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Physics as a Generalized Number Theory
Note: Newest contributions are at the top!
I am not a mathematician and therefore should refrain from consideration of anything related to fundamentals of mathematics. In the discussions with Santeri Satama I could not avoid the temptation to break this rule. I however feel that I must confess my sins and in the following I will do this.
Goedel's problematics involves statements which cannot be proved to be true or false or are simultaneously true and false. This problematics has also a purely geometric analog in terms of set theoretic representation of Boolean algebras when real topology is used but not when p-adic topology is used.
The natural idea is that Boolean algebra is realized in terms of open sets such that the negation of statement corresponds to the complement of the set. In p-adic topologies open sets are simultaneously also closed and there are no boundaries: this makes them and - more generally Stone spaces - ideal for realizing Boolean algebra set theoretically. In real topology the complement of open set is closed and therefore not open and one has a problem.
Could one circumvent the problem somehow?
2. Number theory from quantum theory
Could one define or at least represent the notion of number using the notions of quantum physics? A natural starting point is hierarchy of extensions of rationals defining hierarchy of adeles. Could one obtain rationals and their extensions from simplest possible quantum theory in which one just constructs many particle states by adding or removing particles using creation and annihilation operators?
2.1 How to obtain rationals and their extensions?
Rationals and their extensions are fundamental in TGD. Can one have quantal construction for them?
2.2 Arithmetics of Hilbert spaces
The notions of prime and divisibility and even basic arithmetics emerge also from the tensor product and direct sum for Hilbert spaces. Hilbert spaces with prime dimension do not decompose to tensor products of lower-dimensional Hilbert spaces. One can even perform a formal generalization of the dimension of Hilbert space so that it becomes rational and even algebraic number.
For some years ago I indeed played with this thought but at that time I did not have in mind reduction of number theory to the arithemetics of Hilbert spaces. If this really makes sense, numbers could be replaced by Hilbert spaces with product and sum identified as tensor product and direct sum!
Finite-dimensional Hilbert space represent the analogs of natural numbers. The analogs of integers could be defined as pairs (m,n) of Hilbert spaces with spaces (m,n) and (m+r,n+r) identified (this space would have dimension m-n. This identification would hold true also at the level of states. Hilbert spaces with negative dimension would correspond to pairs with (m-n)<0: the canonical representives for m and -m would be (m,0) and (0,m). Rationals can be defined as pairs (m,n) of Hilbert spaces with pairs (m,n) and (km,kn) identified. These identifications would give rise to kind of gauge conditions and canonical representatives for m and 1/m are (m,1) and (1,m).
What about Hilbert spaces for which the dimension is algebraic number? Algebraic numbers allow a description in terms of partial fractions and Stern-Brocot (S-B) tree (see this and this) containing given rational number once. S-B tree allows to see information about algebraic numbers as constructible by using an algorithm with finite number of steps, which is allowed if one accepts abstraction as basic aspect of cognition. Algebraic number could be seen as a periodic partial fraction defining an infinite path in S-B tree. Each node along this path would correspond to a rational having Hilbert space analog. Hilbert space with algebraic dimension would correspond to this kind of path in the space of Hilbert spaces with rational dimension. Transcendentals allow identification as non-pediodic partial fraction and could correspond to non-periodic paths so that also they could have Hilbert spaces counterparts.
2.3 How to obtain the analogs higher-D spaces?
Algebraic extensions of rationals allow cognitive realization of spaces with arbitrary dimension identified as algebraic dimension of extension of rationals.
3. Could quantum set theory make sense?
In the following my view point is that of quantum physicist fascinated by number theory and willing to reduce set theory to what could be called called quantum set theory. It would follow from physics as generalised number theory (adelic physics) and have ordinary set theory as classical correlate.
What quantum set theory could mean? Suppose that number theory-quantum theory connection really works. What about set theory? Or perhaps its quantum counterpart having ordinary set theory as a classical correlate?
How could one obtain classical set theory?
Here one has two views about infinite set. One has both separable (infinite-D in discrete sense: particle in box with discrete momentum spectrum) and non-separable (infinite-D in real sense: free particle with continuous momentum spectrum) Hilbert spaces. In the latter case the completely localized single particle states would be represented by delta functions divided by infinite normalization factors. They are routinely used in Dirac's bra-ket formalism but problems emerge in quantum field theory.
A possible solution is that one weakens the axiom of choice and accepts that only discrete points set (possibly finite) are cognitively representable and one has wave functions localized to discrete set of points. A stronger assumption is that these points have coordinates in extension of rationals so that one obtains number theoretical universality and adeles. This is TGD view and conforms also with the identification of hyper-finite factors of type II1 as basic algebraic objects in TGD based quantum theory as opposed to wave mechanics (type I) and quantum field theory (type III). They are infinite-D but allow excellent approximation as finite-D objects.
This picture could relate to the notion of non-commutative geometry, where set emerges as spectrum of algebra: the points of spectrum label the ideals of the integer elements of algebra.
See the chapter Unified Number Theoretical Vision or the article Some layman considerations related to fundamentals of mathematics.
The work of Peter Stolze based on the notion of perfectoid has raised a lot of interest in the community of algebraic geometers. One application of the notion relates to the attempt to generalize algebraic geometry by replacing polynomials with analytic functions satisfying suitable restrictions. Also in TGD this kind of generalization might be needed at the level of M4× CP2 whereas at the level of M8 algebraic geometry might be enough. The notion of perfectoid as an extension of p-adic numbers Qp allowing all p:th roots of p-adic prime p is central and provides a powerful technical tool when combined with its dual, which is function field with characteristic p.
Could perfectoids have a role in TGD? The infinite-dimensionality of perfectoid is in conflict with the vision about finiteness of cognition. For other p-adic number fields Qq, q≠ p the extension containing p:th roots of p would be however finite-dimensional even in the case of perfectoid. Furthermore, one has an entire hierarchy of almost-perfectoids allowing powers of pm:th roots of p-adic numbers. The larger the value of m, the larger the number of points in the extension of rationals used, and the larger the number of points in cognitive representations consisting of points with coordinates in the extension of rationals. The emergence of almost-perfectoids could be seen in the adelic physics framework as an outcome of evolution forcing the emergence of increasingly complex extensions of rationals.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article Could the precursors of perfectoids emerge in TGD?.
I had a debate with Santeri Satama about the notion of number leading to the question about what cognitive representability of number could mean. This inspired writing of an articling discussing the notion of cognitive representability. Numbers in the extensions of rationals are assumed to be cognitively representable in terms of points common to real and various p-adic space-time sheets (correlates for sensory and cognitive). One allows extensions of p-adics induced by extension of rationals in question and the hierarchy of adeles defined by them.
One can however argue that algebraic numbers do not allow finite representation as do rational numbers. A weaker condition is that the coding of information about algorithm producing the cognitively representable number contains a finite amount of information although it might take an infinite time to run the algorithm (say containing infinite loops). Furthermore, cognitive representations in TGD sense are also sensory representations allowing to represent algebraic numbers geometrically (21/2) as the diameter of unit square). Stern-Brocot tree associated with partial fractions indeed allows to identify rationals as finite paths connecting the root of S-B tree to the rational in question. Algebraic numbers can be identified as infinite periodic paths so that finite amount of information specifies the path. Transcendental numbers would correspond to infinite non-periodic paths. A very close analogy with chaos theory suggests itself.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article What does cognitive representability really mean?
A proposal that all physics emerges from the notion of number field is made. The first guess for the number field in question would be complexified octonions for which inverse exists except at complexified light-cone boundary: this has interpretation in terms of propagation of signals with light-velocity 8-D sense. The emergence of fermions however requires super-octonions as super variant of number field. Rather surprisingly, it turns out that super-number theory makes perfect sense. One can define the inverse of super-number if its ordinary part is non-vanishing and also the notion of primeness makes sense and construct explicitly the super-primes associated with ordinary primes. The prediction of new number piece of theory can be argued to be a strong support for the integrity of TGD.
See the the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article Super-number fields: does physics emerge from the notion of number?.
Is the quantum leakage between different signatures of the real sectors of the complexified M8 possible?
Complexified octonions have led to a dramatic progress in the understanding of TGD. One cannot however avoid a radical question about fundamentals.
In developing the view about M8-H duality reducing physics to algebraic geometry for complexified octonions at the level of M8, I became aware of trivial looking but amazingly profound observation about the basic arithmetics of complex, quaternion, and octonion number fields.
This extremely simple observation turns out to contain amazingly deep physics.
The entire elementary particle physics emerges from these two simple number theoretic properties for the product of numbers!
Riemann-Roch theorem (RR) is a central piece of algebraic geometry. Atyiah-Singer index theorem is one of its generalizations relating the solution spectrum of partial differential equations and topological data. For instance, characteristic classes classifying bundles associated with Yang-Mills theories have applications in gauge theories and string models.
The advent of octonionic approach to the dynamics of space-time surfaces inspired by M8-H duality (see this and this) gives hopes that dynamics at the level of complexified octonionic M8 could reduce to algebraic equations plus criticality conditions guaranteeing associativity for space-time surfaces representing external particles, in interaction region commutativity and associativity would be broken. The complexification of octonionic M8 replacing norm in flat space metric with its complexification would unify various signatures for flat space metric and allow to overcome the problems due to Minkowskian signature. Wick rotation would not be a mere calculational trick.
For these reasons time might be ripe for applications of possibly existing generalization of RR to TGD framework. In the following I summarize my admittedly unprofessional understanding of RR discussing the generalization of RR for complex algebraic surfaces having real dimension 4: this is obviously interesting from TGD point of view.
I will also consider the possible interpretation of RR in TGD framework. One interesting idea is possible identification of light-like 3-surfaces and curves (string boundaries) as generalized poles and zeros with topological (but not metric) dimension one unit higher than in Euclidian signature.
Atyiah-Singer index theorem (AS) is one of the generalizations of RR and has shown its power in gauge field theories and string models as a method to deduce the dimensions of various moduli spaces for the solutions of field equations. A natural question is whether AS could be useful in TGD and whether the predictions of AS at H side could be consistent with M8-H duality suggesting very simple counting for the numbers of solutions at M8 side as coefficient combinations of polynomials in given extension of rationals satisfying criticality conditions. One can also ask whether the hierarchy of degrees n for octonion polynomials could correspond to the fractal hierarchy of generalized conformal sub-algebras with conformal weights coming as n-multiples for those for the entire algebras.
I wrote a brief summary about basic ideas of enumerative algebraic geometry and proposals for applications to TGD. Here is a short abstract of the article summarizing the new results.
String models and M-theory involve both algebraic and symplectic enumerative geometry. Also in adelic TGD enumerative algebraic geometry emerges. This article gives a brief summary about the basic ideas involved and suggests some applications to TGD.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article About enumerative algebraic geometry in TGD framework.
For last month or so I have been writing an article about reduction of quantum classical TGD to octonionic algebraic geometry. During writing process "octonionic" has been replaced with "super-octonionic" and it has turned out that the formalism of twistor Grassmannian approach could generalize to TGD. The twistorial scattering diagrams have interpretation as cognitive representations with vertices assignable to points of space-time surface in appropriate extension of rationals
. In accordance with the vision that classical theory is exact part of quantum theory, the core part of the calculations of scattering amplitudes would reduce to the determination of zero loci for real and imaginary parts of octonionic polynomials and finding of the points of space-time surface with M8 coordinates in extension of rationals. At fundamental level the physics would reduce to number theory.
A proposal for the description of interactions is discussed in the article. Here is a brief summary.
Could the octonionic level provide an elegant description of fermions in terms of super variant of octonionic algebraic geometry? Could one even construct scattering amplitudes at the level of M8 using the variant of the twistor approach discussed earlier.
The idea about super-geometry is of course very different from the idea that fermionic statistics is realized in terms of the spinor structure of "world of classical worlds" (WCW) but M8-H duality could however map these ideas and also number theoretic and geometric vision to each other. The angel of geometry and the devil of algebra could be dual to each other.
1. Octonionic superspace
Consider now what super version of the octonionic super-space might look like.
Instead of super-fields one would have a super variant of octonionic algebraic geometry.
There are several questions about quantum numbers.
It would be extremely nice if the scattering amplitudes could be computed at the octonionic level by using a generalization of twistor approach in ZEO finding a nice justification at the level of M8. Something rather similar to N=4 twistor Grassmann approach suggests itself.
Octonionic polynomials provide a promising approach to the understanding of zero energy ontology (ZEO) and causal diamonds (CDs) defined as intersections of future and past directed lightcones: CD makes sense both in octonionic (8-D) and quaternionic (4-D) context. Light-like boundary of CD as also light-cone emerge naturally as zeros of octonionic polynomials. This does not yet give CDs and ZEO: one should have intersection of future and past directed light-cones. The intuitive picture is that one has a hierarchy of CDs and that also the space-time surfaces inside different CDs an interact. It turns out that CDs and thus also ZEO emerge naturally both at the level of M8 and M4 .
Remark: In the sequel RE(o) and IM(o) refer to real and imaginary parts of octonions in quaternionic sense: one has o= RE(o)+IM(o)I4, where RE(o) and IM(o) are quaternions.
1. General view about solutions to RE(P)=0 and IM(P)=0 conditions
The first challenge is to understand at general level the nature of RE(P)=0 and IM(P)=0 conditions expected to give 4-D space-time surfaces as zero loci. Appendix shows explicitly for P(o)=o2 that Minkowski signature gives rise to unexpected phenomena. In the following these phenomena are shown to be completely general but not quite what one obtains for P(o)=o2 having double root at origin.
CD defines the basic geometric object in ZEO. It is good to list some basic features of CDS, which appear as both 4-D and 8-D variants.
3. The emergence of CDs
CDs are a key notion of zero energy ontology (ZEO). Could the emergence of CDs be understood in terms of singularities of octonion polynomials located at the light-like boundaries of CDs? In Minkowskian case the complex norm qqci is present in P (c is conjugation changing the sign of quaternionic unit but not that of the commuting imaginary unit i). Could this allow to blow up the singular point to a 3-D boundary of light-cone and allow to understand the emergence of causal diamonds (CDs) crucial in ZEO.
The study of the special properties for zero loci of general polynomial P(o) at light-rays of O indeed demonstrated that both 8-D land 4-D light-cones and their complements emerge naturally, and that the M4 projections of these light-cones and even of their boundaries are 4-D future - or past directed light-cones. What one should understand is how CDs as their intersections, and therefore ZEO, emerge.
4. How could the space-time varieties associated with different CDs interact?
The interaction of space-time surfaces inside given CD is well-defined. Sitation is not so clear for different CDs for which the choice of octonionic coordinate origin is in general different and polynomial bases for different CDs do not commute nor associate.
The intuitive expectation is that 4-D/8-D CDs can be located everywhere in M4/M8. The polynomials with different origins neither commute nor are associative. Their sum is a polynomial whose coefficients are not real. How could one avoid losing the extremely beautiful associative and commutative algebra?
It seems that one cannot form their products and sums and must form the Cartesian product of M8:s with different origins and formulate the interaction at M8 level in this framework. Note that Cayley-Dickson hierarchy does not seem to be relevant since the dimension are powers of 2 rather than multiples of 8.
Should one give up associativity and allow products (but not sums since one should give up the assumption that the coefficients of polynomials are real) of polynomials associated with different CDs as an analog for the formation of free many-particle states. One can still have separate vanishing of the polynomials in separate CDs but how could one describe their interaction?
If one does not give up associativity and commutativity, how can one describe the interactions between space-time surfaces inside different CDs at the level of M8?
All big pieces of quantum TGD are now tightly interlinked.
In the following I give a brief summary about what has been done. I concentrate on M8-H duality since the most significant results are achieved here.
It is fair to say that the new view answers the following a long list of open questions.
I have used last month to develop a detailed vision about M8-H duality and now I dare to speak about genuine breakthrough. I attach below the abstract of the resulting article.
TGD leads to several proposals for the exact solution of field equations defining space-time surfaces as preferred extremals of twistor lift of Kähler action. So called M8-H duality is one of these approaches. The beauty of M8-H duality is that it could reduce classical TGD to algebraic geometry and would immediately provide deep insights to cognitive representation identified as sets of rational points of these surfaces.
In the sequel I shall consider the following topics.
Octonions and quaternions are 20 year old part of TGD: one of the three threads in physics as generalized number theory vision. Second vision is quantum physics as geometry of WCW. The question has been how to fuse geometric and number theory visions. Algebraic geometry woul do it since it is both geometry and algebra and it has been also part of TGD but only now I realized how to get acceess to its enormous power.
Even the proposal discussed now about the algebra of octonionic polynomials with real coefficients was made about two decades ago but only now I managed to formulate it in detail. Here the general wisdom gained from adelic physics helped enormously. I dare say that classical TGD at the most fundamental level is solved exactly.
From the point of pure mathematics the generalization of complex analyticity and linear Cauchy Riemann conditions to multilinear variants for quaternions, octonions and even for the entire hierarchy of algebras obtained by Cayley-Dickson construction is a real breakthrough. Consider only the enormous importance of complex analyticity in mathematics and physics, including string models. I do not believe that this generalization has been discovered: otherwise it would be key part of mathematical physics. Quaternionic and octonionic analyticities will certainly mean huge evolution in mathematics. I had never ended to these discoveries without TGD: TGD forced them.
At these moments I feel deep sadness when knowing that the communication of these results to collegues is impossible in practice. This stupid professional arrogance is something which I find very difficult to accept even after 4 decades. I feel that when society pays a monthly salary for a person for being a scientists, he should feel that his duty is to be keenly aware what is happening in his field. When some idiot proudly tells that he reads only prestigious journals, I get really angry.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.
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.
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.
The p-adic aspects of Topological Geometrodynamics (TGD) will be discussed. Introduction gives a short summary about classical and quantum TGD. This is needed since the p-adic ideas are inspired by TGD based view about physics.
p-Adic mass calculations relying on p-adic generalization of thermodynamics and super-symplectic and super-conformal symmetries are summarized. Number theoretical existence constrains lead to highly non-trivial and successful physical predictions. The notion of canonical identification mapping p-adic mass squared to real mass squared emerges, and is expected to be a key player of adelic physics allowing to map various invariants from p-adics to reals and vice versa.
A view about p-adicization and adelization of real number based physics is proposed. The proposal is a fusion of real physics and various p-adic physics to single coherent whole achieved by a generalization of number concept by fusing reals and extensions of p-adic numbers induced by given extension of rationals to a larger structure and having the extension of rationals as their intersection.
The existence of p-adic variants of definite integral, Fourier analysis, Hilbert space, and Riemann geometry is far from obvious and various constraints lead to the idea of number theoretic universality (NTU) and finite measurement resolution realized in terms of number theory. An attractive manner to overcome the problems in case of symmetric spaces relies on the replacement of angle variables and their hyperbolic analogs with their exponentials identified as roots of unity and roots of e existing in finite-dimensional algebraic extension of p-adic numbers. Only group invariants - typically squares of distances and norms - are mapped by canonical identification from p-adic to real realm and various phases are mapped to themselves as number theoretically universal entities.
Also the understanding of the correspondence between real and p-adic physics at various levels - space-time level, imbedding space level, and level of "world of classical worlds" (WCW) - is a challenge. The gigantic isometry group of WCW and the maximal isometry group of imbedding space give hopes about a resolution of the problems. Strong form of holography (SH) allows a non-local correspondence between real and p-adic space-time surfaces induced by algebraic continuation from common string world sheets and partonic 2-surfaces. Also local correspondence seems intuitively plausible and is based on number theoretic discretization as intersection of real and p-adic surfaces providing automatically finite "cognitive" resolution. he existence p-adic variants of Kähler geometry of WCW is a challenge, and NTU might allow to realize it.
I will also sum up the role of p-adic physics in TGD inspired theory of consciousness. Negentropic entanglement (NE) characterized by number theoretical entanglement negentropy (NEN) plays a key role. Negentropy Maximization Principle (NMP) forces the generation of NE. The interpretation is in terms of evolution as increase of negentropy resources.
For details see the new chapter Philosophy of Adelic Physics.
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.
For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.
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.
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.
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?.
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.
Simple arguments lead to the identification of heff/h=n as a factor of the order of Galois group of extension of rationals.
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.
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?
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.