Wednesday, September 22, 2021

Breakthrough 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 this). 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.

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.

The findings of the group led by Bush and describe in his Youtube lecture (see this) 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.

The article of Bush et al (see this) describes the findings about the analog of quantum corral. The latter involves electrons inside a circular corral defined by negative ions.

" 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.

The key questions are following.

  1. 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.
  2. 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 heff=nh0 labelling phases of ordinary matter located at magnetic body (MB). MB has a hierarchical structure and defines a master slave hierarchy.

    A given level of the hierarchy controls the physics at the lower levels. heff 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 heff requires however the analogy of metabolic energy feed quite generally.

    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 fF= f/2. The Faraday frequency has slow time variation with the frequency f and slightly below fF.

    The system system should be near criticality for the generation of heff 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.

  3. What could the value of heff be? The Faraday wave length λF= 2\pi\sqrt{2ν/μ should be equal to the analog of Compton wavelength λc =ℏeff/m, m the mass of the water droplet. λF 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.

    The only possibility is that one has ℏeff= ℏgr = GMm/v0, where ℏgr is the gravitational Planck constant introduced by Nottale and also appearing in the TGD based model of superconductivity. This would give λF= λ>gr= GM/v0= rs(M)/2v0, where rs(M) is Schartschild radius. M is naturally the mass of Earth. The minimum value of λgr corresponds to v0/c=1 and is λgr= rs/2. Earth's Scwartschild radius is 8.7 mm so that one would have λF= 4.35 mm.

    The value of λF 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 β0=v0/c\simeq .92!

    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 β0. One could measure λF=2\pi (2ν/μ)1/2 for different liquids to see whether v0 codes for the properties of the liquid or whether λF is independent of the liquid so that the classical model for Faraday waves could be wrong.

  4. 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.

    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.

  5. 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 heff=nh0 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 ℏgr enters into the physics of the moving water droplet.
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).

See the article TGD and condensed matter.

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

Articles and other material related to TGD. 

Saturday, September 11, 2021

TGD 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.
  1. 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.

    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.

  2. The understanding of the details of the M8-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 M8 is a region bounded by two mass shells rather than CD. M8-H duality maps the points of cognitive representations as momenta of quarks with fixed mass in M8 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= heff/m, m the mass associated with momentum.
  3. Galois confinement at the level of M8 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.
  4. 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.
See the the article TGD as it is towards end of 2021 or the chapter with the same title.

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

Articles and other material related to TGD.

Wednesday, September 08, 2021

TGD 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.

It is perhaps good to explain what TGD is not and what it is or hoped to be.

  1. "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.

    Standard model symmetries uniquely fix the choice of 8-D space in which space-time surfaces live to H=M4× CP2. 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.

  2. 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=M4× CP2.

    The space-time as 4-surface X4 ⊂ 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.

    Practically any GCI action has the same universal basic extremals: CP2 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 CP2 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.

    "Topological" could refer also to p-adic number fields obeying p-adic local topology differing radically from the real topology.

  3. 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 heff=nh0, 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.

    The concrete realization of the number theoretic vision is based on M8-H duality. The physics in the complexification of M8 is algebraic - field equations as partial differential equations are replaced with algebraic equations associating to a polynomial with rational coefficients a X4 mapped to H by M8-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.

    M8-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).

  4. 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.

    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.

    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).

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.
  1. 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.

    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.

  2. The understanding of the details of the M8-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 M8 is a region bounded by two mass shells rather than CD. M8-H duality maps the points of cognitive representations as momenta of quarks with fixed mass in M8 to either boundary of CD in H.
  3. Galois confinement at the level of M8 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.
  4. 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.
The improved vision allows us to develop the TGD interpretation for various condensed matter notions.
  1. 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.
  2. 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 CP2 topology.
  3. 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. M8 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 heff. What is new is that energy feed makes possible supra-phases and their analogs above the critical temperature.
  4. Fermi surface emerges as a fundamental notion at the level of M8 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 M8 defined by algebraic integers in an extension of rationals. Also the TGD analogs of condensed matter Majorana fermions emerge.
See the article TGD and Condensed Matter or a chapter with the same title.

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

Articles and other material related to TGD.

Wednesday, August 18, 2021

Could 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.

1. Background and motivations for the idea

M8-H-duality (H= M4× CP2) 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 this, this, and this).

The octonionic polynomials are obtained by algebraic continuation of ordinary real polynomials with rational coefficients although one can also consider algebraic coefficients.

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 this). This leads to an action consisting of Kähler action with both CP2 and M4 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 this).

Irreducible polynomials of rational coefficients give rise to algebraic extensions characterized by the Galois group and these notions are central in adelic vision (see this).

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 ∏p(ex-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.

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.

  1. For polynomials as roots of octonionic polynomials space-time surfaces are finite and located inside finite-sized causal diamond (CD).

    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 M4 and CP2 parts and having ground state degeneracy analogous to spin glass degeneracy would be present.

  2. 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.
  3. 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.

    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.

    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.

    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.

2. Intuitive view about the situation

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?

  1. Suppose that the analytic prime decomposes to a product over monomials x-xi with transcendental roots xi such that the Taylor series has rational coefficients. This requires an infinite Taylor series.
  2. One obtains an infinite number of conditions. Each power xn in f has a rational coefficient fn equal to the sum over all possible products ∏k=1n xik of n transcendental roots xik. 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.
  3. The sums of products are generalized symmetric functions of transcendental roots and should have rational values equal to xn. This generalizes the corresponding condition for ordinary polynomials. Symmetric functions for Sn have Sn as a group of symmetries. For a Galois extension of a polynomial of order n, the Galois group is a subgroup of Sn. This suggests that the Galois group is a subgroup of S. S has the simple A 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 this).
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.
  1. 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 fn for the function ∑ fn xn defined by the infinite product?
  2. fn 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.

3. Profinite groups and Galois extensions as inverse limits

Infinite groups indeed appear as Galois groups of infinite extensions. Absolute Galois groups, say Galois groups of algebraic numbers, provide the basic example.

  1. There exists a natural topology, known as Krull topology, which turns Galois group to a profinite group (totally disconnected, Hausdorff topological group) (see this), which is also Stone space (see this).
  2. Profinite groups are not countably infinite but are effectively finite just as hyper-finite factors of type II1 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.
  3. 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 ..Fn→ Fn+1→ .
  4. This kind of iterated extensions are of special interest in the TGD framework and an infinite extension would be obtained at the limit (see this). The naive expectation is that the polynomial of infinite degree is a limit of a composite ...Pn○ Pn-1..○ P1 of rational polynomials. The number of infinite extensions obtained in this manner would be infinite.

    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.

  5. The identification as the iterate ...Pn○ Pn-1○ P1 is indeed suggestive. Infinite cyclic extension defined at the limit by the polynomial xN, N=∞, to be discussed below, has this kind of interpretation. The Galois group of this kind of extension is however not simple.

    Remark: The polynomials in question satisfying P(0)=0 are not irreducible: the composite of N polynomials has xN as a factor and has 0 as N-fold root. The origin of octonionic M8 appears as an isolated root.

  6. Is the infinite-D extension obtained as an inverse limit transcendental or algebraic? In the TGD framework the condition that the polynomial P1○ P2 has the roots of P1 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 Pi(0)=0 also leads to a loss of root inheritance.
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 this). 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.

4. Could infinite extensions of rationals with a simple Galois group exist?

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?

A is simple and could not be seen as an inverse limit. Also the groups PSLn(K) are infinite discrete groups for K=Z or Q. A further example is provided by Tarski monster groups (see this) having only cyclic groups Zp for a fixed p as subgroups and existing or p>1075. For these primes, there is a continuum of these monsters.

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.

In fact, there is also a theorem stating that every Galois group is profinite (see this). 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.

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.

Two examples

In the following two examples are consider to test whether the notion of a polynomial of infinite order might work.

5.1 Cyclic extension as an example

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.

Consider as an example cyclic extension defined by a polynomial XN, which can be regarded as a composite of polynomials xpi for ∏ pi=N. This is perhaps the simplest possible extension than one can imagine.

  1. The roots are now powers of roots of unity. The notion of the root of unity as ei2π/N does not make sense at the limit N→ ∞. One can however consider the roots ei2π M/N and its powers such that the limit M/N → α is irrational. The powers of ei nα give a dense subset of the circle S1 consisting of irrational points. Note that one obtains an infinite number of extensions labelled by irrational values of α.
  2. The polynomial should correspond to the limit PN(x)=xN-1, N→ ∞. For each finite value of N, one has PN(x)= ∏n=1N(x-Un)-1, U= ei2π/N . The reduction to P=xN-1 follows from the vanishing of all terms involving lower powers of x than xN.
  3. If these conditions hold true at the limit N→ ∞, one obtains the same result. The coefficient of xN equals to 1 trivially. The coefficient of xN-1 is the sum over all roots and should vanish. This is also assumed in Fourier analysis ∑n eiα n=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 f1.
  4. The Galois group is in this case the cyclic group U∞,α defined by the powers of Uα.
5.2 Infinite iteration yield contimuum or roots

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 LimN→ ∞ (1+1/N)N=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 M8-H duality in (see this).

Consider polynomials PN= QN○ R, where R with Q(0)=0 is fixed polynomial and QN=Q○N 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 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.

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.

In the TGD framework these roots correspond to values of M4 time (or actually energy!) in M8 mapped to actual time values in H by M8-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 this, this). One might perhaps say that at this limit subjective time consisting of these moments becomes continuous in the interval [0,T].

See the article Does the notion of polynomial of infinite order make sense? or the chapter About the role of Galois groups in TGD framework.

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

Articles and other material related to TGD.

Friday, August 13, 2021

Empirical 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.

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.

See the article Empirical support for the Expanding Earth Model and TGD view about classical gauge fields.

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

Articles and other material related to TGD. 

Thursday, August 12, 2021

Expanding Venus model

News about unexpected findings relating to the physics of astrophysical objects emerge on an almost daily basis. The most recent news relates to Venus.

Has Venus turned itself inside-out?

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.

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.

  1. The TGD explanation for Cambrian Explosion relies on Expanding Earth model (see this, this, this, and this).

    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.

  2. 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.

  3. There are many objections to be circumvented. For instance, how photosynthesis could evolve in the underground ocean. Here TGD views dark matter as heff=nh0 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 this).
  4. 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.

Why does Venus not possess a magnetic field?

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 this). The solar dynamo mechanism would suggest its presence.

  1. TGD predicts that there are two kinds of flux tubes carrying Earth's magnetic field BE 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 M4× CP2. 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 BE.
  2. 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.

    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 (this). The standard dynamo model is part of this model.

  3. 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!

    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?

    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.

  4. 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.

    This could provide a totally new insight to the Meissner effect and loss of superconductivity. In TGD the based model (see this), 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!

  5. 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.

    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 heff 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 heff.

    Could the superconductivity be forced by the thermal energy feed from the interior of Venus? Superconductivity means in the TGD framework large heff and therefore complexity, intelligence, and long quantum coherence length (see this). 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.

    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 here.

    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.

See the article Updated version of Expanding Earth model or the chapter Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life.

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

Articles and other material related to TGD.

Tuesday, August 10, 2021

The conjectured duality between number theory and geometry from TGD point of view

There was a Quanta Magazine article about the link between geometry and numbers (see this). In the following I consider this idea from the TGD view point.

1. M8-M4× CP2 duality

What makes the proposed connection so interesting from the TGD point of view is that in TGD M8-M4× CP2 duality (see for instance this, this and this) 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.

  1. M8 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.
  2. H= M4× CP2 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 (this).

    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 (this).

    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 (this).

It would be very nice if this duality between number theory and geometry would be present at the level of mathematics itself.

2. Adelic physics as unified description of sensory experience and cognition

Adelic physics involves both real and p-adic number fields (see for instance this). 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.

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.

3. Langlands corresponds and TGD

Also Langlands correspondence, which I have tried to understand several times with my tiny physicist's brain, is involved.

  1. Global Langlands correspondence (GLC) states that there is a connection between representations of continuous groups and Galois groups of extensions of rationals.
  2. Local LC states (LLC) states this in the case of p-adics.
There is a nice interpretation for the two LCs in terms of sensory experience and cognition in the TGD inspired theory of consciousness.
  1. 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.
  2. Sensory world, "real" world corresponds to representation of continuous groups/Galois groups of rationals: this would be GLC.
  3. "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.
  4. 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.

4. Galois groups, physics and cognition

TGD provides a geometrization for the action of Galois groups (see this and this).

  1. 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.
  2. The order n of the Galois group over rationals corresponds to an effective Planck constant heff= nh0 so that there is a direct connection to a generalization of quantum physics (see for instance this). The phases of ordinary matter with heff=nh0 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.

    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.

    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.

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

    Articles and other material related to TGD.