Objection against the idea about theoretician friendly Mother Nature

One of the key ideas behind the TGD view of dark matter is that Nature is theoretician friendly (see this). When the coupling strength proportional to ℏ_eff becomes so large that perturbation series ceases to converge, a phase transition increasing the value of h_eff takes place so that the perturbation series converges.

One can however argue that this argument is quantum field-theoretic and does not apply in TGD since holography changes the very concept of perturbation theory. There is no path integral to worry about. Path integral is indeed such a fundamental concept that one expects it to have some approximate counterpart also in the TGD Universe. Bohr orbits are not completely deterministic: could the sum over the Bohr orbits however translate to an approximate description as a path integral at the QFT limit? The dynamics of light-like partonic orbits is indeed non-deterministic and could give rise to an analog of path integral as a finite sum.

1. The dynamics implied by Chern-Simons-Kähler action assignable to the partonic 3-surface with light-one coordinate in the role of time, is very topological in that the partonic orbits is light-like 3-surface and has 2-D CP₂ and M⁴ projections unless the induced M⁴ and CP₂ Kähler forms sum up to zero. The light-likeness of the projection is a very loose condition and and the sum over partonic orbits as possible representation of holographic data analogous to initial values (light-likeness!) is therefore analogous to the sum over all paths appearing as a representation of Schrödinger equation in wave mechanics.

   One would have an analog of 1-D QFT. This means that the infinities of quantum field theories are absent but for a large enough coupling strength g²/4πℏ the perturbation series fails to converge. The increase of h_eff would resolve the problem. For instance,   Dirac equation in atomic physics makes unphysical predictions when the value of nucler charge is larger than Z≈ 137.

2. I have also considered a discretized variant of this picture. The light-like orbits would consist of pieces of light-like geodesics. The points at which the direction of segment changes would correspond to points at which energy and momentum transfer between the partonic orbit and environment takes place. This kind of quantum number transfer might occur at least for the fermionic lines as boundaries of string world sheets. They could be described quantum mechanically as interactions with classical fields in the same way as the creation of fermion pairs as a fundamental vertex (see this). The same universal 2-vertex would be in question.
3. What is intriguing, that the light-likeness of the projection of the CP₂ type extremals in M⁴ leads to Virasoro conditions assignable to M⁴ coordinates and this eventually led to the idea of conformal symmetries as isometries as WCW. In the case of the partonic orbits, the light-like curve would be in M⁴× CP₂ but it would not be surprising if the generalization of the Virasoro conditions would emerge also now.

   One can write M⁴ and CP₂ coordinates for the light-like curve as Fourier expansion in powers of exp(it), where t is the light-like coordinate. This gives h^k= ∑ h^k_n exp(int). If the CP₂ projection of the orbits of the partonic 2-surface is geodesic circle, CP₂ metric s_kl is constant, the light-likeness condition h_kl∂_th^k∂l_th^l=0 gives Re(h_kl∑_m h^k_n-mh^l_m=0). This does not give Virasoro conditions.

   The condition d/dt(h_kl∂_th^k∂_th^l=0)=0 however gives the standard Virasoro conditions stating that the normal ordered operators L_n= Re(h_kl∑_m (n-m) h^k_n-mh^l_m) annihilate the physical states. What is interesting is that the latter condition also allows time-like (and even space-like) geodesics.

4. Could massivation mean a failure of light-likeness? For piecewise light-like geodesics the light-likeness condition would be true only inside the segments. By taking Fourier transform one expects to obtain Virasoro conditions with a cutoff analogous to the momentum cutoff in condensed matter physics for crystals. For piecewise light-like geodesics the condition would be trivially true inside the segments and therefore discretized. By taking Fourier transform one expects to obtain Virasoro conditions with a cutoff analogous to the momentum cutoff in condensed matter physics for crystals.
5. In TGD the Virasoro, Kac-Moody algebras and symplectic algebras are replaced by half-algebras and the gauge conditions are satisfied for conformal weights which are n-multiples of fundamentals with with n larger than some minimal value. This would dramatically reduce the effects of the non-determinism and could make the sum over all paths allowed by the light-likeness manifestly finite and reduce it to a sum with a finite number of terms. This cutoff in degrees of freedom would correspond to a genuinely physical cutoff due to the finite measurement resolution coded to the number theoretical anatomy of the space-time surfaces. This cutoff is analogous to momentum cutoff and could at the space-time picture correspond to finite minimum length for the light-like segments of the orbit of the partoic 2-surface.

See the articles About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe, Holography and Hamilton-Jacobi Structure as 4-D generalization of 2-D complex structure, and Symmetries and Geometry of the "World of Classical Worlds".

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The Great Narrative of theoretical physics

I have used a lot of time in pondering the Great Narrative of theoretical physics. It has two components. The stubborn belief on the idea of a continuous progress that arose during the Enlightenment: it has proven impossible to admit that huge mistakes have been made at the level of basic postulates so that progress has become illusory.

The second component comes from colonialism: the development of theoretical physics is seen as a series of wars of conquest. Reductionism encapsulates how conquest-war progresses. Each conquered area corresponds to a new field of physics. The community is still unable to see that the triumphal march of reductionism is a very similar illusion as colonialism was.

Wars of conquest have progressed in the direction of both short and long length scales. We have progressed from the planetary system to astrophysical and cosmic scales. The narrative of cosmology has started to crack more and more: there is a crisis related to the understanding of dark matter and energy and inflationary theory has been in crisis from the beginning. As a matter of fact, a state of stagnation here might be a better word than crisis since crisis means criticality and a promise for something new.

The observations already made earlier, and especially James Webb, have now once and for all destroyed the grand cosmological narrative. The Big Bang remains, but Webb's observations call into question the concept of time at the base of cosmology (galaxies older than the universe), the assumption that coherence is only possible on short scales (correlations on cosmic scales), the assumptions about how signals propagate (or rather do not propagate) on cosmological scales, and also the existing view of the formation of astro-physical objects.

On the other hand, progress has been made in both directions by starting from atomic physics which was a real triumph, but molecular physics is already just phenomenology without any real theory (for example, the concept of a chemical bond is not understood on a basic level at all). In biochemistry, biocatalysis remains a complete mystery.

The troops marched also in the direction of nuclear physics, but it was necessary to decide that it is a completely separate area from atomic physics, even though correlations were noticed very early on. The march proceeded to electroweak interactions: this was a real success and also to hadron physics and QCD. It was agreed that hadrons have been understood even though color confinement remained a complete mystery. Standard model emerged and all that was left was the jump to the Planck length scales. The GUTs were a leap into the void producing nothing, but were accepted as a part of the great narrative, partly for reasons related to funding.

Finally, the super string model was built as a theory that was supposed to unify the standard model and quantum gravity. The trial was based on two theories, both of which have a huge gap. The gaps were already noticed a hundred years ago.

In Einstein's theory, conservation laws of the Special Relativity are lost, but perhaps because the discoverer of the gap was Emmy Noether, a woman and a Jew, this discovery was not allowed to mess with the unfolding Grand Narrative.

The basic paradox of quantum measurement theory was the big gap of quantum mechanics. In the spirit of pragmatism, even that was not allowed to interfere with the development of the Great Narrative so that an endless variety of interpretations were invented. So it's no wonder that the superstring theory built above these two great gaps eventually collapsed.

When one thinks about reductionist wars of conquest, one can't avoid comparisons to Alexander the Great's victories and the rapid collapse of the empire that followed. The Colonial Wars is another point of comparison. In between all the areas of the physics landscape agreed to be conquered, there are white areas on the map, about which nothing is actually known. The last hundred years of theoretical physics will probably be seen as the greatest intellectual self-deception in human intellectual history.

I remember the novel, was it the core of Darkness, which told about a similar illusion related to colonialism. It told about a commander of a British base in Africa, a drunkard who desperately tried to maintain the illusion that the situation was under control after all. In the same way the community of theoretical physicists tries to preserve its Grand Illusions. Internalized censorship takes care that new ideas challenging the basic dogmas are neither published in "prestigious" journals nor funded.

Can one find reasons for the recent situation? Sloppy thinking is certainly one basic reason. Colleagues must respect the rules of logic when they write computer code but when it comes to the consistency between fundamental assumptions and mathematics and empirical facts, the basic rules of logic are given up: the justification for this deadly sin of theoretician comes from "pragmatism".

It has been said that great narratives are dead. I do not agree with this. Great Narrative of theoretical physics is possible but it can be developed only by a continual challenging of the basic assumptions of the existing narrative. This has not been done for a century.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Kundalini phenomenon from TGD point of view

There was a very interesting post by Hunter Glenn in Qualia Computing Network about the Kundalini phenomenon. The proposal of Glenn was that the charge separations relate somehow to the Kundalini phenomenon involving both the bliss and the dark night of the soul suggesting that (quantum) criticality is involved.

In bioelectromagnetism it has been known for decades that the sign of electric gradient along the longitudinal axis of the body correlates with the state of consciousness. The sign of the electric gradient changes when one falls asleep. This sign matters also at the level of brain hemispheres in horizontal direction. At the neuronal level the membrane potential changes temporarily sign during the nerve pulses. At the axonal microtubular level the sign of gradient matters and the tubule is in (quantum?) critical state in the sense that it is decaying and re-assembling all the time.

This suggests that the electric gradient along the spine correlates with the contents of consciousness and has a lot to do with the kundalini phenomenon. The appearance of chills in the spine could reflect the generation of electrical gradients. In my own Great Experience around 1985I experienced these chills and the subsequent "whole body consciousness" completely free from the usual "thermal noise". The attempt to understand this experience led to the development of TGD inspired theory of consciousness.

Charges are needed to create these gradients and the natural question is where these charges reside. Between what kind of systems the charge separations are generated?

Electric gradients and charge separations seem to be fundamental. In the TGD inspired quantum model of the nerve pulse, the cell membrane is regarded as a Josephson junction. Standard physics does not of course allow this: according to the Hodgkin-Huxley model the currents are ohmic currents. There is very intriguing experimental evidence in conflict with the assumption of Ohmic currents as cause of nerve pulse: they could be of course caused by it. This evidence justifies TGD inspired model of nerve pulse discussed here. The model involves the Pollack effect as a way to generate charge separations. In presence of suitable energy feed and gel phase, water develops negatively charged regions with very high charge. 1/4:th of protons of water molecules to somewhere, "outside" the system in some sense. This generates electric gradients and all electric gradients in living matter could be created in this way by metabolic energy feed.

Where could the protons go? In the TGD Universe they would go to the magnetic body (MB), the TGD geometric counterpart for Maxwellian magnetic fields, and form a dark phase there. This would mean that they have non-standard and very large values of effective Planck constant so that they form a large-scale quantum coherent phase at MB: this would induce the coherence of ordinary biomatter as forced coherence. The MB in question would be a gravitational magnetic body and the value of gravitational Planck constant was proposed already by Nottale. Charge separations would reduce to those between the biological body (cell membrane, etc) and corresponding (gravitational) MB and this would allow us to understand how the electric gradients develop. Also nerve pulse would be based on the Pollack effect.

The spiritual aspect would come to play via the magnetic body representing higher level consciousness (an entire hierarchy of them is predicted). In Kundalini a connection to some magnetic body would be created (or lost) and could also give rise to the experience of becoming God.

See the article Some new aspects of the TGD inspired model of the nerve pulse or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see 1 comment:

Symmetries and Geometry of the "World of Classical Worlds"

Still about the symmetries of WCW

I have been analyzing the basic visions of TGD trying to identify weak points. WCW geometry exists only if it has maximal isometries. I have proposed that WCW could be regarded as a union of generalized symmetric spaces labelled by zero modes which do not contribute to the metric. The induced Kähler field is invariant under symplectic transformations of CP₂ and would therefore define zero mode degrees of freedom if one assumes that WCW metric has symplectic transformations as isometries. In particular, Kähler magnetic fluxes would define zero modes and are quantized closed 2-surfaces. The induced metric appearing in Kähler action is however not zero mode degree of freedom. If the action contains volume term, the assumption about union of symmetric spaces is not well-motivated.

Symplectic transformations are not the only candidates for the isometries of WCW. The basic picture about what these maximal isometries could be, is partially inspired by string models.

1. A weaker proposal is that the symplectomorphisms of H define only symplectomorphisms of WCW. Extended conformal symmetries define also a candidate for isometry group. Remarkably, light-like boundary has an infinite-dimensional group of isometries which are in 1-1 correspondence with conformal symmetries of S²⊂ S²× R₊= δ M⁴₊.
2. Extended Kac Moody symmetries induced by isometries of δ M⁴₊ are also natural candidates for isometries. The motivation for the proposal comes from physical intuition deriving from string models. Note they do not include Poincare symmetries, which act naturally as isometries in the moduli space of causal diamonds (CDs) forming the "spine" of WCW.
3. The light-like orbits of partonic 2-surfaces might allow separate symmetry algebras. One must however notice that there is exchange of charges between interior degrees of freedom and partonic 2-surfaces. The essential point is that one can assign to these surface conserved charges when the dual light-like coordinate defines time coordinate. This picture also assumes a slicing of space-time surface by by the partonic orbits for which partonic orbits associated with wormrhole throats and boundaries of the space-time surface would be special. This slicing would correspond to Hamilton-Jacobi structure.
4. Fractal hierarchy of symmetry algebras with conformal weights, which are non-negative integer multiples of fundamental conformal weights, is essential and distinguishes TGD from string models. Gauge conditions are true only the isomorphic subalgebra and its commutator with the entire algebra and the maximal gauge symmetry to a dynamical symmetry with generators having conformal weights below maximal value. This view also conforms with p-adic mass calculations.
5. The realization of the symmetries for 3-surfaces at the boundaries of CD and for light-like orbits of partonic 2-surfaces is known. The problem is how to extend the symmetries to the interior of the space-time surface. It is natural to expect that the symmetries at partonic orbits and light-cone boundary extend to the same symmetries.

Could generalized holomorphy allow to sharpen the existing views?

This picture is rather speculative, allows several variants, and is not proven. There is now however a rather convincing ansatz for the general form of preferred extremals. This proposal relies on the realization of holography as generalized 4-D holomorphy. Could it help to make the picture more precise?

1. Explicit solution of field equations in terms of the generalized holomorphy is now known. The solution ansatz is independent of action as long it is general coordinate invariance depending only on the induced geometric structures. Space-time surfaces would be minimal surfaces apart from lower-dimensional singular surfaces at which the field equations involve the entire action. Only the singularities, classical charges and positions of topological interaction vertices depend on the choice of the action (see this). Kähler action plus volume term is the choice of action forced by twistor lift making the choice of H unique.
2. Hamilton-Jacobi structures emerge naturally as generalized conformal structures of space-time surfaces and M⁴ (see this). This inspires a proposal for a generalization of modular invariance and of moduli spaces as subspaces of Teichmüller spaces.
3. One can assign to holomorphy conserved Noether charges. The conservation reduces to the algebraic conditions satisfied for the same reason as field equations, i.e. the conservation conditions involving contractions of complex tensors of type (1,1) with tensors of type (2,0) and (0,2). The charges have the same form as Noether charges but it is not completely clear whether the action remains invariant under these transformations. This point is non-trivial since Noether theorem says that invariance of the action implies the existence of conserved charges but not vice versa. Could TGD represent a situation in which the equivalence between symmetries of action and conservation laws fails?

   Also string models have conformal symmetries but in this case 2-D area form suffers conformal scaling. Also the fact that holomorphic ansatz is satisfied for such a large class of actions apart from singularities suggests that the action is not invariant.

4. The action should define Kähler function for WCW identified as the space of Bohr orbits. WCW Kähler metric is defined in terms of the second derivatives of the Kähler action of type (1,1) with respect to complex coordinates of WCW. Does the invariance of the action under holomorphies imply a trivial Kähler metric and constant Kähler function?

   Here one must be very cautious since by holography the variations of the space-time surface are induced by those of 3-surface defining holographic data so that the entire space-time surface is modified and the action can change. The presence of singularities, analogous to poles and cuts of an analytic function and representing particles, suggests that the action represents the interactions of particles and must change. Therefore the action might not be invariant under holomorphies. The parameters characterizing the singularities should affect the value of the action just as the positions of these singularities in 2-D electrostatistics affect the Coulomb energy.

   Generalized conformal charges and supercharges define a generalization of Super Virasoro algebra of string models. Also Kac-Moody algebras assignable to the isometries of δ M⁴₊× CP₂ and light-like 3 surfaces generalize trivially.

5. An absolutely essential point is that generalized holomorphisms are not symmetries of Kähler function since otherwise Kähler metric involving second derivatives of type (1,1) with respect to complex coordinates of WCW is non-trivial if defined by these symmetry generators as differential operators. If Kähler function is equal to Kähler action, as it seems, Kähler action cannot be invariant under generalized holomorphies.

   Noether's theorem states that the invariance of the action under a symmetry implies the conservation of corresponding charge but does not claim that the existence of conserved Noether currents implies invariance of the action. Since Noether currents are conserved now, one would have a concrete example about the situation in which the inverse of Noether's theorem does not hold true. In a string model based on area action, conformal transformations of complex string coordinates give rise to conserved Noether currents as one easily checks. The area element defined by the induced metric suffers a conformal scaling so that the action is not invariant in this case.

Challenging the existing picture of WCW geometry

These findings make it possible to challenge and perhaps sharpen the existing speculations concerning the metric and isometries of WCW.

I have considered the possibility that also the symplectomorphisms of δ M⁴+× CP₂ could define WCW isometries. This actually the original proposal. One can imagine two options.

1. The continuation of symplectic transformations to transformations of the space-time surface from the boundary of light-cone or from the orbits partonic 2-surfaces should give rise to conserved Noether currents but it is not at all obvious whether this is the case.
2. One can assign conserved charges to the time evolution of the 3-D boundary data defining the holographic data: the time coordinate for the evolution would correspond to the light-like coordinate of light-cone boundary or partonic orbit. This option I have not considered hitherto. It turns out that this option works!

The conclusion would be that generalized holomorphies give rise to conserved charges for 4-D time evolution and symplectic transformations give rise to conserved charged for 3-D time evolution associated with the holographic data.

About extremals of Chern-Simons-Kähler action

Let us look first the general nature of the solutions to the extremization of Chern-Simons-Kähler action.

1.  The light-likeness of the partonic orbits requires Chern-Simons action, which is equivalent to the topological action J∧ J, which is total divergence and   is a symplectic in variant.  The field equations at the boundary cannot involve  induced metric so that only induced symplectic structure remains. The 3-D holographic data   at partonic orbits would extremize Cherns-Simons-Kähler action. Note that at the ends of the space-time surface about boundaries of CD one cannot pose any dynamics.
2. If the induced Kähler form has only the CP₂ part, the variation of Chern-Simons-Kähler form would give equations  satisfied if the CP₂ projection is at most 2-dimensional and Chern-Simons action would vanish and imply that instanton number vanishes.
3. If the action is the sum of M⁴ and CP₂  parts, the field equations in M⁴ and CP₂ degrees of freedom would give the same result. If the induced Kähler form is  identified as the sum of the M⁴ and CP₂ parts, the equations also allow solutions for which the induced M⁴ and CP₂ Kähler forms sum up to zero.  This phase would involve a map identifying M⁴ and CP₂ projections and force induce Kähler forms to be identical. This would force magnetic charge in M⁴ and the question is whether the line connecting the tips of the CD makes non-trivial homology possible.  The homology charges and the 2-D ends of the partonic orbit cancel each other so that partonic surfaces can have monopole charge.

   The conditions at the partonic orbits do not pose conditions on the interior and should allow generalized holomorphy. The following considerations show that besides homology charges as Kähler magnetic fluxes also Hamiltonian fluxes are conserved in Chern-Simons-Kähler dynamics.

Can one assign conserved charges with symplectic transformations or partonic orbits and 3-surfaces at light-cone boundary?

The geometric picture is that symplectic symmetries are Hamiltonian flows along the light-like partonic orbits generated by the projection A_t of the Kähler gauge potential in the direction of the light-like time coordinate. The physical picture is that the partonic 2-surface is a Kähler charged particle that couples to the Hamilton H=A_t. The Hamiltonians H_A are conserved in this time evolution and give rise to conserved Noether currents. The corresponding conserved charge is integral over the 2-surface defined by the area form defined by the induced Kähler form.

Let's examine the change of the Chern-Simons-Kähler action in a deformation that corresponds, for example, to the CP₂ symplectic transformation generated by Hamilton H_A. M⁴ symplectic transformations can be treated in the same way:here however M⁴ Kähler form would be involved, assumed to accompany Hamilton-Jacobi structure as a dynamically generated structure.

1. Instanton density for the induced Kähler form reduces to a total divergence and gives Chern-Simons-Kähler action, which is TGD analog of topological action. This action should change in infinitesimal symplectic transformations by a total divergence, which should vanish for extremals and give rise to a conserved current. The integral of the divergence gives a vanishing charge difference between the ends of the partonic orbit. If the symplectic transformations define symmetries, it should be possible to assign to each Hamiltonian H_A a conserved charge. The corresponding quantal charge would be associated with the modified Dirac action.

2. The conserved charge would be an integral over X². The surface element is not given by the metric but by the symplectic structure, so that it is preserved in symplectic transformations. The 2-surface of the time evolution should correspond to the Hamiltonian time transformation generated by the projection A_α=A_k ∂_αs^k of the Kähler gauge potential A_k to the direction of light-like time coordinate xα== t.

3. The effect of the generator j_A^k= J^kl∂_lH_A on the Kähler potential A_l is given by j^k_A∂_kA_l. This can be written as ∂_kA_l=J_kl + ∂_lA_k. The first term gives the desired total divergence ∂_α (ε^αβγJ_βγ H_A).

   The second term is proportional to the term ∂_αH_A- {A_α,H}. Suppose that the induced Kähler form is transversal to the light-like time coordinate t, i.e. the induced Kähler form does not have components of form J_tμ. In this kind of situation the only possible choice for α corresponds to the time coordinate t. In this situation one can perform the replacement ∂_αH_A-{A_α,H}→ dH_A/dt-{A_t,H}. This corresponds to a Hamiltonian time evolution generated by the projection A_t acting as a Hamiltonian. If this is really a Hamiltonian time evolution, one has dH_A/dt-{A,H}=0. Because the Poisson bracket represents a commutator, the Hamiltonian time evolution equation is analogous to the vanishing of a covariant derivative of H_A along light-like curves: ∂_tH_A +[A,H_A]= 0. The physical interpretation is that the partonic surface develops like a particle with a Kähler charge. As a consequence the change of the action reduces to a total divergence.

   An explicit expression for the conserved current J_A^α=H_A ε^αβγJ_βγ can be derived from the vanishing of the total divergence. Symplectic transformations on X² generate an infinite-dimensional symplectic algebra. The charge is given by the Hamiltonian flux Q_A =∫ H_A J_αβdx^α∧dx^β.

4. If the projection of the partonic path CP₂ or M⁴ is 2-D, then the light-like geodesic line corresponds to the path of the parton surface. If A_l can be chosen parallel to the surface, its projection in the direction of time disappears and one has A_t=0. In the more general case, X² could, for example, rotate in CP₂. In this case A_t is nonvanishing. If J is transversal (no Kähler electric field), charge conservation is obtained.

Do the above observations apply at the boundary of the light-cone?

1. Now the 3-surface is space-like and Chern-Simons-Kähler action makes sense. It is not necessary but emerges from the "instanton density" for the Kähler form. The symplectic transformations of δ M⁴₊× CP₂ are the symmetries. The most time evolution associated with the radial light-like coordinate would be from the tip of the light-cone boundary to the boundary of CD. Conserved charges as homological invariants defining symplectic algebra would be associated with the 2-D slices of 3-surfaces. For closed 3-surfaces the total charges from the sheets of 3-space as covering of δ M⁴₊ must sum up to zero.
2. Interestingly, the original proposal for the isometries of WCW was that the Hamiltonian fluxes assignable to M⁴ and CP₂ degrees of freedom at light-like boundary act define the charges associated with the WCW isometries as symplectic transformations so that a strong form of holography would have been be realized and space-time surface would have been effectively 2-dimensional. The recent view is that these symmetries pose conditions only on the 3-D holographic data. The holographic charges would correspond to additional isometries of WCW and would be well-defined for the 3-surfaces at the light-cone boundary.

To sum up, one can imagine many options but the following picture is perhaps the simplest one and is supported by physical intuition and mathematical facts. The isometry algebra ofδ M⁴₊× CP₂ consists of generalized conformal and KM algebras at 3-surfaces in δ M⁴₊× CP₂ and symplectic algebras at the light cone boundary and 3-D light-like partonic orbits. The latter symmetries give constraints on the 3-D holographic data. It is still unclear whether one can assign generalized conformal and Kac-Moody charges to Chern-Simons-K\"ahler action. The isomorphic subalgebras labelled by a positive integer and their commutators with the entire algebra would annihilate the physical states.

The TGD counterparts of the gauge conditions of string models

The string model picture forces to ask whether the symplectic algebras and the generalized conformal and Kac-Moody algebras could act as gauge symmetries.

1. In string model picture conformal invariance would suggest that the generators of the generalized conformal and KM symmetries act as gauge transformations annihilate the physical states. In the TGD framework, this does not however make sense physically. This also suggests that the components of the metric defined by supergenerators of generalized conformal and Kac Moody transformations vanish. If so, the symplectomorphisms δ M⁴₊× CP₂ localized with respect to the light-like radial coordinate acting as isometries would be needed. The half-algebras of both symplectic and conformal generators are labelled by a non-negative integer defining an analog of conformal weight so there is a fractal hierarchy of isomorphic subalgebras in both cases.
2. TGD forces to ask whether only subalgebras of both conformal and Kac-Moody half algebras, isomorphic to the full algebras, act as gauge algebras. This applies also to the symplectic case. Here it is essential that only the half algebra with non-negative multiples of the fundamental conformal weights is allowed. For the subalgebra annihilating the states the conformal weights would be fixed integer multiples of those for the full algebra. The gauge property would be true for all algebras involved. The remaining symmetries would be genuine dynamical symmetries of the reduced WCW and this would reflect the number theoretically realized finite measurement resolution. The reduction of degrees of freedom would also be analogous to the basic property of hyperfinite factors assumed to play a key role in thee definition of finite measurement resolution.
3. For strong holography, the orbits of partonic 2-surfaces and boundaries of the spacetime surface at δ M⁴₊ would be dual in the information theoretic sense. Either would be enough to determine the space-time surface.

See the articles About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe, Holography and Hamilton-Jacobi Structure as 4-D generalization of 2-D complex structure, Symmetries and Geometry of the "World of Classical Worlds" and the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds".

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Still about the symmetries of WCW

I have been analyzing the basic visions of TGD trying to identify weak points. WCW geometry exists only if it has maximal isometries. I have proposed that WCW could be regarded as a union of generalized symmetric spaces labelled by zero modes which do not contribute to the metric. The induced Kähler field is invariant under symplectic transformations of CP₂ and would therefore define zero mode degrees of freedom if one assumes that WCW metric has symplectic transformations as isometries. In particular, Kähler magnetic fluxes would define zero modes and are quantized closed 2-surfaces. The induced metric appearing in Kähler action is however not zero mode degree of freedom. If the action contains volume term, the assumption about union of symmetric spaces is not well-motivated.

Symplectic transformations are not the only candidates for the isometries of WCW. The basic picture about what these maximal isometries could be, is partially inspired by string models.

1. A weaker proposal is that the symplectomorphisms of H define only symplectomorphisms of WCW. Extended conformal symmetries define also a candidate for isometry group. Remarkably, light-like boundary has an infinite-dimensional group of isometries which are in 1-1 correspondence with conformal symmetries of S²⊂ S²× R₊= δ M⁴₊.
2. Extended Kac Moody symmetries induced by isometries of δ M⁴₊ are also natural candidates for isometries. The motivation for the proposal comes from physical intuition deriving from string models. Note they do not include Poincare symmetries, which act naturally as isometries in the moduli space of causal diamonds (CDs) forming the "spine" of WCW.
3. The light-like orbits of partonic 2-surfaces might allow separate symmetry algebras. One must however notice that there is exchange of charges between interior degrees of freedom and partonic 2-surfaces. The essential point is that one can assign to these surface conserved charges when the dual light-like coordinate defines time coordinate. This picture also assumes a slicing of space-time surface by by the partonic orbits for which partonic orbits associated with wormrhole throats and boundaries of the space-time surface would be special. This slicing would correspond to Hamilton-Jacobi structure.
4. Fractal hierarchy of symmetry algebras with conformal weights, which are non-negative integer multiples of fundamental conformal weights, is essential and distinguishes TGD from string models. Gauge conditions are true only the isomorphic subalgebra and its commutator with the entire algebra and the maximal gauge symmetry to a dynamical symmetry with generators having conformal weights below maximal value. This view also conforms with p-adic mass calculations.
5. The realization of the symmetries for 3-surfaces at the boundaries of CD and for light-like orbits of partonic 2-surfaces is known. The problem is how to extend the symmetries to the interior of the space-time surface. It is natural to expect that the symmetries at partonic orbits and light-cone boundary extend to the same symmetries.

Could generalized holomorphy allow to sharpen the existing views?

This picture is rather speculative, allows several variants, and is not proven. There is now however a rather convincing ansatz for the general form of preferred extremals. This proposal relies on the realization of holography as generalized 4-D holomorphy. Could it help to make the picture more precise?

1. Explicit solution of field equations in terms of the generalized holomorphy is now known. The solution ansatz is independent of action as long it is general coordinate invariance depending only on the induced geometric structures. Space-time surfaces would be minimal surfaces apart from lower-dimensional singular surfaces at which the field equations involve the entire action. Only the singularities, classical charges and positions of topological interaction vertices depend on the choice of the action (see this). Kähler action plus volume term is the choice of action forced by twistor lift making the choice of H unique.
2. Hamilton-Jacobi structures emerge naturally as generalized conformal structures of space-time surfaces and M⁴ (see this). This inspires a proposal for a generalization of modular invariance and of moduli spaces as subspaces of Teichmüller spaces.
3. One can assign to holomorphy conserved Noether charges. The conservation reduces to the algebraic conditions satisfied for the same reason as field equations, i.e. the conservation conditions involving contractions of complex tensors of type (1,1) with tensors of type (2,0) and (0,2). The charges have the same form as Noether charges but it is not completely clear whether the action remains invariant under these transformations. This point is non-trivial since Noether theorem says that invariance of the action implies the existence of conserved charges but not vice versa. Could TGD represent a situation in which the equivalence between symmetries of action and conservation laws fails?

   Also string models have conformal symmetries but in this case 2-D area form suffers conformal scaling. Also the fact that holomorphic ansatz is satisfied for such a large class of actions apart from singularities suggests that the action is not invariant.

4. The action should define Kähler function for WCW identified as the space of Bohr orbits. WCW Kähler metric is defined in terms of the second derivatives of the Kähler action of type (1,1) with respect to complex coordinates of WCW. Does the invariance of the action under holomorphies imply a trivial Kähler metric and constant Kähler function?

   Here one must be very cautious since by holography the variations of the space-time surface are induced by those of 3-surface defining holographic data so that the entire space-time surface is modified and the action can change. The presence of singularities, analogous to poles and cuts of an analytic function and representing particles, suggests that the action represents the interactions of particles and must change. Therefore the action might not be invariant under holomorphies. The parameters characterizing the singularities should affect the value of the action just as the positions of these singularities in 2-D electrostatistics affect the Coulomb energy.

   Generalized conformal charges and supercharges define a generalization of Super Virasoro algebra of string models. Also Kac-Moody algebras assignable to the isometries of δ M⁴₊× CP₂ and light-like 3 surfaces generalize trivially.

5. An absolutely essential point is that generalized holomorphisms are not symmetries of Kähler function since otherwise Kähler metric involving second derivatives of type (1,1) with respect to complex coordinates of WCW is non-trivial if defined by these symmetry generators as differential operators. If Kähler function is equal to Kähler action, as it seems, Kähler action cannot be invariant under generalized holomorphies.

   Noether's theorem states that the invariance of the action under a symmetry implies the conservation of corresponding charge but does not claim that the existence of conserved Noether currents implies invariance of the action. Since Noether currents are conserved now, one would have a concrete example about the situation in which the inverse of Noether's theorem does not hold true. In a string model based on area action, conformal transformations of complex string coordinates give rise to conserved Noether currents as one easily checks. The area element defined by the induced metric suffers a conformal scaling so that the action is not invariant in this case.

Challenging the existing picture of WCW geometry

These findings make it possible to challenge and perhaps sharpen the existing speculations concerning the metric and isometries of WCW.

I have considered the possibility that also the symplectomorphisms of δ M⁴+× CP₂ could define WCW isometries. This actually the original proposal. One can imagine two options.

1. The continuation of symplectic transformations to transformations of the space-time surface from the boundary of light-cone or from the orbits partonic 2-surfaces should give rise to conserved Noether currents but it is not at all obvious whether this is the case.
2. One can assign conserved charges to the time evolution of the 3-D boundary data defining the holographic data: the time coordinate for the evolution would correspond to the light-like coordinate of light-cone boundary or partonic orbit. This option I have not considered hitherto. It turns out that this option works!

The conclusion would be that generalized holomorphies give rise to conserved charges for 4-D time evolution and symplectic transformations give rise to conserved charged for 3-D time evolution associated with the holographic data.

About extremals of Chern-Simons-Kähler action

Let us look first the general nature of the solutions to the extremization of Chern-Simons-Kähler action.

1.  The light-likeness of the partonic orbits requires Chern-Simons action, which is equivalent to the topological action J∧ J, which is total divergence and   is a symplectic in variant.  The field equations at the boundary cannot involve  induced metric so that only induced symplectic structure remains. The 3-D holographic data   at partonic orbits would extremize Cherns-Simons-Kähler action. Note that at the ends of the space-time surface about boundaries of CD one cannot pose any dynamics.
2. If the induced Kähler form has only the CP₂ part, the variation of Chern-Simons-Kähler form would give equations  satisfied if the CP₂ projection is at most 2-dimensional and Chern-Simons action would vanish and imply that instanton number vanishes.
3. If the action is the sum of M⁴ and CP₂  parts, the field equations in M⁴ and CP₂ degrees of freedom would give the same result. If the induced Kähler form is  identified as the sum of the M⁴ and CP₂ parts, the equations also allow solutions for which the induced M⁴ and CP₂ Kähler forms sum up to zero.  This phase would involve a map identifying M⁴ and CP₂ projections and force induce Kähler forms to be identical. This would force magnetic charge in M⁴ and the question is whether the line connecting the tips of the CD makes non-trivial homology possible.  The homology charges and the 2-D ends of the partonic orbit cancel each other so that partonic surfaces can have monopole charge.

   The conditions at the partonic orbits do not pose conditions on the interior and should allow generalized holomorphy. The following considerations show that besides homology charges as Kähler magnetic fluxes also Hamiltonian fluxes are conserved in Chern-Simons-Kähler dynamics.

Can one assign conserved charges with symplectic transformations or partonic orbits and 3-surfaces at light-cone boundary?

The geometric picture is that symplectic symmetries are Hamiltonian flows along the light-like partonic orbits generated by the projection A_t of the Kähler gauge potential in the direction of the light-like time coordinate. The physical picture is that the partonic 2-surface is a Kähler charged particle that couples to the Hamilton H=A_t. The Hamiltonians H_A are conserved in this time evolution and give rise to conserved Noether currents. The corresponding conserved charge is integral over the 2-surface defined by the area form defined by the induced Kähler form.

Let's examine the change of the Chern-Simons-Kähler action in a deformation that corresponds, for example, to the CP₂ symplectic transformation generated by Hamilton H_A. M⁴ symplectic transformations can be treated in the same way:here however M⁴ Kähler form would be involved, assumed to accompany Hamilton-Jacobi structure as a dynamically generated structure.

1. Instanton density for the induced Kähler form reduces to a total divergence and gives Chern-Simons-Kähler action, which is TGD analog of topological action. This action should change in infinitesimal symplectic transformations by a total divergence, which should vanish for extremals and give rise to a conserved current. The integral of the divergence gives a vanishing charge difference between the ends of the partonic orbit. If the symplectic transformations define symmetries, it should be possible to assign to each Hamiltonian H_A a conserved charge. The corresponding quantal charge would be associated with the modified Dirac action.

2. The conserved charge would be an integral over X². The surface element is not given by the metric but by the symplectic structure, so that it is preserved in symplectic transformations. The 2-surface of the time evolution should correspond to the Hamiltonian time transformation generated by the projection A_α=A_k ∂_αs^k of the Kähler gauge potential A_k to the direction of light-like time coordinate xα== t.

3. The effect of the generator j_A^k= J^kl∂_lH_A on the Kähler potential A_l is given by j^k_A∂_kA_l. This can be written as ∂_kA_l=J_kl + ∂_lA_k. The first term gives the desired total divergence ∂_α (ε^αβγJ_βγ H_A).

   The second term is proportional to the term ∂_αH_A- {A_α,H}. Suppose that the induced Kähler form is transversal to the light-like time coordinate t, i.e. the induced Kähler form does not have components of form J_tμ. In this kind of situation the only possible choice for α corresponds to the time coordinate t. In this situation one can perform the replacement ∂_αH_A-{A_α,H}→ dH_A/dt-{A_t,H}. This corresponds to a Hamiltonian time evolution generated by the projection A_t acting as a Hamiltonian. If this is really a Hamiltonian time evolution, one has dH_A/dt-{A,H}=0. Because the Poisson bracket represents a commutator, the Hamiltonian time evolution equation is analogous to the vanishing of a covariant derivative of H_A along light-like curves: ∂_tH_A +[A,H_A]= 0. The physical interpretation is that the partonic surface develops like a particle with a Kähler charge. As a consequence the change of the action reduces to a total divergence.

   An explicit expression for the conserved current J_A^α=H_A ε^αβγJ_βγ can be derived from the vanishing of the total divergence. Symplectic transformations on X² generate an infinite-dimensional symplectic algebra. The charge is given by the Hamiltonian flux Q_A =∫ H_A J_αβdx^α∧dx^β.

4. If the projection of the partonic path CP₂ or M⁴ is 2-D, then the light-like geodesic line corresponds to the path of the parton surface. If A_l can be chosen parallel to the surface, its projection in the direction of time disappears and one has A_t=0. In the more general case, X² could, for example, rotate in CP₂. In this case A_t is nonvanishing. If J is transversal (no Kähler electric field), charge conservation is obtained.

Do the above observations apply at the boundary of the light-cone?

1. Now the 3-surface is space-like and Chern-Simons-Kähler action makes sense. It is not necessary but emerges from the "instanton density" for the Kähler form. The symplectic transformations of δ M⁴₊× CP₂ are the symmetries. The most time evolution associated with the radial light-like coordinate would be from the tip of the light-cone boundary to the boundary of CD. Conserved charges as homological invariants defining symplectic algebra would be associated with the 2-D slices of 3-surfaces. For closed 3-surfaces the total charges from the sheets of 3-space as covering of δ M⁴₊ must sum up to zero.
2. Interestingly, the original proposal for the isometries of WCW was that the Hamiltonian fluxes assignable to M⁴ and CP₂ degrees of freedom at light-like boundary act define the charges associated with the WCW isometries as symplectic transformations so that a strong form of holography would have been be realized and space-time surface would have been effectively 2-dimensional. The recent view is that these symmetries pose conditions only on the 3-D holographic data. The holographic charges would correspond to additional isometries of WCW and would be well-defined for the 3-surfaces at the light-cone boundary.

To sum up, one can imagine many options but the following picture is perhaps the simplest one and is supported by physical intuition and mathematical facts. The isometry algebra ofδ M⁴₊× CP₂ consists of generalized conformal and KM algebras at 3-surfaces in δ M⁴₊× CP₂ and symplectic algebras at the light cone boundary and 3-D light-like partonic orbits. The latter symmetries give constraints on the 3-D holographic data. It is still unclear whether one can assign generalized conformal and Kac-Moody charges to Chern-Simons-K\"ahler action. The isomorphic subalgebras labelled by a positive integer and their commutators with the entire algebra would annihilate the physical states.

The TGD counterparts of the gauge conditions of string models

The string model picture forces to ask whether the symplectic algebras and the generalized conformal and Kac-Moody algebras could act as gauge symmetries.

1. In string model picture conformal invariance would suggest that the generators of the generalized conformal and KM symmetries act as gauge transformations annihilate the physical states. In the TGD framework, this does not however make sense physically. This also suggests that the components of the metric defined by supergenerators of generalized conformal and Kac Moody transformations vanish. If so, the symplectomorphisms δ M⁴₊× CP₂ localized with respect to the light-like radial coordinate acting as isometries would be needed. The half-algebras of both symplectic and conformal generators are labelled by a non-negative integer defining an analog of conformal weight so there is a fractal hierarchy of isomorphic subalgebras in both cases.
2. TGD forces to ask whether only subalgebras of both conformal and Kac-Moody half algebras, isomorphic to the full algebras, act as gauge algebras. This applies also to the symplectic case. Here it is essential that only the half algebra with non-negative multiples of the fundamental conformal weights is allowed. For the subalgebra annihilating the states the conformal weights would be fixed integer multiples of those for the full algebra. The gauge property would be true for all algebras involved. The remaining symmetries would be genuine dynamical symmetries of the reduced WCW and this would reflect the number theoretically realized finite measurement resolution. The reduction of degrees of freedom would also be analogous to the basic property of hyperfinite factors assumed to play a key role in thee definition of finite measurement resolution.
3. For strong holography, the orbits of partonic 2-surfaces and boundaries of the spacetime surface at δ M⁴₊ would be dual in the information theoretic sense. Either would be enough to determine the space-time surface.

See the articles About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe and Holography and Hamilton-Jacobi Structure as 4-D generalization of 2-D complex structure)

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

4-D generalization of holomorphies and Kac-Moody symmetries as isometries of the "world of classical worlds"?

Quite recently, I learned that the generalized holomorphies of space-time surfaces define non-trivial conserved charges. Also a generalization of Super-Kac-Moody charges associated with certain embedding space isometries emerges. This suggests a very close connection with string models and provides a possibility to provide answers to the longstanding questions relating to the identification of the isometry group of the "world of classical worlds" (WCW). Generalized holomorphy not only solves explicitly the equations of motion but, as found quite recently, also gives corresponding conserved Noether currents and charges.

1. Generalized holomorphy algebra generalizes the Super-Virasoro algebra and the Super-Kac-Moody algebra related to the conformal invariance of the string model. The corresponding Noether charges  are conserved. Modified Dirac action allows to construct the supercharges having interpretation as WCW gamma matrices. This suggests an answer to a longstanding question related to the isometries of the "world of the classical worlds" (WCW).
2. Either the generalized holomorphies or the symplectic symmetries of H=M⁴× CP₂ or both together define WCW isometries and corresponding super algebra. It would seem that symplectic symmetries induced from H are not necessarily needed and might actually correspond to symplectic symmetries of WCW. This would give a close similarity with the string model, except that one has half-algebra for which conformal weights are proportional to non-negative integers and gauge conditions only apply to an isomorphic subalgebra. These are labeled by positive integers and one obtains a hierarchy.
3. By their light-likeness, the light cone boundary and orbits of partonic 2-surfaces allow an infinite-dimensional isometry group. This is possible only in dimension four. Its transformations are generalized conformal transformations of 2-sphere (partonic 2-surface) depending on light-like radial coordinate such that the radial scaling compensates for the usual conformal scaling of the metric. The WCW isometries would thus correspond to the isometries of the parton orbit and of the boundary of the light cone! These two representations could provide alternative representations for the charges if the strong form of holography holds true and would realize a strong form of holography. Perhaps these realizations deserve to be called inertial and gravitational charges.

   Can these transformations leave the action invariant? For the light-cone boundary, this looks obvious if the light-cone is sliced by a surface parallel to the light-cone boundary. Note however that the tip of this surface might produce problems. A slicing defined by the Hamilton-Jacobi structure would be naturally associated with partonic orbits.

4. What about Poincare symmetries? They would act on the center of mass coordinates of causal diamonds (CDs) as found already earlier (see this). CDs form the "spine" of WCW, which can be regarded as fiber space with fiber for a given CD containing as a fiber the space-time surfaces inside it.

The super-symmetric counterparts of holomorphic charges for the modified Dirac action and bilinear in fermionic oscillator operators associated with the second quantization of free spinor fields in H, define gamma matrices of WCW. Their anticommutators define the Kähler metric of WCW. There is no need to calculate either the action defining the classical Kähler action defining the Kähler function or its derivatives with respect to WCW complex coordinates and their conjugates. What is important is that this makes it possible to speak about WCW metric also for number theoretical discretization of WCW with space-time surfaces replaced with their number theoretic discretizations.

See the article About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe or the chapter About Platonization of Nuclear String Model and of Model of Atoms or .

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

About the proposal of Phil Gibbs that energy is conserved in General Relativity

I have been analyzing the basic visions of TGD trying to identify weak points. Symmetries are central in TGD. The basic motivation for TGD was the loss of Poincare invariance in GRT and in the following I will analyze the claim of Phil Gibbs that one obtains energy conservation in GRT.

The following considerations are inspired by discussions with Marko Manninen and related to whether in general relativity it could be possible to define conserved quantities associated with at least some general coordinate transformations as proposed by Phil Gibbs (see this). This is certainly in conflict with the general idea that the choice of coordinates cannot have any physical effect and personally I am skeptic. I however decided to analyze the proposal in detail and found that it relates to a possible generalization of the notion of Newtonian gravitational flux, which gives the gravitational mass of the system.

1. Gibbs' proposal for Noether charges associated with general coordinate transformations says nothing detailed about charges and the straightforward application of the basic formula gives vanishing charges since these currents turn out to be proportional to T-G which vanishes by Einstein's equations. However, the action includes a term containing second derivatives of the metric. Could this give an anomalous contribution to the Noether charge?
2. In electrodynamics and gauge theories, charges are obtained in connection with gauge transformations that become constant at a distance. The gauge charge density is a total divergence and gauge charge can be expressed as an electric flux across a very large sphere. On the other hand, in Newton's theory, the gravitational flux far enough from the system gives its mass. Could mass correspond to a time translation as a symmetry? Could the transformation of the charge into total divergence generalize to other general coordinate transformations?
3. Einstein action (curvature scalar) contains terms proportional to the second order partial derivatives of the metric: these terms come from the part of the curvature scalar linear in Riemann connection, which serves as analogs of non-abelian gauge potentials. However, this does not give third derivatives to the equations of motion. The reason is that the second derivatives occur linearly. If the square of the curvature tensor would define the action as an analog of Yang-Mills action, the situation would be different. This term is analogous to a dissipative and might relate to the general features of GRT dynamics (blackholes as asymptotic states).

   Is the divergence term taken into account automatically in the straightforward Noetherian guess for the conserved currents? Or could the charges associated with the general coordinate transformations emerge as analogs of electric charge as a flux integral over a very large sphere. This would certainly contradict the fact that general coordinate transformations do nothing to the system, so that they cannot relate physically non-equivalent configurations.

4. The deduction of field equations involves transformation of the terms containing derivatives for the variation of the metric so that only terms involving only the variation of the metric remains besides total divergences, which must vanish for symmetries leaving the action invariant. This gives an explicit formula for the conserved Noether currents. In the case of the curvature scalar, the first term in the conserved current comes from the variation of the first derivatives of the metric. The second term comes from the variation of the second derivatives of the metric tensor and an explicit expression can be deduced for it. This gives a total divergence. Is this term automatically included in the term proportional to T-G? This seems very likely.

Just for curiosity, let us consider the possibility that the total divergence term is not included in T-G and must be included as an additional term.

1. As a total divergence this term can be transformed into a surface integral and is proportional to the vector field generating the transformation. This term could give a non-vanishing contribution as an integral over the boundary at infinity which can be regarded as an infinitely large sphere. If the space-time is asymptotically Minkowskian, the counterparts of 4-momentum, angular momentum and also charges associated with Poincare transformation are obtained. Also the charges associated with arbitrary general coordinate transformations are obtained but these are not in general conserved.
2. The explicit form for the conserved current associated with infinitesimal general coordinate transformation generated by the vector field j^μ is

   J^μ(j)= L^αβμ Dg_αβ + L^αβμν ∂_νDg_αβ

   =L^αβμDg_αβ +∂_νL^αβμ Dg_αβ - ∂_ν[L^αβμDg_αβ],

   where one has

   L^αβμ= ∂ L/∂(∂_μ g_αβ),
   L^αβμν= ∂ L/∂(∂_μνg_αβ),
   Dg_αβ=j^ρ∂_ρ g_αβ .

   The third term at the second line is a total divergence and this contribution, call it Q₃(j) to the expression for the charge as a 3-D integral of the μ=t component of the current can be transformed to a surface integral.

   Q₃(j)= -∫_S²[∂ L/∂(∂_tr ∂_ρ g_αβ)] ∂_ρ g_αβj^ρ]dS .

3. In the stationary case, the g_tt component of the metric includes the gravitational potential and its radial derivative gives a 1/r² term whose flux over the spherical surface is non-vanishing and gives the same result as gravitational flux in Newton's theory. Therefore there is a 1/r² term in the curvature tensor, which is analogous to the electric field. This interpretation requires that the space is asymptotically Minkowski space, so it is possible to talk about Poincare symmetry as an asymptotic symmetry. Constant time shift corresponds to mass.
4. The flux contribution to the charge must be linear in Christoffel symbols and involve the indices t and r. A good guess is that the charge is proportional to

   Q₃(j)= ∫_S² C(t,rρ) j^ρdS .

   where C(t,rρ) denotes Christoffel symbol. For Schwarzschild metric this gives Newtonian gravitational flux for time translation j^ρ=δ^ρ,t.

5. One must pose additional conditions guaranteeing that these charges do not flow radially out of the infinite sphere. This becomes a condition that the second derivatives of the metric with respect to the radial coordinate r approach zero faster than 1/r². This holds true very generally. Note however that the flux associated with arbitrary j need not be conserved. Consider as an example generalized coordinate transformations which approach trivial transformations in the future and non-trivial transformations in the past.
6. Year or two ago there was a lot of talk about an infinite number of charges that can be connected in this way as asymptotic charges to conformal transformations of an infinitely large sphere. These charges could be a special case of pseudo charges described above.

To conclude, there are two options. Personally I am convinced that the T-G option is the right one. For the T-G=0 option, the total charges related to general coordinate transformations are therefore zero. One could however say that the total Noether charges are always zero but that they can be divided into interior and flux parts according to holography and cancel out each other. Flux part would correspond to what is called gravitational charge. In this sense the charges related to general coordinate transformations or at least Poincare transformations can be assigned with the system via holography as flux integrals. This could perhaps be interpreted within the framework of holography. These fluxes would characterize the asymptotic behavior giving in turn information about the dynamics in the interior.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The age of the Universe is twice the usual estimate for the age suggests James Webb: what does this mean?

James Webb telescope has reported stars and galaxies older than the Universe. This finding is new but cannot be put under the rug anymore. It has been proposed (see this that the age of the Universe is about 26.7 billion years and rather precisely twice the standard age about 13.2 billion years.

In TGD the time arrow changes in ordinary "big" state function reductions (BSFRs) which can take place in arbitrarily long time scales. This means that the system lives forth and back in time. One must distinguish between ordinary age and developmental age.

Remarkably, the total evolutionary time spent per given ordinary time interval is roughly twice(!) this time interval! This view explains stars and galaxies older than the Universe and might also explain why the researchers have concluded that the age of the universe is twice the standard age.

The most dramatic implications relate to living systems. BSFR means death and reincarnation with a reversed arrow of time. Similar doublings might occur in biology: for instance, the developmental age of the genome could be twice the age deduced from, say fossiles.

See the article TGD view of the paradoxical findings of the James Webb telescope.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Questions related to the generalized holomorphies and fundamental vertices according to TGD

We had very inspiring discussions with Marko Manninen at a birthday meal with wine. During the trip back home some questions and ideas emerged. Could the 4-D generalization of holomorphy realizing holography allow an infinite hierarchy of conserved charges generalizing the Super Virasoro algebra. Could the only particle vertex in TGD correspond to a creation of fermion-antifermion pair: in this 2-vertex fermion state and fermionic line, partonic orbit, or Bohr orbit turns back in time.? Can one identify the graviton emission vertex?

4-D generalization of the holomorphy allows conserved charges associated with the generalized holomorphies

Does the 4-D analogy of holomorphy as a realization of holography give rise to conserved quantities? Now the symmetries would not be isometries, nor some other symmetries of the action, but dynamic symmetries satisfied only by the Bohr orbits. A little calculation that one can do in your head shows that one obtains conserved currents: the reason is the same as in the case of field equations. The divergence of the Noether current is a contraction of tensors with no common index pairs for the generalization of complex coordinates.

Unlike those associated with the general coordinate invariance, these conserved quantities do not vanish. They correspond to the 4-D generalization of conformal transformations and give rise to a generalization of the Virasoro algebra and also of Super Virasoro algebra realized in terms of the modified Dirac action for the induced spinor fields obtained from the free second quantized spinor fields of H.

In the string model, these conformal charges are assumed to annihilate the physical states. In TGD, I have proposed that only a subalgebra that is isomorphic to the whole algebra, having conformal weights which are integer multiples of the entire algebra, does this. In TGD framework, the conformal weights are necessarily non-negative and ZEO allows this. One obtains a whole hierarchy of subalgebras and a sub-hierarchy of algebras for which conformal symmetry as gauge symmetry is "broken" to dynamical Lie symmetries for physical states having conformal weight below some maximum value. These hierarchies could correspond to the hierarchies of algebraic extensions for rationals defined by composite polynomials.

Are fermionic 2-vertices all that is needed in TGD?

In quantum field theories, already the interaction vertex for 3 particles leads to divergences. In a typical 3-vertex, fermion emits a boson or boson decays to a fermion-antifermion pair. In TGD, the situation changes.

1. Fermions are the only fundamental particles in TGD. Since fundamental bosons are missing, there is no vertex representing emission of a fundamental boson emission from fermion or a vertex producing fermion antifermion pair from a fundamental boson. In TGD, bosons as elementary particles (distinguished from fundamental bosons) are fermion-antifermion pairs, and the emission of elementary bosons is possible. However, the problem is that the total fermion and antifermion numbers are separately conserved. Unless it is possible to create fermion pairs from classical fields!
2. In the standard theory fermion-antifermion pairs can be indeed created in classical gauge fields. This creation is an experimental fact but it is thought that this description is only a convenient approximation. In TGD however, the classical fields associated with the Bohr orbits of 3-surfaces are an exact part of quantum theory. Could this description be accurate in TGD? In the classical induced fields associated with particles, pairs could arise. Approximation would become exact in TGD.

A 2-vertex for creation of fermion-antifermion pair (or corresponding boson) is needed. In this vertex, the fermion turns must turn backwards in time.

1. I managed to identify the fermionic 2-vertex was specified towards the end of this year as I realized the connection to the problem of general relativity, which arises from the existence of GRT space-times for which the 4-D diffeo structure is non-standard. There are a lot of these. For an exotic diffeo structure, the standard diffeo structure can be said to have point-like defects analogous to lattice defects.
2. Remarkably, this problem is encountered only in the space-time dimension 4 (see this)! Physical intuition suggests that it must be possible to turn this problem from a disaster to victory. In TGD, this is what actually happens: these point-like diffeo-defects can be identified as interaction vertices, the fermion turns back in the direction of time. Pair creation would be possible only in space-time dimension 4!

   A generalization of the classical fermion pair creation vertex has the same general form as in QFT. As a special case the pair can correspond to a boson as a fermion-antifermion bound state. This vertex also has geometric variants in different dimensions. A fermion line, string world sheet, the orbit of a partonic 2-surface and also the Bohr orbit of 3 surface can turn backwards in time and the fermion states associated with the induced spinor fields do the same.

This inspires two questions.

1. Is the creation of a pair actually the only vertex or is it possible to have a geometric 3-vertex and is it really needed? At the fermion level only the 2-vertex described above is not possible, but for the topological reactions of surfaces one could think of 3-vertices and in the earlier picture I thought these are needed. They do not seem to be necessary however.

   If so, the theory would be extremely simple compared to quantum field theories. There dangerous genuine 3-vertices would be absent and diffeo defects defining 2 vertices, which give all that is needed! At the geometric level, monopole fluxes would replicate and break and join. Intriguingly, this is what would happen at the magnetic bodies of DNA and induce similar reactions at the level of DNA molecules! Maybe biology has been doing its best to tell us what the fundamental particle dynamics is!

2. Since only the induced electroweak gauge potentials couple to fermions, the question arises whether color and strong interactions are obtained. How is it possible to have strong interactions without parity violation when basic vertices involve weak parity violation? I have already discussed this question (see this).

Vertex for graviton emission

There is still one crucially important question left. Is it possible and what would happen in it? Can one obtain a vertex, where the analog for a contraction T^αβδ g_αβ of energy-momentum tensor with the deviation of the metric from the Minkowski metric appears?

1. In TGD all elementary particles, also gravitons, are identified as closed 2-sheeted monopole flux tubes with two wormhole contacts at its "ends" and opposite wormhole throats carrying fermions and antifermions (see this and this). For gravitation one has 1 fermion or antifermion for each wormhole throat.

   The graviton emission vertex should correspond to a splitting of flux tubes. Mopole flux tubes with fermion-antifermion pairs assignable to both wormhole contacts should appear. The fermion and antifermion should reside at the opposite throats of each wormhole contact. This should happen in the splitting of a monopole flux tube and second monopole flux tube would correspond to graviton. That two bosonic vertices are involved with the emission, brings to mind the proposal that gravitation is in some sense a square of gauge theory.

2. The vertex is the same as for gauge boson emission and for a creation of a fermion-antifermion pair. The definition of the modified gamma matrices as Γ^α= T^α_kΓ^k appearing in the modified Dirac action (see this), involving the modified Dirac operator Γ^μD_μ makes it possible to identify the gravitational part of the vertex. Here the quantities T^α_k=∂ L/∂(∂_αh^k) are canonical momentum currents associated with the action defining the space-time surface and also the analog of the energy-momentum tensor.

   Modified gamma matrices are required by hermiticity forcing the vanishing of the divergence of the vector Γ^α giving classical field equations for space-time surfaces. This implies a supersymmetry between the dynamics of fermions and 3-surfaces. The gravitational interaction would correspond to the deviation of the induced metric from the induced metric defined by induced CP₂ metric. CP₂ radius must correspond to Planck length l_P. This requires that the CP₂ as R≈ 10⁴l_P must correspond to h= nh₀, n≈ 10⁷ as found already earlier.

3. The cosmological term in GRT has coefficient 1/8π GΛ== 1/R⁴ so that the modified gamma matrices would contain a term proportional to 1/R⁴ plus a term coming from the Kähler action. In the TGD framework (see this and this) cosmological constant Λ depends on the p-adic length scale, which is assumed to correspond to a ramified prime for an extension of rationals associated with the polynomial P determining to high degree the space-time surface and approaches to zero in cosmic scales. The cosmological value corresponds to R≈ 10^-4 meters, i.e. cell length scale and a scale near neutrino Compton length.

   In the general coordinate invariant formalism, one does not assign dimension to the coordinates or to covariant derivative D_α. Metric has dimension 2. The scale dimension of T^α_kg^1/2 is the same dimension of Lg^1/2 and thus vanishes. Γ^α has scale dimension -1. The modified Dirac action must be dimensionless so that the induced spinors must have scale dimension 1/2.

4. The cosmological constant as the coefficient of the action depends on the p-adic length scale unlike. This term contributes to the string tension of string-like objects an additional term, which among other things can explain hadronic string tension. This term is visible also in the interaction vertices. The Kähler part of the bosonic action terms comes from the deviation of the induced metric from the flat metric and should give the usual gravitational interactions with matter.
5. Holomorphy hypothesis allows any general coordinate invariant action constructible in terms of the induced geometry. Although preferred extremals are always minimal surfaces, the properties of the action are visible via classical conservation laws, via the field equation at singular 3-surfaces involving the entire action, and via the vertices.

See the article About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe or the chapter About Platonization of Nuclear String Model and of Model of Atoms

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Gravitational hum and the precise form of M8-H duality

The precise form of M⁸-H duality (see this, this, and this) has remained open since one consider several variants for the duality in M⁴⊂ M⁸=M⁴× E⁴ degrees of freedom mapped to M⁴⊂ M⁴× CP₂ degrees of freedom.

The model for gravitational hum involves diffraction in the tessellation of H³ formed by stars or rather, by their magnetic bodies (see this). Reciprocal lattice is closely related to diffraction and equals to the lattice only in the case of cubic lattice. This is expected to be true also for tessellations and suggests that M8-H duality mapping momentum 4-surfaces of M⁸ to space-time surfaces of H maps the tessellation of H³⊂ M⁴⊂ M⁸ to a reciprocal tessellation of H³⊂ M⁴⊂ H.

The first problem is that the momenta at the M⁸ side are complex unlike the space-time points at the H side. The basic condition comes from the Uncertainty Principle in semiclassical form but does not complettessellationely fix the duality.

1. If the momenta are real, the simplest option is that the mass shell is mapped to a time shell a=h_eff/m, where a is light-cone proper time. For physical states the momenta are real by Galois confinement and have integer components when the momentum scale is defined by causal diamond (CD). For virtual fermions, the momenta are assumed to be algebraic integers and can be complex. The question is whether one should apply M⁸-H a duality only too the real momenta of physical states or also the virtual momenta.
2. For virtual momenta the M⁸-H duality must be consistent with that for real momenta and the simplest option is that one projects the real part of the virtual momentum and applies M⁸-H duality to it. The square m_R² for the real part Re(p) of momentum however varies for the points on the complex mass shell since only the real part Re(m²) of mass squared is constant at the complex mass shell. If 3-momenta are real, one has (Re(p))²= Re(p₀)² -p₃²=m_R² and is not constant and in general larger than Re(p²)=Re(p₀)²-Im(p₀)² -p₃²=Re(m²). Should one use m_R² or Re(m²) in M⁸-H duality?

   Re(m²) is constant at M⁸ side in accordance with the definition of mass shell. The value of a²= h_eff²(m_R²)/Re(m²)² at H side varies and has a width defined by the variation of Im(p₀) at the points of the mass shell. m_R² is not constant at the M⁸ side. This might relate to the fact that particle masses have a width and would relate Im(p₀ to a physical observable. The time shell is given by a²= h_eff²/m_R² and is genuine H³.

The model for the gravitational hum (see this) provides another problem, which can serve as an additional guideline.

1. The model was based on gravitational diffraction in the tessellation defined by a discrete subgroup of SL(2,C). This tessellation is a hyperbolic analog of a lattice in E³ with a discrete translation group replaced with a discrete subgroup Γ of the Lorentz group or its covering SL(2,C). The matrix elements of the matrices in Γ should belong to the extension of rationals defined by the polynomial P defining the space-time surface by M⁸-H duality.
2. For ordinary lattices, the reciprocal lattice assigns to a spatial lattice a momentum space lattice, which automatically satisfies the constraint from the Uncertainty Principle. Could the notion of the reciprocal lattice generalize to H³? What is needed are 3 basis vectors (at least) characterizing the position of a fundamental region and having components that must belong to the algebraic extension of rationals considered. The application of Γ would then produce the entire lattice. In this case a linear superposition of lattice vectors is not possible.
3. The (at least) 3 basic 3-vectors p_3,i need not be orthogonal or have the same length. They should have components, which are algebraic integers in the extension of rationals defined by P. M⁴⊃ H³ is a subspace of complexified quaternions with the space-like part of momentum vector, which is imaginary with respect to commuting imaginary unit i to transform the algebraic scalar product (no conjugation with respect to i). The ordinary cross product appearing in the definition of the reciprocal lattice appears in the quaternionic product. This suggests that the (at least) 3 reciprocal vectors p_3,i as M⁴ projections of four-momentum vectors are proportional to the cross products of the basis vectors apart from a normalization factor determined by the condition that the light-cone proper time is proportional to the inverse of mass. One would have x₃ⁱ∝ ε^ijkp_3,j× p_3,k.

   Physical intuition suggests that the components of spatial momentum are real for the basis vectors p_3,i so that only the energy has imaginary part. For their discrete Lorentz books by Γ this cannot be the case in M⁸_c.

4. Mass shell condition p_i²= M² must be replaced with x²_i= h_eff/M². The precise identification of M² will be considered below. The image of the real part of the energy p_0,i is the time coordinate t⁰_i= h_effRe(p_0,i)/M². For the naive option considered earlier, the time shell condition is satisfied if the dual position vectors x₃ⁱ are of form x₃ⁱ =h_eff [(p²_i)^1/2/M²]eⁱ, where eⁱ is a unit vector in the direction of p_i. This option is correct if the momenta p_i are orthogonal since in this case the reciprocal unit vectors and vectors co-insider. In the general case, one must replace the unit vector eⁱ with the unit vector associated with the vector Xⁱ= ε^ijkp_3,j p_3,j given by eⁱ=Xⁱ/((Xⁱ)²)^1/2 so that the formula for x³_i remains otherwise the same.
5. The conditions have a similar form independently of whether one takes the mass squared parameter M² to be M²=Re(m²) or M²=m_R². The time components of the momentum vectors p_i associated with p_3,i, which are assumed to be real, are determined by the mass shell condition Re(p_0,i)²-Im(p_0,i)²- Re(p_i²) =Re(m²). Spatial coordinates in M⁴ must obey similar formula, which implies the length of the image vector is r= h_eff×p_3,i/M² so that time shell condition t²-r²= h_eff²/M² conforms with the Uncertainty Principle.
6. Which option is correct: M²=Re(m²) or M²=m_R²? For the first option the discretized real mass shell m_R² is deformed and might be essential for having a non-trivial number theoretical holography implying by M⁸-H duality a non-trivial holography at H side. One can however defend the second option by non-trivial holography at H side.

Note that the proposed definition of M⁸-H duality is indirect in that it is applied only to the (at least) 3 basic vectors and the action of Γ gives the tessellation in H³⊂ H. One can also apply the entire Lorentz group to these vectors to obtain the time shell.

The proposed construction assumed that the basis of 3 vectors is essential for the definition of tessellation and that it is possible to assign a set of reciprocal vectors to it in the proposed way involving cross product, which is essentially 3-D notion and relates to quaternions. Is this really the case for all tessellations of H³?

1. In Euclidian 3-space E³ only cubic lattice defines a regular tessellation and for the reciprocal lattices is well-defined. In this case, the linear combinations of 3 basic vectors define the lattice. Note that Platonic solids have duals but this duality has nothing to do with the reciprocal lattice.
2. In hyperbolic 3-space H³ one can have cubic tessellation, 2 icosahedral tessellations, and dodecahedral tessellation as regular tessellations. There is also icosa tetrahedral tessellation involving both tetrahedra, octahedra and icosahedra (see this). Linear combinations of the basic vectors do not exist now. If 3 basic vectors of the tessellation are known, it would be their orbit under the discrete group Γ which defines the tessellation. If Γ is transitive, a single point in principle defines the tessellation as the orbit of Γ.
3. One can assign to tessellations of H³ what might be called a fundamental tetrahedron as a 4-simplex, which is not a regular tetrahedron in the general case. Could the loci of its vertices with respect to a selected vertex define the fundamental tetrahedron? For a cube in E³ this tetrahedron would correspond to a tetrahedron defined by the 3 nearest vertices of a selected vertex of the cube. At least in E³ one can select the vertex by requiring that the distances of the neighbouring vertices from the selected vertex are minimal. In this case, the remaining 3 vertices form an orbit under Z³ .
4. Could one reduce the situation from H³ to E³ by considering the 3 basic vectors for the projection of the fundamental tetrahedron from H³ to t=constant hyperplane E³?The 3 basic vectors would be from the selected vertex defining the origin to neighboring vertices: note that here the submanifold property H³⊂ M⁴ is essential.

   Could one assign to this triplet a reciprocal in the same way as in the Euclidian case using the cross product induced by the quaternion structure? The notion of dual basis is also behind the bra-ket formalism of quantum mechanics and is based on the notion of vector space and its dual? The following considerations rely on this optimistic assumption.

See the article The TGD view of the recently discovered gravitational hum as gravitational diffraction, the articles Magnetic Bubbles in TGD Universe: Part I and Magnetic Bubbles in TGD Universe: Part II or the chapter Quantum Astrophysics .

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.