours Option 2)</B>
</p><p>
In an attempt to select between these two options, one can take space-time picture as a guideline. The study of the imbeddings of the space-time surfaces with spherically symmetric metric carried out for almost 4 decades ago suggested that CP<sub>2</sub> radius R could naturally correspond to Planck length l<sub>P</sub>. The argument is described in detail in Appendix and shows that the l<sub>P</sub>=R option with h<sub>eff</sub>=h used in the classical theory to determine α<sub>K</sub> appearing in the mass formula is the most natural.
</p><p>
<B>Deduction of the value of ℏ/ℏ<sub>0</sub></B>
</p><p>
Assuming Option 2), the questions are following.
<OL>
<LI> Could l<sub>P</sub>=R be true apart from some numerical constant so that CP<sub>2</sub> mass M(CP<sub>2</sub>) would be given by M(CP<sub>2</sub>)<sup>2</sup>= ℏ<sub>0</sub>/l<sub>P</sub><sup>2</sup>, where ℏ<sub>0</sub>≈ 2.4× 10<sup>-7</sup> ℏ (ℏ corresponds to ℏ<sub>eff</sub>(2)) is the minimal value of ℏ<sub>eff</sub>(2). The value of h<sub>0</sub> would be fixed by the requirement that classical theory is consistent with quantum theory! It will be assumed that ℏ<sub>0</sub> is also the minimal value of ℏ<sub>eff</sub>(1) both ℏ<sub>eff</sub>(2).
<LI> Could ℏ(2)/ℏ<sub>0</sub>(2)=n<sub>0</sub> correspond to the order of the product of identical Galois groups for two Minkowskian space-time sheets connected by the wormhole contact serving as a building brick of elementary particles and be therefore be given as n<sub>0</sub>=m<sup>2</sup>?
</OL>
Assume that one has n<sub>0</sub>=m<sup>2</sup>.
<OL>
<LI> The natural assumption is that Galois symmetry of the ground state is maximal so that m corresponds to the order a maximal Galois group - that is permutation group S<sub>k</sub>, where k is the degree of polynomial.
</p><p>
This condition fixes the value k to k=7 and gives m=k!=7! = 5040 and gives n<sub>0</sub>= (k!)<sup>2</sup>= 25401600=2.5401600 × 10<sup>7</sup>. The value of ℏ<sub>0</sub>(2)/ℏ(2)=m<sup>-2</sup> would be rather small as also the value of ℏ<sub>0</sub>(1)ℏ(1). p-Adic mass calculations lead to the estimate m<sub>Pl</sub>/m(CP<sub>2</sub>)= m<sup>1/2</sup> m(CP<sub>2</sub>)=4.2× 10<sup>3</sub>, which is not far from m=5040.
<LI> The interpretation of the product structure S<sub>7</sub> × S<sub>7</sub> would be as a failure of irreducibility so that the polynomial decomposes into a product of polynomials - most naturally defined for causally isolated Minkowskian space-time sheets connected by a wormhole contact with Euclidian signature of metric representing a basic building brick of elementary particles.
</p><p>
Each sheet would decompose to 7 sheets. ℏ<sub>gr</sub> would be 2-particle Planck constant h<sub>eff</sub>(2) to be distinguished from the ordinary Planck constant, which is single particle Planck constant and could be denoted by h<sub>eff</sub>(1).
</p><p>
The normal subgroups of S<sub>7</sub> × S<sub>7</sub> S<sub>7</sub>× A<sub>7</sub> and A<sub>7</sub>× A<sub>7</sub>, S<sub>7</sub>, A<sub>7</sub> and trivial group. A<sub>7</sub> is simple group and therefore does not have any normal subgroups expect the trivial one. S<sub>7</sub> and A<sub>7</sub> could be regarded as the Galois group of a single space-time sheet assignable to elementary particles. One can consider the possibility that in the gravitational sector all EQs are extensions of this extension so that ℏ becomes effectively the unit of quantization and m<sub>Pl</sub> the fundamental mass unit. Note however that for very small values of α<sub>K</sub> in long p-adic length scales also the values of h<sub>eff</sub><h, even h<sub>0</sub>, are in principle possible.
</p><p>
The large value of α<sub>K</sub> ∝ 1/ℏ<sub>eff</sub> for Galois groups with order not considerably smaller than m=(7!)<sup>2</sup> suggests that very few values of h<sub>eff</sub>(2)<h are realized. Perhaps only S<sub>7</sub> × S<sub>7</sub> S<sub>7</sub>× A<sub>7</sub> and A<sub>7</sub>× A<sub>7</sub> are allow by perturbation theory. Now however that in the "stringy phase" for which super-conformal invariance holds true, h<sub>0</sub> might be realized as required by p-adic mass calculations. The alternative interpretation is that ordinary particles correspond to dark phase with R identified dark scale.
<LI> A<sub>7</sub> is the only normal subgroup of S<sub>7</sub> and also a simple group and one has S<sub>7</sub>/A<sub>7</sub>= Z<sub>2</sub>. S<sub>7</sub>× S<sup>7</sup> has S<sub>7</sub>× S<sup>7</sup>/A<sub>7</sub>× A<sub>7</sub>= Z<sub>2</sub>× Z<sub>2</sub> with n=n<sub>0</sub>/4 and S<sub>7</sub>× S<sup>7</sup>/A<sub>7</sub>× S<sub>7</sub>= Z<sub>2</sub> with n=n<sub>0</sub>/2. This would allow the values ℏ/2 and ℏ/4 as exotic values of Planck constant.
</p><p>
The atomic energy levels scale like 1/ℏ<sup>2</sup> and would be scaled up by factor 4 or 16 for these two options. It is not clear whether ℏ→ ℏ/2 option can explain all findings of Randel Mills in TGD framework, which effectively scale down the principal quantum number n from n to n/2.
<LI> The product structure of the Nottale formula suggests
</p><p>
n=n<sub>1</sub>× n<sub>2</sub> = k<sub>1</sub>k<sub>2</sub>m<sup>2</sup> .
</p><p>
Equivalently, n<sub>i</sub> would be a multiple of m. One could say that M<sub>Pl</sub>=(ℏ/ℏ<sub>0</sub>)<sup>1/2</sup>M(CP<sub>2</sub>) effectively replaces M(CP<sub>2</sub>) as a mass unit. At the level of polynomials this would mean that polynomials are composites P○ P<sub>0</sub> where P<sub>0</sub> is ground state polynomial and has a Galois group with degree n<sub>0</sub>. Perhaps S<sub>7</sub> could be called the gravitational or ground state Galois group.
</OL>
</p><p>
See the article <a HREF= "http://tgdtheory.fi/public_html/articles/ccheff.pdf">Questions about coupling constant evolution</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/ccheff.pdf"> chapter</A> with the same title.
</p><p>
For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
</p><p>
<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0tag:blogger.com,1999:blog-10614348.post-32072078936169163452021-08-02T21:30:00.003-07:002021-08-02T21:30:52.016-07:00TGD view of the engine powering jets from active galactic nuclei
The identification of the energy source (central engine) explaining the energy loss associated with the jets from active galactic nuclei (AGNs) is a long-standing problem of astrophysics. In the model of Blandford and Znajek (BZ model) for the central engine as a blackhole, the Penrose process would provide the energy. The energy would come basically from the blackhole mass.
</p><p>
Empirical support for the BZ model emerges from the study of the supermassive blackhole associated with a galaxy known as Messier 87 (M87). The finding is that the magnetic field associated with the jet structure is tightly wound helical structure and so strong that it would control the dynamics of the matter from falling to blackhole except by occasional leakages. Electron-positron pairs created in the annihilation of photons would accelerate in the force-free helical electromagnetic field having also an electric component.
</p><p>
The TGD based model involves several aspects of the new physics predicted by TDG. TGD leads to a model of galaxies and other astrophysical structures. Inflaton decay is replaced with the thickening of cosmic strings to flux tubes liberating as ordinary matter. Hierarchy of Planck constants h<sub>eff</sub>=nh<sub>0</sub>, in particular Nottale's hypothesis predicts quantum coherence in the exterior of in scales at least of order Schwartschild radius of the blackhole-like entity. Zero energy ontology (ZEO) predicts that the arrow of time changes in ordinary state function reductions. TGD replaces black-holes with blackhole-like entities (BHs) and white-holes with their time reversals (WHs) allowed in ZEO.
</p><p>
BH (WH) would be a volume filling flux tube but with a relatively small value of h<sub>eff</sub>. In the case of WH, it would provide "metabolic energy" for jets and take care that the value of h<sub>eff</sub> is preserved (the analogy with living systems is very strong). The jets would be analogous to laser beams/supracurrents with a huge value of h<sub>eff</sub>=h<sub>gr</sub>. The model would also explain the ultrahigh energy cosmic rays. The force-free fields would be generalized Beltrami fields associated with flux tubes and identifiable as minimal surfaces in the Minkowskian regions of space-time surface. The absence of classical dissipation would be a correlate for the absence of dissipation for supra-currents and dark photon laser beams.
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See the article <a HREF= "http://tgdtheory.fi/public_html/articles/galjets.pdf">TGD view of the engine powering jets from active galactic nuclei</A> or the <a HREF= "http://tgdtheory.fi/pdfpool/galjets.pdf">chapter</A> with the same title.
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For a summary of earlier postings see <a HREF= "http://tgdtheory.fi/public_html/articles/progress.pdf">Latest progress in TGD</A>.
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<A HREF="http://www.tgdtheory.fi/tgdmaterial.html">Articles and other material related to TGD</A>.
Matti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.com0