What's new in

TGD: Physics as Infinite-Dimensional Geometry

Note: Newest contributions are at the top!

Year 2016

Generalization of Riemann zeta to Dedekind zeta and adelic physics?

A further insight to adelic physics comes from the possible physical interpretation of the L-functions appearing also in Langlands program (see this. The most important L-function would be generalization of Riemann zeta to extension of rationals. I have proposed several roles for ζ, which would be the simplest L-function assignable to rational primes, and for its zeros.

  1. Riemann zeta itself could be identifiable as an analog of partition function for a system with energies given by logarithms of prime. In ZEO this function could be regarded as complex square root of thermodynamical partition function in accordance with the interpretation of quantum theory as complex square root of thermodynamics.
  2. The zeros of zeta could define the conformal weights for the generators of super-symplectic algebra so that the number of generators would be infinite. The rough idea - certainly not correct as such except at the limit of infinitely large CD - is that corresponding functions would correspond to functions of radial light-like coordinate rM of light-cone boundary (boundary of causal diamond) of form (rM/r0)sn, sn=1/2+iy, sn would be radial conformal weight. Periodic boundary conditions for CD do not allow all possible zeros as conformal weights so that for given CD only finite subset corresponds to generators of supersymplectic algebra. Conformal confinement would hold true in the sense that the sum sn for physical states would be integer. Roots and their conjugates should appear as pairs in physical states.
  3. On basis of numerical evidence Dyson (see this) has conjectured that the Fourier transform for the set formed by zeros of zeta consists of primes so that one could regard zeros as one-dimensional quasi-crystal. This hypothesis makes sense if the zeros of zeta decompose into disjoint sets such that each set corresponds to its own prime (and its powers) and one has piy= Um/n=exp(i2π m/n) (see the appendix of this). This hypothesis is motivated by number theoretical universality.
  4. I have considered the possibility (see this) that the inverse of the electro-weak U(1) coupling constant for a gauge field assignable to the Kähler form of CP2 corresponds to poles of the fermionic zeta ζF(s)= ζ(s)/ζ(2s) coming from sn/2 (denominator) and pole at s=1 (numerator) zeros of zeta assignable to rational primes. Here one can consider scaling of argument of ζF(s). More general coupling constant evolutions could correspond to ζF(m(s)) where m(s)= (as+b)/(cs+d) is Möbius transformation performed for the argument mapping upper complex plane to itself so that a,b,c,d are real and also rational by number theoretical universality.
Suppose for a moment that more precise formulations of these physics inspired conjectures hold true and even that their generalization for extensions K/Q of rationals holds true. This would solve quite a portion of adelic physics! Not surprisingly, the generalization of zeta function was proposed already by Dedekind (see this).
  1. The definition of Dedekind zeta function ζK relies on the product representation and analytic continuation allows to deduce ζK elsewhere. One has a product over prime ideals of K/Q of rationals with the factors 1/(1-p-s) associated with the ordinary primes in Riemann zeta replaced with the factors X(P) =1/(1-NK/Q(P)-s), where P is prime for the integers O(K) of extension and NK/Q(P) is the norm of P in the extension. In the region s>1 where the product converges, ζK is non-vanishing and s=1 is a pole of ζK. The functional identifies of ζ hold true for ζK as well. Riemann hypothesis is generalized for ζK.
  2. It is possible to interpret ζK in terms of a physical picture. By the general results (see this) one NK/Q(P)= pr, r>0 integer. One can deduce for r a general expression. This implies that one can arrange in ζK all primes P for which the norm is power or given p in the same group. The prime ideals p of ordinary integers decompose to products of prime ideals P of the extension: one has p= ∏r=1g Prer, where er is so called ramification index. One can say that each factor of ζ decomposes to a product of factors associated with corresponding primes P with norm a power of p. In the language of physics, the particle state represented by p decomposes in an improved resolution to a product of many-particle states consisting of er particles in states Pr, very much like hadron decomposes to quarks.

    The norms of NK/Q(Pr) = pdr satisfy the condition ∑r=1g dr er= n. Mathematician would say that the prime ideals of Q modulo p decompose in n-dimensional extension K to products of prime power ideals Prer and that Pr corresponds to a finite field G(p,dr) with algebraic dimension dr. The formula ∑r=1g dr er = n reflects the fact the dimension n of extension is same independent of p even when one has g<n and ramification occurs.

    Physicist would say that the number of degrees of freedom is n and is preserved although one has only g<n different particle types with er particles having dr internal degrees of freedom. The factor replacing 1/(1-p-s) for the general prime p is given by ∏r=1g 1/(1-p-erdrs).

  3. There are only finite number of ramified primes p having er>1 for some r and they correspond to primes dividing the so called discriminant D of the irreducible polynomial P defining the extension. D mod p obviously vanishes if D is divisible by p. For second order polynomials P=x2+bx+c equals to the familiar D=b2-4c and in this case the two roots indeed co-incide. For quadratic extensions with D= b2-4c>0 the ramified primes divide D.

    Remark: Resultant R(P,Q) and discriminant D(P)= R(P,dP/dx) are elegant tools used by number theorists to study extensions of rationals defined by irreducible polynomials. From Wikipedia articles one finds an elegant proof for the facts that R(P,Q) is proportional to the product of differences of the roots of P and Q, and D to the product of squares for the differences of distinct roots. R(P,Q)=0 tells that two polynomials have a common root. D mod p=0 tells that polynomial and its derivative have a common root so that there is a degenerate root modulo p and the prime is indeed ramified. For modulo p reduction of P the vanishing of D(P) mod p follows if D is divisible by p. There exist clearly only a finite number of primes of this kind.

    Most primes are unramified. If one has maximum number n of factors in the decomposition and er=1, maximum splitting of p occurs. The factor 1/(1-p-s) is replaced with its n:th power 1/(1-p-s)n. The geometric interpretation is that space-time sheet is replaced with n-fold covering and each sheet gives one factor in the power. It is also possible to have a situation in which no splitting occurs and one as er=1 for one prime Pr=p. The factor is in this case equal to 1/(1-p-ns).

From Wikipedia one learns that for Galois extensions L/K the ratio ζLK is so called Artin L-function of the regular representation (group algebra) of Galois group factorizing in terms of irreps of Gal(L/K) is holomorphic (no poles!) so that ζL must have also the zeros of ζK. This holds in the special case K=Q. Therefore extension of rationals can only bring new zeros but no new poles!
  1. This result is quite far reaching if one accepts the hypothesis about super-symplectic conformal weights as zeros of ζK and the conjecture about coupling constant evolution. In the case of ζF,K this means new poles meaning new conformal weights due to increased complexity and a modification of the conjecture for the coupling constant evolution due to new primes in extension. The outcome looks physically sensible.
  2. Quadratic field Q(m1/2) serves as example. Quite generally, the factorization of rational primes to the primes of extension corresponds to the factorization of the minimal polynomial for the generating element θ for the integers of extension and one has p= Piei, where ei is ramification index. The norm of p factorizes to the produce of norms of Piei.

    Rational prime can either remain prime in which case x2-m does not factorize mod p, split when x2-m factorizes mod P, or ramify when it divides the discriminant of x2-m = 4m. From this it is clear that for unramfied primes the factors in ζ are replaced by either 1/(1-p-s)2 or 1/(1-p-2s)= 1/(1-p-s)(1+p-s). For a finite number of unramified primes one can have something different.

    For Gaussian primes with m=-1 one has er=1 for p mod 4=3 and er=2 for p=~mod~4=1. zK therefore decomposes into two factors corresponding to primes p ~mod~4=3 and p mod 4=1. One can extract out Riemann zeta and the remaining factor

    p mod 4=3 1/(1-p-s) × ∏p mod 4=1 1/(1+p-s)

    should be holomorphic and without poles but having possibly additional zeros at critical line. That ζK should possess also the poles of ζ as poles looks therefore highly non-trivial.

See the article p-Adization and adelic physics or the chapter Unified Number Theoretical Vision.

Are Preferred Extremals Quaternion-Analytic in Some Sense?

A generalization of 2-D conformal invariance to its 4-D variant is strongly suggestive in TGD framework, and leads to the idea that for preferred extremals of action space-time regions have (co-)associative/(co-)quaternionic tangent space or normal space. The notion of M8-H correspondence allows to formulate this idea more precisely. The beauty of this notion is that it does not depend on the signature of Minkowski space M4 representable as sub-space of of complexified quaternions M4c, which in turn can be seen as sub-space of complexified octonions M8c.

The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. This notion is however not so straightforward even in Euclidian signature, and the generalization to Minkowskian signature brings in further problems. The Cauchy-Riemann-Fuerter conditions make however sense also in Minkowskian quaternionic situation and the problem is whether they allow the physically expected solutions. One should also show that the possible generalization is consistent with (co)-associativity.

In this article these problems are considered. Also a comparison with Igor Frenkel's ideas about hierarchy of Lie algebras, loop, algebras and double look algebras and their quantum variants is made: it seems that TGD as a generalization of string models replacing string world sheets with space-time surfaces gives rise to the analogs of double loop algebras and they quantum variants and Yangians. The straightforward generalization of double loop algebras seems to make sense only at the light-like boundaries of causal diamonds and at light-like orbits of partonic 2-surfaces but that in the interior of space-time surface the simple form of the conformal generators is not preserved. The twistor lift of TGD in turn corresponds nicely to the heuristic proposal of Frenkel for the realization of double loop algebras.

See the article Are Preferred Extremals Quaternion-Analytic in Some Sense? or the chapter Unified Number Theoretical Vision.

Why would primes near powers of two (or small primes) be important?

The earlier What's New was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.

Also new insights about how preferred p-adic primes identified as ramified primes of extension emerge. The picture suggests strong resemblance with the evolution of genetic code with conserved genes having ramified primes as their analogs. Category theoretic thinking in turn suggests that the positions of fermions at partonic 2-surfaces correspond to singularities of the Galois covering so that the number of sheets of covering is not maximal and that the singularities has as their analogs what happens for ramified primes.

p-Adic length scale hypothesis states that physically preferred p-adic primes come as primes near prime powers of two and possibly also other small primes. Does this have some analog to complexity theory, period doubling, and with the super-stability associated with period doublings?

Also ramified primes characterize the extension of rationals and would define naturally preferred primes for a given extension.

  1. Any rational prime p can be decomposes to a product of powers Pki of primes Pi of extension given by p= ∏i Piki, ∑ ki=n. If one has ki≠ 1 for some i, one has ramified prime. Prime p is Galois invariant but ramified prime decomposes to lower-dimensional orbits of Galois group formed by a subset of Piki with the same index ki . One might say that ramified primes are more structured and informative than un-ramified ones. This could mean also representative capacity.
  2. Ramification has as its analog criticality leading to the degenerate roots of a polynomial or the lowering of the rank of the matrix defined by the second derivatives of potential function depending on parameters. The graph of potential function in the space defined by its arguments and parameters if n-sheeted singular covering of this space since the potential has several extrema for given parameters. At boundaries of the n-sheeted structure some sheets degenerate and the dimension is reduced locally . Cusp catastrophe with 3-sheets in catastrophe region is standard example about this.

    Ramification also brings in mind super-stability of n-cycle for the iteration of functions meaning that the derivative of n:th iterate f(f(...)(x)== fn)(x) vanishes. Superstability occurs for the iteration of function f= ax+bx2 for a=0.

  3. I have considered the possibility that that the n-sheeted coverings of the space-time surface are singular in that the sheet co-incide at the ends of space-time surface or at some partonic 2-surfaces. One can also consider the possibility that only some sheets or partonic 2-surfaces co-incide.

    The extreme option is that the singularities occur only at the points representing fermions at partonic 2-surfaces. Fermions could in this case correspond to different ramified primes. The graph of w=z1/2 defining 2-fold covering of complex plane with singularity at origin gives an idea about what would be involved. This option looks the most attractive one and conforms with the idea that singularities of the coverings in general correspond to isolated points. It also conforms with the hypothesis that fermions are labelled by p-adic primes and the connection between ramifications and Galois singularities could justify this hypothesis.

  4. Category theorists love structural similarities and might ask whether there might be a morphism mapping these singularities of the space-time surfaces as Galois coverings to the ramified primes so that sheets would correspond to primes of extension appearing in the decomposition of prime to primes of extension.

    Could the singularities of the covering correspond to the ramification of primes of extension? Could this degeneracy for given extension be coded by a ramified prime? Could quantum criticality of TGD favour ramified primes and singularities at the locations of fermions at partonic 2-surfaces?

    Could the fundamental fermions at the partonic surfaces be quite generally localize at the singularities of the covering space serving as markings for them? This also conforms with the assumption that fermions with standard value of Planck constants corresponds to 2-sheeted coverings.

  5. What could the ramification for a point of cognitive representation mean algebraically? The covering orbit of point is obtained as orbit of Galois group. For maximal singularity the Galois orbit reduces to single point so that the point is rational. Maximally ramified fermions would be located at rational points of extension. For non-maximal ramifications the number of sheets would be reduced but there would be several of them and one can ask whether only maximally ramified primes are realized. Could this relate at the deeper level to the fact that only rational numbers can be represented in computers exactly.
  6. Can one imagine a physical correlate for the singular points of the space-time sheets at the ends of the space-time surface? Quantum criticality as analogy of criticality associated with super-stable cycles in chaos theory could be in question. Could the fusion of the space-time sheets correspond to a phenomenon analogous to Bose-Einstein condensation? Most naturally the condensate would correspond to a fractionization of fermion number allowing to put n fermions to point with same M4 projection? The largest condensate would correspond to a maximal ramification p= Pin.
Why ramified primes would tend to be primes near powers of two or of small prime? The attempt to answer this question forces to ask what it means to be a survivor in number theoretical evolution. One can imagine two kinds of explanations.
  1. Some extensions are winners in the number theoretic evolution, and also the ramified primes assignable to them. These extensions would be especially stable against further evolution representing analogs of evolutionary fossils. As proposed earlier, they could also allow exceptionally large cognitive representations that is large number of points of real space-time surface in extension.
  2. Certain primes as ramified primes are winners in the sense the further extensions conserve the property of being ramified.
    1. The first possibility is that further evolution could preserve these ramified primes and only add new ramified primes. The preferred primes would be like genes, which are conserved during biological evolution. What kind of extensions of existing extension preserve the already existing ramified primes. One could naively think that extension of an extension replaces Pi in the extension for Pi= Qikki so that the ramified primes would remain ramified primes.
    2. Surviving ramified primes could be associated with a exceptionally large number of extensions and thus with their Galois groups. In other words, some primes would have strong tendency to ramify. They would be at criticality with respect to ramification. They would be critical in the sense that multiple roots appear.

      Can one find any support for this purely TGD inspired conjecture from literature? I am not a number theorist so that I can only go to web and search and try to understand what I found. Web search led to a thesis (see this) studying Galois group with prescribed ramified primes.

      The thesis contained the statement that not every finite group can appear as Galois group with prescribed ramification. The second statement was that as the number and size of ramified primes increases more Galois groups are possible for given pre-determined ramified primes. This would conform with the conjecture. The number and size of ramified primes would be a measure for complexity of the system, and both would increase with the size of the system.

    3. Of course, both mechanisms could be involved.
Why ramified primes near powers of 2 would be winners? Do they correspond to ramified primes associated with especially many extension and are they conserved in evolution by subsequent extensions of Galois group. But why? This brings in mind the fact that n=2k-cycles becomes super-stable and thus critical at certain critical value of the control parameter. Note also that ramified primes are analogous to prime cycles in iteration. Analogy with the evolution of genome is also strongly suggestive.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.

heff/h=n hypothesis and Galois group

The previous What's New was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.

I ended up to rather interesting information theoretic interpretation about the understanding of effective Planck constant assigned to flux tubes mediating as gravitational/electromagnetic/etc... interactions. The real surprise was that this leads to a proposal how mono-cellulars and multicellulars differ! The emergence of multicellulars would have meant emergence of systems with mass larger than critical mass making possible gravitational quantum coherence. Penrose's vision about the role of gravitation would be correct although Orch-OR as such has little to do with reality!

The natural hypothesis is that heff/h=n equals to the order of Galois group in the case that it gives the number of sheets of the covering assignable to the space-time surfaces. The stronger hypothesis is that heff/h=n is associated with flux tubes and is proportional to the quantum numbers associated with the ends.

  1. The basic idea is that Mother Nature is theoretician friendly. As perturbation theory breaks down, the interaction strength expressible as a product of appropriate charges divided by Planck constant, is reduced in the phase transition hbar→ hbareff.
  2. In the case of gravitation GMm→ = GMm (h/heff). Equivalence Principle is satisfied if one has hbareff=hbargr = GMm/v0, where v0 is parameter with dimensions of velocity and of the order of some rotation velocity associated with the system. If the masses move with relativistic velocities the interaction strength is proportional to the inner product of four-momenta and therefore to Lorentz boost factors for energies in the rest system of the entire system. In this case one must assume quantization of energies to satisfy the constraint or a compensating reduction of v0. Interactions strength becomes equal to β0= v0/c having no dependence on the masses: this brings in mind the universality associated with quantum criticality.
  3. The hypothesis applies to all interactions. For electromagnetism one would have the replacements Z1Z2α→ Z1Z2α (h/ hem) and hbarem=Z1Z2α/&beta0 giving Universal interaction strength. In the case of color interactions the phase transition would lead to the emergence of hadron and it could be that inside hadrons the valence quark have heff/h=n>1. In this case one could consider a generalization in which the product of masses is replaced with the inner product of four-momenta. In this case quantization of energy at either or both ends is required. For astrophysical energies one would have effective energy continuum.
This hypothesis suggests the interpretation of heff/h=n as either the dimension of the extension or the order of its Galois group. If the extensions have dimensions n1 and n2, then the composite system would be n2-dimensional extension of n1-dimensional extension and have dimension n1× n2. This could be also true for the orders of Galois groups. This would be the case if Galois group of the entire system is free group generated by the G1 and G2. One just takes all products of elements of G1 and G2 and assumes that they commute to get G1× G2. Consider gravitation as example.
  1. The order of Galois group should coincide with hbareff/hbar=n= hbargr/hbar= GMm/v0hbar. The transition occurs only if the value of hbargr/hbar is larger than one. One can say that the order of Galois group is proportional the product of masses using as unit Planck mass. Rather large extensions are involved and the number of sheets in the Galois covering is huge.

    Note that it is difficult to say how larger Planck constants are actually involved since by gravitational binding the classical gravitational forces are additive and by Equivalence principle same potential is obtained as sum of potentials for splitting of masses into pieces. Also the gravitational Compton length λgr= GM/v0 for m does not depend on m at all so that all particles have same λgr= GM/v0 irrespective of mass (note that v0 is expressed using units with c=1).

    The maximally incoherent situation would correspond to ordinary Planck constant and the usual view about gravitational interaction between particles. The extreme quantum coherence would mean that both M and m behave as single quantum unit. In many-sheeted space-time this could be understood in terms of a picture based on flux tubes. The interpretation for the degree of coherence is in terms of flux tube connections mediating gravitational flux.

  2. hgr/h would be order of Galois group, and there is a temptation to associated with the product of masses the product n=n1n2 of the orders ni of Galois groups associated masses M and m. The order of Galois group for both masses would have as unit mP01/2, β0=v0/c, rather than Planck mass mP. For instance, the reduction of the Galois group of entire system to a product of Galois groups of parts would occur if Galois groups for M and m are cyclic groups with orders with have no common prime factors but not generally.

    The problem is that the order of the Galois group associated with m would be smaller than 1 for masses m<mP01/2. Planck mass is about 1.3 × 1019 proton masses and corresponds to a blob of water with size scale 10-4 meters - size scale of a large neuron so that only above these scale gravitational quantum coherence would be possible. For v0<1 it would seem that even in the case of large neurons one must have more than one neurons. Maybe pyramidal neurons could satisfy the mass constraint and would represent higher level of conscious as compared to other neurons and cells. The giant neurons discovered by the group led by Christof Koch in the brain of of mouse having axonal connections distributed over the entire brain might fulfil the constraint (see this).

  3. It is difficult to avoid the idea that macroscopic quantum gravitational coherence for multicellular objects with mass at least that for the largest neurons could be involved with biology. Multicellular systems can have mass above this threshold for some critical cell number. This might explain the dramatic evolutionary step distinguishing between prokaryotes (mono-cellulars consisting of Archaea and bacteria including also cellular organelles and cells with sub-critical size) and eukaryotes (multi-cellulars).
  4. I have proposed an explanation of the fountain effect appearing in super-fluidity and apparently defying the law of gravity. In this case m was assumed to be the mass of 4He atom in case of super-fluidity to explain fountain effect. The above arguments however allow to ask whether anything changes if one allows the blobs of superfluid to have masses coming as a multiple of mP01/2. One could check whether fountain effect is possible for super-fluid volumes with mass below mP01/2.
What about hem? In the case of super-conductivity the interpretation of hem/h as product of orders of Galois groups would allow to estimate the number N= Q/2e of Cooper pairs of a minimal blob of super-conducting matter from the condition that the order of its Galois group is larger than integer. The number N=Q/2e is such that one has 2N(α/β0)1/2=n. The condition is satisfied if one has α/β0=q2, with q=k/2l such that N is divisible by l. The number of Cooper pairs would be quantized as multiples of l. What is clear that em interaction would correspond to a lower level of cognitive consciousness and that the step to gravitation dominated cognition would be huge if the dark gravitational interaction with size of astrophysical systems is involved. Many-sheeted space-time allows this in principle.

These arguments support the view that quantum information theory indeed closely relates not only to gravitation but also other interactions. Speculations revolving around blackhole, entropy, and holography, and emergence of space would be replaced with the number theoretic vision about cognition providing information theoretic interpretation of basic interactions in terms of entangled tensor networks (see this). Negentropic entanglement would have magnetic flux tubes (and fermionic strings at them) as topological correlates. The increase of the complexity of quantum states could occur by the "fusion" of Galois groups associated with various nodes of this network as macroscopic quantum states are formed. Galois groups and their representations would define the basic information theoretic concepts. The emergence of gravitational quantum coherence identified as the emergence of multi-cellulars would mean a major step in biological evolution.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.

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

Articles and other material related to TGD.

What could be the role of complexity theory in TGD?

Chaotic (or actually extremely complex and only apparently chaotic) systems seem to be the diametrical opposite of completely integrable systems about which TGD is a possible example. There is however also something common: in completely integrable classical systems all orbits are cyclic and in chaotic systems they form a dense set in the space of orbits. Furthermore, in chaotic systems the approach to chaos occurs via steps as a control parameter is changed. Same would take place in adelic TGD fusing the descriptions of matter and cognition.

In TGD Universe the hierarchy of extensions of rationals inducing finite-dimensional extension of p-adic number fields defines a hierarchy of adelic physics and provides a natural correlate for evolution. Galois groups and ramified primes appear as characterizers of the extensions. The sequences of Galois groups could characterize an evolution by phase transitions increasing the dimension of the extension associated with the coordinates of "world of classical worlds" (WCW) in turn inducing the extension used at space-time and Hilbert space level. WCW decomposes to sectors characterized by Galois groups G3 of extensions associated with the 3-surfaces at the ends of space-time surface at boundaries of causal diamond (CD) and G4 characterizing the space-time surface itself. G3 (G4) acts on the discretization and induces a covering structure of the 3-surface (space-time surface). If the state function reduction to the opposite boundary of CD involves localization into a sector with fixed G3, evolution is indeed mapped to a sequence of G3s.

Also the cognitive representation defined by the intersection of real and p-adic surfaces with coordinates of points in an extension of rationals evolve. The number of points in this representation becomes increasingly complex during evolution. Fermions at partonic 2-surfaces connected by fermionic strings define a tensor network, which also evolves since the number of fermions can change.

The points of space-time surface invariant under non-trivial subgroup of Galois group define singularities of the covering, and the positions of fermions at partonic surfaces could correspond to these singularities - maybe even the maximal ones, in which case the singular points would be rational. There is a temptation to interpret the p-adic prime characterizing elementary particle as a ramified prime of extension having a decomposition similar to that of singularity so that category theoretic view suggests itself.

One also ends up to ask how the number theoretic evolution could select preferred p-adic primes satisfying the p-adic length scale hypothesis as a survivors in number theoretic evolution, and ends up to a vision bringing strongly in mind the notion of conserved genes as analogy for conservation of ramified primes in extensions of extension. heff/h=n has natural interpretation as the order of Galois group of extension. The generalization of hbargr= GMm/v0=hbareff hypothesis to other interactions is discussed in terms of number theoretic evolution as increase of G3, and one ends up to surprisingly concrete vision for what might happen in the transition from prokaryotes to eukaryotes.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD? of the article What could be the role of complexity theory in TGD?.

Progress in adelic physics

The preparation of an article about number theoretic aspects of TGD forced to go through various related ideas and led to a considerable integration of the ideas. In this note ideas related directly to adelic TGD are discussed.

  1. Both hierarchy of Planck constant and preferred p-adic primes are now understood number theoretically.
  2. The realization of number theoretical universality (NTU) for functional integral seems like a formidable problem but the special features of functional integral makes the challenge tractable. NTU of functional integral is indeed suggested by the need to describe also cognition quantally.
  3. Strong form of holography (SH) is now understood. 2-D surfaces (string world sheets and possibly partonic 2-surfaces) are not quite enough: also number theoretic discretization of space-time surface is required. This allows to understand the distinction between imagination in terms of p-adic space-time surfaces and reality in terms of real space-time surface. The number of imaginations is much larger than realities (p-adic pseudo-constants).
  4. The localization of spinor modes to string world sheets can be understand only as effective: this resolves several interpretational problems. These spinors give all information needed to construct 4-D spinor modes. Also 2-D modified Dirac action and area action are enough to construct scattering amplitudes once number theoretic discretization of space-time surface responsible for dark matter is given. This means enormous simplification of the theory.
Galois group of number theoretic discretization and hierarchy of Planck constants

Simple arguments lead to the identification of heff/h=n as a factor of the order of Galois group of extension of rationals.

  1. The strongest form of NTU would require that the allowed points of imbedding space belonging an extension of rationals are mapped as such to corresponding extensions of p-adic number fields (no canonical identification). At imbedding space level this correspondence would be extremely discontinuous. The "spines" of space-time surfaces would however contain only a subset of points of extension, and a natural resolution length scale could emerge and prevent the fluctuation. This could be also seen as a reason for why space-times surfaces must be 4-D. The fact that the curve xn+yn=zn has no rational points for n>2, raises the hope that the resolution scale could emerge spontaneously.
  2. The notion of monadic geometry - discussed in detail here would realize this idea. Define first a number theoretic discretization of imbedding space in terms of points, whose coordinates in group theoretically preferred coordinate system belong to the extension of rationals considered. One can say that these algebraic points are in the intersection of reality and various p-adicities. Overlapping open sets assigned with this discretization define in the real sector a covering by open sets. In p-adic sector compact-open-topology allows to assign with each point 8th Cartesian power of algebraic extension of p-adic numbers. These compact open sets define analogs for the monads of Leibniz and p-adic variants of field equations make sense inside them.

    The monadic manifold structure of H is induced to space-time surfaces containing discrete subset of points in the algebraic discretization with field equations defining a continuation to space-time surface in given number field, and unique only in finite measurement resolution. This approach would resolve the tension between continuity and symmetries in p-adic--real correspondence: isometry groups would be replaced by their sub-groups with parameters in extension of rationals considered and acting in the intersection of reality and p-adicities.

    The Galois group of extension acts non-trivially on the "spines" of space-time surfaces. Hence the number theoretical symmetries act as physical symmetries and define the orbit of given space-time surface as a kind of covering space. The coverings assigned to the hierarchy of Planck constants would naturally correspond to Galois coverings and dark matter would represent number theoretical physics.

    This would give rise to a kind of algebraic hierarchy of adelic 4-surfaces identifiable as evolutionary hierarchy: the higher the dimension of the extension, the higher the evolutionary level.

  3. But how does quantum criticality relate to number theory and adelic physics? heff/h=n has been identified as the number of sheets of space-time surface identified as a covering space of some kind. Number theoretic discretization defining the "spine for a monadic space-time surface defines also a covering space with Galois group for an extension of rationals acting as covering group. Could n be identifiable as the order for a sub-group of Galois group?

    If this is the case, the proposed rule for heff changing phase transitions stating that the reduction of n occurs to its factor would translate to spontaneous symmetry breaking for Galois group and spontaneous - symmetry breakings indeed accompany phase transitions.

Ramified primes as referred primes for a given extension

The intuitive feeling is that the notion of preferred prime is something extremely deep and to me the deepest thing I know is number theory. Does one end up with preferred primes in number theory? This question brought to my mind the notion of ramification of primes (more precisely, of prime ideals of number field in its extension), which happens only for special primes in a given extension of number field, say rationals. Ramification is completely analogous to the degeneracy of some roots of polynomial and corresponds to criticality if the polynomial corresponds to criticality (catastrophe theory of Thom is one application). Could this be the mechanism assigning preferred prime(s) to a given elementary system, such as elementary particle? I have not considered their role earlier also their hierarchy is highly relevant in the number theoretical vision about TGD.

  1. Stating it very roughly (I hope that mathematicians tolerate this sloppy language of physicist): as one goes from number field K, say rationals Q, to its algebraic extension L, the original prime ideals in the so called integral closure over integers of K decompose to products of prime ideals of L (prime ideal is a more rigorous manner to express primeness). Note that the general ideal is analog of integer.

    Integral closure for integers of number field K is defined as the set of elements of K, which are roots of some monic polynomial with coefficients, which are integers of K having the form xn+ an-1xn-1 +...+a0. The integral closures of both K and L are considered. For instance, integral closure of algebraic extension of K over K is the extension itself. The integral closure of complex numbers over ordinary integers is the set of algebraic numbers.

    Prime ideals of K can be decomposed to products of prime ideals of L: P= ∏ Piei, where ei is the ramification index. If ei>1 is true for some i, ramification occurs. Pi:s in question are like co-inciding roots of polynomial, which for in thermodynamics and Thom's catastrophe theory corresponds to criticality. Ramification could therefore be a natural aspect of quantum criticality and ramified primes P are good candidates for preferred primes for a given extension of rationals. Note that the ramification make sense also for extensions of given extension of rationals.

  2. A physical analogy for the decomposition of ideals to ideals of extension is provided by decomposition of hadrons to valence quarks. Elementary particles becomes composite of more elementary particles in the extension. The decomposition to these more elementary primes is of form P= ∏ Pie(i), the physical analog would be the number of elementary particles of type i in the state (see this). Unramified prime P would be analogous a state with e fermions. Maximally ramified prime would be analogous to Bose-Einstein condensate of e bosons. General ramified prime would be analogous to an e-particle state containing both fermions and condensed bosons. This is of course just a formal analogy.
  3. There are two further basic notions related to ramification and characterizing it. Relative discriminant is the ideal divided by all ramified ideals in K (integer of K having no ramified prime factors) and relative different for P is the ideal of L divided by all ramified Pi:s (product of prime factors of P in L). These ideals represent the analogs of product of preferred primes P of K and primes Pi of L dividing them. These two integers ideals would characterize the ramification.
Ramified primes for preferred extensions as preferred p-adic primes?

In TGD framework the extensions of rationals (see this) and p-adic number fields (see this) are unavoidable and interpreted as an evolutionary hierarchy physically and cosmological evolution would gradually proceed to more and more complex extensions. One can say that string world sheets and partonic 2-surfaces with parameters of defining functions in increasingly complex extensions of prime emerge during evolution. Therefore ramifications and the preferred primes defined by them are unavoidable. For p-adic number fields the number of extensions is much smaller for instance for p>2 there are only 3 quadratic extensions.

How could ramification relate to p-adic and adelic physics and could it explain preferred primes?

  1. Ramified p-adic prime P=Pie would be replaced with its e:th root Pi in p-adicization. Same would apply to general ramified primes. Each un-ramified prime of K is replaced with e=K:L primes of L and ramified primes P with #{Pi}<e primes of L: the increase of algebraic dimension is smaller. An interesting question relates to p-adic length scale. What happens to p-adic length scales. Is p-adic prime effectively replaced with e:th root of p-adic prime: Lp∝ p1/2L1 → p1/2eL1? The only physical option is that the p-adic temperature for P would be scaled down Tp=1/n → 1/ne for its e:th root (for fermions serving as fundamental particles in TGD one actually has Tp=1). Could the lower temperature state be more stable and select the preferred primes as maximimally ramified ones? What about general ramified primes?
  2. This need not be the whole story. Some algebraic extensions would be more favored than others and p-adic view about realizable imaginations could be involved. p-Adic pseudo constants are expected to allow p-adic continuations of string world sheets and partonic 2-surfaces to 4-D preferred extremals with number theoretic discretization. For real continuations the situation is more difficult. For preferred extensions - and therefore for corresponding ramified primes - the number of real continuations - realizable imaginations - would be especially large.

    The challenge would be to understand why primes near powers of 2 and possibly also of other small primes would be favored. Why for them the number of realizable imaginations would be especially large so that they would be winners in number theoretical fight for survival?

NTU for functional integral

Number theoretical vision relies on NTU. In fermionic sector NTU is necessary: one cannot speak about real and p-adic fermions as separate entities and fermionic anti-commutation relations are indeed number theoretically universal.

What about NTU in case of functional integral? There are two opposite views.

  1. One can define p-adic variants of field equations without difficulties if preferred extremals are minimal surface extremals of Kähler action so that coupling constants do not appear in the solutions. If the extremal property is determined solely by the analyticity properties as it is for various conjectures, it makes sense independent of number field. Therefore there would be no need to continue the functional integral to p-adic sectors. This in accordance with the philosophy that thought cannot be put in scale. This would be also the option favored by pragmatist.
  2. Consciousness theorist might argue that also cognition and imagination allow quantum description. The supersymmetry NTU should apply also to functional integral over WCW (more precisely, its sector defined by CD) involved with the definition of scattering amplitudes.
The general vision involves some crucial observations.
  1. Only the expressions for the scatterings amplitudes should should satisfy NTU. This does not require that the functional integral satisfies NTU.
  2. Since the Gaussian and metric determinants cancel in WCW Kähler metric the contributions form maxima are proportional to action exponentials exp(Sk) divided by the ∑k exp(Sk). Loops vanish by quantum criticality.
  3. Scattering amplitudes can be defined as sums over the contributions from the maxima, which would have also stationary phase by the double extremal property made possible by the complex value of αK. These contributions are normalized by the vacuum amplitude.

    It is enough to require NTU for Xi=exp(Si)/∑k exp(Sk). This requires that Sk-Sl has form q1+q2 iπ + q3log(n). The condition brings in mind homology theory without boundary operation defined by the difference Sk-Sl. NTU for both Sk and exp(Sk) would only values of general form Sk=q1+q2 iπ + q3log(n) for Sk and this looks quite too strong a condition.

  4. If it is possible to express the 4-D exponentials as single 2-D exponential associated with union of string world sheets, vacuum functional disappears completely from consideration! There is only a sum over discretization with the same effective action and one obtains purely combinatorial expression.

See the chapter Unified Number Theoretic Vision or the article p-Adicization and adelic physics.

About some unclear issues of TGD

TGD has been in the middle of palace revolution during last two years and it is almost impossible to keep the chapters of the books updated. Adelic vision and twistor lift of TGD are the newest developments and there are still many details to be understood and errors to be corrected. The description of fermions in TGD framework has contained some unclear issues. Hence the motivation for the following brief comments.

Adelic vision and symmetries

In the adelic TGD SH is weakened: also the points of the space-time surface having imbedding space coordinates in an extension of rationals (cognitive representation) are needed so that data are not precisely 2-D. I have believed hitherto that one must use preferred coordinates for the imbedding space H - a subset of these coordinates would define space-time coordinates. These coordinates are determined apart from isometries. Does the number theoretic discretization imply loss of general coordinate invariance and also other symmetries?

The reduction of symmetry groups to their subgroups (not only algebraic since powers of e define finite-dimensional extension of p-adic numbers since ep is ordinary p-adic number) is genuine loss of symmetry and reflects finite cognitive resolution. The physics itself has the symmetries of real physics.

The assumption about preferred imbedding space coordinates is actually not necessary. Different choices of H-coordinates means only different and non-equivalent cognitive representations. Spherical and linear coordinates in finite accuracy do not provide equivalent representations.

Quantum-classical correspondence for fermions

Quantum-classical correspondence (QCC) for fermions is rather well-understood but deserves to be mentioned also here.

QCC for fermions means that the space-time surface as preferred extremal should depend on fermionic quantum numbers. This is indeed the case if one requires QCC in the sense that the fermionic representations of Noether charges in the Cartan algebras of symmetry algebras are equal to those to the classical Noether charges for preferred extremals.

Second aspect of QCC becomes visible in the representation of fermionic states as point like particles moving along the light-like curves at the light-like orbits of the partonic 2-surfaces (curve at the orbit can be locally only light-like or space-like). The number of fermions and antifermions dictates the number of string world sheets carrying the data needed to fix the preferred extremal by SH. The complexity of the space-time surface increases as the number of fermions increases.

Strong form of holography for fermions

It seems that scattering amplitudes can be formulated by assigning fermions with the boundaries of strings defining the lines of twistor diagrams. This information theoretic dimensional reduction from D=4 to D=2 for the scattering amplitudes can be partially understood in terms of strong form of holography (SH): one can construct the theory by using the data at string worlds sheets and/or partonic 2-surfaces at the ends of the space-time surface at the opposite boundaries of causal diamond (CD).

4-D modified Dirac action would appear at fundamental level as supersymmetry demands but would be reduced for preferred extremals to its 2-D stringy variant serving as effective action. Also the value of the 4-D action determining the space-time dynamics would reduce to effective stringy action containing area term, 2-D Kähler action, and topological Kähler magnetic flux term. This reduction would be due to the huge gauge symmetries of preferred extremals. Sub-algebra of super-symplectic algebra with conformal weigths coming as n-multiples of those for the entire algebra and the commutators of this algebra with the entire algebra would annihilate the physical states, and thecorresponding classical Noether charges would vanish.

One still has the question why not the data at the entire string world sheets is not needed to construct scattering amplitudes. Scattering amplitudes of course need not code for the entire physics. QCC is indeed motivated by the fact that quantum experiments are always interpreted in terms of classical physics, which in TGD framework reduces to that for space-time surface.

The relationship between spinors in space-time interior and at boundaries between Euclidian and Minkoskian regions

Space-time surface decomposes to interiors of Minkowskian and Euclidian regions. At light-like 3-surfaces at which the four-metric changes, the 4-metric is degenerate. These metrically singular 3-surfaces - partonic orbits- carry the boundaries of string world sheets identified as carriers of fermionic quantum numbers. The boundaries define fermion lines in the twistor lift of TGD. The relationship between fermions at the partonic orbits and interior of the space-time surface has however remained somewhat enigmatic.

So: What is the precise relationship between induced spinors ΨB at light-like partonic 3-surfaces and ΨI in the interior of Minkowskian and Euclidian regions? Same question can be made for the spinors ΨB at the boundaries of string world sheets and ΨI in interior of the string world sheets. There are two options to consider:

  • Option I: ΨB is the limiting value of ΨI .
  • Option II: ΨB serves as a source of ΨI .
For Opion I it is difficult to understand the preferred role of ΨB.

I have considered Option II already years ago but have not been able to decide.

  1. That scattering amplitudes could be formulated only in terms of sources only, would fit nicely with SH, twistorial amplitude construction, and also with the idea that scattering amplitudes in gauge theories can be formulated in terms of sources of boson fields assignable to vertices and propagators. Now the sources would become fermionic.
  2. One can take gauge theory as a guideline. One adds to free Dirac equation source term γkkAkΨ. Therefore the natural boundary term in the action would be of the form (forgetting overall scale factor)

    SB=ΨIΓα(C-S)AαΨB+ c.c .

    Here the modified gamma matrix is Γα(C-S) (contravariant form is natural for light-like 3-surfaces) is most naturally defined by the boundary part of the action - naturally Chern-Simons term for Kähler action. A denotes the Kähler gauge potential.

  3. The variation with respect to ΨB gives


    at the boundary so that the C-S term and interaction term vanish. This does not however imply vanishing of the source term! This condition can be seen as a boundary condition.

The same argument applies also to string world sheets.

About second quantization of the induced spinor fields

The anti-commutation relations for the induced spinors have been a long-standing issue and during years I have considered several options. The solution of the problem looks however stupifuingly simple. The conserved fermion currents are accompanied by super-currents obtained by replacing Ψ with a mode of the induced spinor field to get unΓαΨ or ΨΓαun with the conjugate of the mode. One obtains infinite number of conserved super currents. One can also replace both Ψ and Ψ in this manner to get purely bosonic conserved currents ΨmΓαun to which one can assign a conserved bosonic charges Qmn.

I noticed this years ago but did not realize that these bosonic charges define naturally anti-commutators of fermionic creation and annihilation operators! The ordinary anti-commutators of quantum field theory follow as a special case! By a suitable unitary transformation of the spinor basis one can diagonalize the hermitian matrix defined by Qmn and by performing suitable scalings one can transform anti-commutation relations to the standard form. An interesting question is whether the diagonalization is needed, and whether the deviation of the diagonal elements from unity could have some meaning and possibly relate to the hierarchy heff=n× h of Planck constants - probably not.

Is statistical entanglement "real" entanglement?

The question about the "reality" of statistical entanglement has bothered me for years. This entanglement is maximal and it cannot be reduced by measurement so that one can argue that it is not "real". Quite recently I learned that there has been a longstanding debate about the statistical entanglement and that the issue still remains unresolved.

The idea that all electrons of the Universe are maximally entangled looks crazy. TGD provides several variants for solutions of this problem. It could be that only the fermionic oscillator operators at partonic 2-surfaces associated with the space-time surface (or its connected component) inside given CD anti-commute and the fermions are thus indistinguishable. The extremist option is that the fermionic oscillator operators belonging to a network of partonic 2-surfaces connected by string world sheets anti-commute: only the oscillator operators assignable to the same scattering diagram would anti-commute.

What about QCC in the case of entanglement. ER-EPR correspondence introduced by Maldacena and Susskind for 4 years ago proposes that blackholes (maybe even elementary particles) are connected by wormholes. In TGD the analogous statement emerged for more than decade ago - magnetic flux tubes take the role of wormholes in TGD. Magnetic flux tubes were assumed to be accompanied by string world sheets. I did not consider the question whether string world sheets are always accompanied by flux tubes.

What could be the criterion for entanglement to be "real"? "Reality" of entanglement demands some space-time correlate. Could the presence of the flux tubes make the entanglement "real"? If statistical entanglement is accompanied by string connections without magnetic flux tubes, it would not be "real": only the presence of flux tubes would make it "real". Or is the presence of strings enough to make the statistical entanglement "real". In both cases the fermions associated with disjoint space-time surfaces or with disjoint CDs would not be indistinguishable. This looks rather sensible.

The space-time correlate for the reduction of entanglement would be the splitting of a flux tube and fermionic strings inside it. The fermionic strings associated with flux tubes carrying monopole flux are closed and the return flux comes back along parallel space-time sheet. Also fermionic string has similar structure. Reconnection of this flux tube with shape of very long flattened square splitting it to two pieces would be the correlate for the state function reduction reducing the entanglement with other fermions and would indeed decouple the fermion from the network.

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds".

To the index page