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Physics as a Generalized Number Theory
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As found in the article Boolean algebra, Stone spaces, and TGD, one can generalize Boolean logic to a logic in finite field G(p) with p elements. p-Logics have very nice features. For a given set the p-Boolean algebra can be represented as maps having values in finite field G(p). The subsets with a given value 0≤ k<p define subsets of a partition and one indeed obtains p subsets some of which are empty unless the map is surjection.
The basic challenges are following: generalize logical negation and generalize Boolean operations AND and OR. I have considered several options but the one based on category theoretical thinking seems to be the most promising one. One can imbed p1-Boolean algebras to p-Boolean algebra by considering functions which have values in G(p1)⊂ G(p). One can also project G(p) valued functions to G(p1) by mod p1 operation. The operations should respect the logical negation and p-Boolean operations if possible.
The lowest level of the algebraic structure generalizes as such also to p-adic-valued functions in discrete or even continuous set. The negation fails to have an obvious generalization and the second level of the hierarchy would require defining functions in the infinite-D space of p-adic-valued functions.
I received a link to an interesting the article "Brain Computation Is Organized via Power-of-Two-Based Permutation Logic" by Kun Xie et al in Frontiers in Systems Neuroscience (see this).
The proposed model is about how brain classifies neuronal inputs and suggests that the classification is based on Boolean algebra represents as subsets of n-element set for n inputs. The following represents my attempt to understand the model of the article.
The Facebook discussion with Stephen King about Stone spaces led to a highly interesting development of ideas concerning Boolean, algebras, Stone spaces, and p-adic physics. I have discussed these ideas already earlier but the improved understanding of the notion of Stone space helped to make the ideas more concrete. The basic ideas are briefly summarized.
p-adic integers/numbers correspond to the Stone space assignable to Boolean algebra of natural numbers/rationals with p=2 assignable to Boolean logic. Boolean logic generalizes for n-valued logics with prime values of n in special role. The decomposition of set to n subsets defined by an element of n-Boolean algebra is obtained by iterating Boolean decomposition n-2 times. n-valued logics could be interpreted in terms of error correction allowing only bit sequences, which correspond to n<p<2k in k-bit Boolean algebra. Adelic physics would correspond to the inclusion of all p-valued logics in single adelic logic.
The Stone spaces of p-adics, reals, etc.. have huge size and a possible identification (in absence of any other!) is in terms of concept of real number assigning to real/p-adic/etc... number a fiber space consisting of all units obtained as ratios of infinite primes. As real numbers they are just units but has complex number theoretic anatomy and would give rise to what I have assigned the terms algebraic holography and number theoretic Brahman = Atman.
Langlands correspondence is for mathematics what unified theories are for physics. The number theoretic vision about TGD has intriguing resemblances with number theoretic Langlands program. There is also geometric variant of Langlands program. I am of course amateur and do not have grasp about the mathematical technicalities and can only try to understand the general ideas and related them to those behind TGD. Physics as geometry of WCW ("world of classical worlds") and physics as generalized number theory are the two visions about quantum TGD: this division brings in mind geometric and number theoretic Langlands programs. This motivates re-consideration of Langlands program from TGD point of view. I have written years ago a chapter about this earlier but TGD has evolved considerably since then so that it is time for a second attempt to understand what Langlands is about.
By Langlands correspondence the representations of semi-direct product of G and Galois group Gal and G should correspond to each other. This suggests that he representations of G should have G-spin such that the dimension of this representation is same as the representation of non-commutative Galois group. This would conform with the vision about physics as generalized number theory. Could this be the really deep physical content of Langlands correspondence?
p-Adization of quantum TGD is one of the long term projects of TGD. In the sequel the recent view about the situation is discussed. The notion of finite measurement resolution reducing to number theoretic existence in p-adic sense is the fundamental notion. p-Adic geometries replace discrete points of discretization with p-adic analogs of monads of Leibniz making possible to construct differential calculus and formulate p-adic variants of field equations allowing to construct p-adic cognitive representations for real space-time surfaces.
This leads to a beautiful construction for the hierarchy of p-adic variants of imbedding space inducing in turn the construction of p-adic variants of space-time surfaces. Number theoretical existence reduces to conditions demanding that all ordinary (hyperbolic) phases assignable to (hyperbolic) angles are expressible in terms of roots of unity (roots of e).
For SU(2) one obtains as a special case Platonic solids and regular polygons as preferred p-adic geometries assignable also to the inclusions of hyperfinite factors. Platonic solids represent idealized geometric objects ofthe p-adic world serving as a correlate for cognition as contrast to the geometric objects of the sensory world relying on real continuum.
In the case of causal diamonds (CDs) - the construction leads to the discrete variants of Lorentz group SO(1,3) and hyperbolic spaces SO(1,3)/SO(3). The construction gives not only the p-adicizable discrete subgroups of SU(2) and SU(3) but applies iteratively for all classical Lie groups meaning that the counterparts of Platonic solids are countered also for their p-adic coset spaces. Even the p-adic variants of WCW might be constructed if the general recipe for the construction of finite-dimensional symplectic groups applies also to the symplectic group assignable to Δ CD× CP2.
The emergence of Platonic solids is very remarkable also from the point of view of TGD inspired theory of consciousness and quantum biology. For a couple of years ago I developed a model of music harmony relying on the geometries of icosahedron and tetrahedron. The basic observation is that 12-note scale can be represented as a closed curve connecting nearest number points (Hamiltonian cycle) at icosahedron going through all 12 vertices without self intersections. Icosahedron has also 20 triangles as faces. The idea is that the faces represent 3-chords for a given harmony characterized by Hamiltonian cycle. Also the interpretation terms of 20 amino-acids identifiable and genetic code with 3-chords identifiable as DNA codons consisting of three letters is highly suggestive.
One ends up with a model of music harmony predicting correctly the numbers of DNA codons coding for a given amino-acid. This however requires the inclusion of also tetrahedron. Why icosahedron should relate to music experience and genetic code? Icosahedral geometry and its dodecahedral dual as well as tetrahedral geometry appear frequently in molecular biology but its appearance as a preferred p-adic geometry is what provides an intuitive justification for the model of genetic code. Music experience involves both emotion and cognition. Musical notes could code for the points of p-adic geometries of the cognitive world. The model of harmony in fact generalizes. One can assign Hamiltonian cycles to any graph in any dimension and assign chords and harmonies with them. Hence one can ask whether music experience could be a form of p-adic geometric cognition in much more general sense.
The geometries of biomolecules brings strongly in mind the geometry p-adic space-time sheets. p-Adic space-time sheets can be regarded as collections of p-adic monad like objects at algebraic space-time points common to real and p-adic space-time sheets. Monad corresponds to p-adic units with norm smaller than unit. The collections of algebraic points defining the positions of monads and also intersections with real space-time sheets are highly symmetric and determined by the discrete p-adicizable subgroups of Lorentz group and color group. When the subgroup of the rotation group is finite one obtains polygons and Platonic solids. Bio-molecules typically consists of this kind of structures - such as regular hexagons and pentagons - and could be seen as cognitive representations of these geometries often called sacred! I have proposed this idea long time ago and the discovery of the recipe for the construction of p-adic geometries gave a justification for this idea.
See the chapter Number Theoretical Vision or the article p-Adicizable discrete variants of classical Lie groups and coset spaces in TGD framework..
Number theoretical feats of some mathematicians like Ramanujan remain a mystery for those believing that brain is a classical computer. Also the ability of idiot savants - lacking even the idea about what prime is - to factorize integers to primes challenges the idea that an algorithm is involved. In this article I discuss ideas about how various arithmetical feats such as partitioning integer to a sum of integers and to a product of prime factors might take place. The ideas are inspired by the number theoretic vision about TGD suggesting that basic arithmetics might be realized as naturally occurring processes at quantum level and the outcomes might be "sensorily perceived". One can also ask whether zero energy ontology (ZEO) could allow to perform quantum computations in polynomial instead of exponential time.
The indian mathematician Srinivasa Ramanujan is perhaps the most well-known example about a mathematician with miraculous gifts. He told immediately answers to difficult mathematical questions - ordinary mortals had to to hard computational work to check that the answer was right. Many of the extremely intricate mathematical formulas of Ramanujan have been proved much later by using advanced number theory. Ramanujan told that he got the answers from his personal Goddess. A possible TGD based explanation of this feat relies on the idea that in zero energy ontology (ZEO) quantum computation like activity could consist of steps consisting quantum computation and its time reversal with long-lasting part of each step performed in reverse time direction at opposite boundary of causal diamond so that the net time used would be short at second boundary.
The adelic picture about state function reduction in ZEO suggests that it might be possible to have direct sensory experience about prime factorization of integers (see this). What about partitions of integers to sums of primes? For years ago I proposed that symplectic QFT is an essential part of TGD. The basic observation was that one can assign to polygons of partonic 2-surface - say geodesic triangles - Kähler magnetic fluxes defining symplectic invariance identifiable as zero modes. This assignment makes sense also for string world sheets and gives rise to what is usually called Abelian Wilson line. I could not specify at that time how to select these polygons. A very natural manner to fix the vertices of polygon (or polygons) is to assume that they correspond ends of fermion lines which appear as boundaries of string world sheets. The polygons would be fixed rather uniquely by requiring that fermions reside at their vertices.
The number 1 is the only prime for addition so that the analog of prime factorization for sum is not of much use. Polygons with n=3,4,5 vertices are special in that one cannot decompose them to non-degenerate polygons. Non-degenerate polygons also represent integers n>2. This inspires the idea about numbers 3,4,5 as "additive primes" for integers n>2 representable as non-degenerate polygons. These polygons could be associated many-fermion states with negentropic entanglement (NE) - this notion relate to cognition and conscious information and is something totally new from standard physics point of view. This inspires also a conjecture about a deep connection with arithmetic consciousness: polygons would define conscious representations for integers n>2. The splicings of polygons to smaller ones could be dynamical quantum processes behind arithmetic conscious processes involving addition.
For details see the chapter Unified Number Theoretical Vision or the article Number Theoretical Feats and TGD Inspired Theory of Consciousness.
Mersenne primes are central in TGD based world view. p-Adic thermodynamics combined with p-adic length scale hypothesis stating that primes near powers of two are physically preferred provides a nice understanding of elementary particle mass spectrum. Mersenne primes Mk=2k-1, where also k must be prime, seem to be preferred. Mersenne prime labels hadronic mass scale (there is now evidence from LHC for two new hadronic physics labelled by Mersenne and Gaussian Mersenne), and weak mass scale. Also electron and tau lepton are labelled by Mersenne prime. Also Gaussian Mersennes MG,k=(1+i)k-1 seem to be important. Muon is labelled by Gaussian Mersenne and the range of length scales between cell membrane thickness and size of cell nucleus contains 4 Gaussian Mersennes!
What gives Mersenne primes so special physical status in TGD Universe? I have considered this problem many times during years. The key idea is that natural selection is realized in much more general sense than usually thought, and has chosen them and corresponding p-adic length scales. Particles characterized by p-adic length scales should be stable in some well-defined sense.
Since evolution in TGD corresponds to generation of information, the obvious guess is that Mersenne primes are information theoretically special. Could the fact that 2k-1 represents almost k bits be of significance? Or could Mersenne primes characterize systems, which are information theoretically especially stable?
In the following a more refined TGD inspired quantum information theoretic argument based on stability of entanglement against state function reduction, which would be fundamental process governed by Negentropy Maximization Principle (NMP) and requiring no human observer, will be discussed.
How to achieve stability against state function reductions?
TGD provides actually several ideas about how to achieve stability against state function reductions. This stability would be of course marvellous fact from the point of view of quantum computation since it would make possible stable quantum information storage. Also living systems could apply this kind of storage mechanism.
How to realize Mk=2k-1-dimensional Hilbert space physically?
One can imagine at least three physical realizations of Mk=2k-1-dimensional Hilbert space.
Returning to the original question "Why Mersenne primes are so special?". A possible explanation is that elementary particle or hadron characterized by a p-adic length scale p= Mk=2k-1 both stores and processes information with maximal effectiveness. This would not be surprising if p-adic physics defines the physical correlates of cognition assumed to be universal rather than being restricted to human brain.
In adelic physics p-dimensional Hilbert space could be naturally associated with the p-adic adelic sector of the system. Information storage could take place in p=Mk=2k-1 phase and information processing (cognition) would take place in 2k-dimensional state space. This state space would be reached in a phase transition p=2k-1→ 2 changing effective p-adic topology in real sector and genuine p-adic topology in p-adic sector and replacing padic length scale ∝ p1/2≈ 2k/2 with k-nary 2-adic length scale ∝ 2k/2.
Electron is characterized by the largest not completely super-astrophysical Mersenne prime M127 and corresponds to k=127 bits. Intriguingly, the secondary p-adic time scale of electron corresponds to .1 seconds defining the fundamental biorhythm of 10 Hz.
This proposal suffers from deficiencies. It does not explain why also Gaussian Mersennes are special. Gaussian Mersennes correspond ordinary primes near power of 2 but not so near as Mersenne primes are. Neither does it explain why also more general primes p≈ 2k seem to be preferred. Furthermore, p-adic length scale hypothesis generalizes and states that primes near powers of at least small primes q: p≈ qk are special at least number theoretically. For instance, q=3 seems to be important for music experience and also q=5 might be important (Golden Mean). .
Could the proposed model relying on criticality generalize. There would be p<2k-dimensional state space allowing isometric imbedding to 2k-dimensional space such that the bit configurations orthogonal to the image would be unstable in some sense. Say against a phase transition changing the direction of magnetization. One can imagine the variants of above described mechanism also now. For q>2 one should consider pinary digits instead of bits but the same arguments would apply (except in the case of Boolean logic).