# Physics as a Generalized Number Theory

Note: Newest contributions are at the top!

 Year 2016

### p-Adic logic and hierarchy of partition algebras

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.

1. The basic question is how to define logical negation. Since 2-Boolean algebra is imbeddable to any p-Boolean algebra, it is natural to require that also in p-Boolean case the operation permute 0 and 1. These elements are also preferred elements algebraically since they are neutral elements for sum and product. This condition could be satisfied by simply defining negation as an operation leaving other elements of G(p) un-affected. An alternative definition would be as shift k→ k-1. This is an attractive option since it corresponds to a cyclic symmetry.For G(p) also higher powers of this operation would define analogs of negation in accordance with p-valuedness.

I have considered also the possibility that for p>2 the analog of logical negation could be defined as an additive inverse k→ p-k in G(p) and k=p-1 would be mapped to k=1 as one might expect. The non-allowed value k=0 is mapped to k=p=0. k=0 would be its own negation. This would suggest that k=0 corresponds to an ill-defined truth value for p>2. For p=2 k=0 must however correspond to false. This option is not however consistent with category theory inspired thinking.

2. For G(p)-valued functions f, one can define the p-analogs of both XOR (excluded or [(A OR B) but not (A AND B)] and AND using local sum and product for the everywhere-non-vanishing G(p)-valued functions. One can also define the analog of OR in terms of f1+f2-f1f2 for arbitrary G(p)-valued functions. Note that minus sign is essential as one can see by considering p=3 case (1+1-1× 1=1 and 1+1+1× 1=0). For p=2 this would give ordinary OR and it would be obviously non-vanishing unless both functions are identically zero. For p>2 A OR B defined in this manner f1 +f2- f1f2 for functions having no zeros can however have zeros. The mod p1 projection from G(p)→ G(p1) indeed commutes with these operations.

Could 3-logic with 0 interpreted as ill-defined logical value serve as a representation of Boolean logic? This is not the case: 1× 2=2 would correspond to 1× 0=0 but 2× 2=1 does not correspond to 0× 0=0.

3. It would be nice to have well-defined inverse of Boolean function giving additional algebra structure for the partitions. For non-vanishing values of f(x) one would have (1/f)(x)=1/f(x). How to define (1/f)(x) for f(x)=0? One can consider three options.
1. Option I: If 0 is interpreted as ill-defined value of p-Boolean function, there is a temptation to argue that the value of 1/f is also ill defined: (1/f)(x)=0 for f(x)=0. That function values would be replaced with their inverses only at points, where they are no-vanishing would conform with how ill-defined Boolean values are treated in computation. This leads to a well-defined algebra structure but the inverse defined in this manner is only local inverse. One has f (f-1) (x))=1 only for f(x)>0. One has algebra but not a field.
2. Option II: One could consider the extension of G(p) by the inverse of 0, call it ∞, satisfying 0× ∞=1 ("false" AND ∞ = "true"!). Arithmetic intuition would suggest k× ∞ = ∞ for k>0 and k+∞ = ∞ for all k.

On the other hand, the interpretation of + as XOR would suggest that k+∞ corresponds to [(k OR ∞) but not (k AND ∞)=∞] suggesting k+∞= k so that 0 and ∞ would be in completely symmetrical position with respect to product and sum (k+∞=k and k+0=k; k× ∞=∞ and k× 0=0). It would be nice to have a logical interpretation for the inverse and for the element ∞. Especially so in 2-Boolean case. A plausible looking interpretation of ∞ would be as "ill-defined" implying that k OR ∞ and k AND ∞ is also "ill-defined". ["false" AND "ill-defined"]="true" sounds however strange.

For a set with N elements this would give a genuine field with (p+1)N elements. For the more convincing arithmetic option the outcome is completely analogous to the addition of point ∞ to real or complex numbers.

3. Option III: One could also consider functions, which are non-vanishing at all points of the set are allowed. This function space is not however closed under summation.
4. For these three options one would have K(N)=pN, K(N)=(p+1)N and K(N)=(p-1)N different maps of this kind having additive and multiplicative inverses. This hierarchy of statements about statements continues ad infinitum with K(n)=K(K(n-1)). For Option II this gives K(n)= (p+1)K(n-1) so that one does not obtain finite field G(p,N) with pN elements but function field.
5. One can also consider maps for which values are in the range 0<k<p. This set of maps would be however closed with respect to OR and would not obtain hierarchy of finite fields. In this case the interpretation of 0 would be is un-determined and for p=2 this option would be trivial. For p=3 one would have effectively two well-defined logic values but the algebra would not be equivalent with ordinary Boolean algebra.
The outcome for Option II would be a very nice algebraic structure having also geometric interpretation possibly interesting from the point of view of logic. p-Boolean algebra provides p-partitions with generalizations of XOR, OR, AND, negation, and finite field structure at each level of the hierarchy: kind of calculus for p-partitions.

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.

See the chapter Infinite Primes and Motives or the article Boolean algebra, Stone spaces, and TGD.

### Brain and Mersenne integers

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.

1. One can consider a situation in which one has n inputs identifiable as bits: bit could correspond to neuron firing or not. The question is however to classify various input combinations. The obvious criterion is how many bits are equal to 1 (corresponding neuron fires). The input combinations in the same class have same number of firing neurons and the number of subsets with k elements is given by the binomial coefficient B(n,k)= n!/k!(n-k)!. There are clearly n-1 different classes in the classification since no neurons firing is not a possible observation. The conceptualization would tell how many neurons fire but would not specify which of them.
2. To represent these bit combinations one needs 2n-1 neuron groups acting as unit representing one particular firing combination. These subsets with k elements would be mapped to neuron cliques with k firing neutrons. For given input individual firing neurons (k=1) would represent features, lowest level information. The n cliques with k=2 neurons would represent a more general classification of input. One obtains Mn=2n-1 combinations of firing neurons since the situations in which no neurons are firing is not counted as an input.
3. If all neurons are firing then all the however level cliques are also activated. Set theoretically the subsets of set partially ordered by the number of elements form an inclusion hierarchy, which in Boolean algebra corresponds to the hierarchy of implications in opposite direction. The clique with all neurons firing correspond to the most general statement implying all the lower level statements. At k:th level of hierarchy the statements are inconsistent so that one has B(n,k) disjoint classes.
The Mn=2n-1 (Mersenne number) labelling the algorithm is more than familiar to me.
1. For instance, electron's p-adic prime corresponds to Mersenne prime M127 =2127-1, the largest not completely super-astrophysical Mersenne prime for which the mass of particle would be extremely small. Hadron physics corresponds to M107 and M89 to weak bosons and possible scaled up variant of hadron physics with mass scale scaled up by a factor 512 (=2(107-89)/2). Also Gaussian Mersennes seem to be physically important: for instance, muon and also nuclear physics corresponds to MG,n= (1+i)n-1, n=113.
2. In biology the Mersenne prime M7= 27-1 is especially interesting. The number of statements in Boolean algebra of 7 bits is 128 and the number of statements that are consistent with given atomic statement (one bit fixed) is 26= 64. This is the number of genetic codons which suggests that the letters of code represent 2 bits. As a matter of fact, the so called Combinatorial Hierarchy M(n)= MM(n-1) consists of Mersenne primes n=3,7,127, 2127-1 and would have an interpretation as a hierarchy of statements about statements about ... It is now known whether the hierarchy continues beyond M127 and what it means if it does not continue. One can ask whether M127 defines a higher level code - memetic code as I have called it - and realizable in terms of DNA codon sequences of 21 codons (see this).
3. The Gaussian Mersennes MG,n n=151,157,163,167, can be regarded as a number theoretical miracles since the these primes are so near to each other. They correspond to p-adic length scales varying between cell membrane thickness 10 nm and cell nucleus size 2.5 μm and should be of fundamental importance in biology. I have proposed that p-adically scaled down variants of hadron physics and perhaps also weak interaction physics are associated with them.
I have made attempts to understand why Mersenne primes Mn and more generally primes near powers of 2 seem to be so important physically in TGD Universe.
1. The states formed from n fermions form a Boolean algebra with 2n elements, but one of the elements is vacuum state and could be argued to be non-realizable. Hence Mersenne number Mn=2n-1. The realization as algebra of subsets contains empty set, which is also physically non-realizable. Mersenne primes are especially interesting as sine the reduction of statements to prime nearest to Mn corresponds to the number Mn-1 of physically representable Boolean statements.
2. Quantum information theory suggests itself as explanation for the importance of Mersenne primes since Mn would correspond the number of physically representable Boolean statements of a Boolean algebra with n-elements. The prime p≤ Mn could represent the number of elements of Boolean algebra representable p-adically (see this).
3. In TGD Fermion Fock states basis has interpretation as elements of quantum Boolean algebra and fermionic zero energy states in ZEO expressible as superpositions of pairs of states with same net fermion numbers can be interpreted as logical implications. WCW spinor structure would define quantum Boolean logic as "square root of Kähler geometry". This Boolean algebra would be infinite-dimensional and the above classification for the abstractness of concept by the number of elements in subset would correspond to similar classification by fermion number. One could say that bosonic degrees of freedom (the geometry of 3-surfaces) represent sensory world and spinor structure (many-fermion states) represent that logical thought in quantum sense.
4. Fermion number conservation would seem to represent an obstacle but in ZEO it can circumvented since zero energy states can be superpositions of pair of states with opposite fermion number F at opposite boundaries of causal diamond (CD) in such a manner that F varies. In state function reduction however localization to single value of F is expected to happen usually. If superconductors carry coherent states of Cooper pairs, fermion number for them is ill defined and this makes sense in ZEO but not in standard ontology unless one gives up the super-selection rule that fermion number of quantum states is well-defined.
One can of course ask whether primes n defining Mersenne primes (see this) could define preferred numbers of inputs for subsystems of neurons. This would predict n=2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257,.. define favoured numbers of inputs. n=127 would correspond to memetic code.

See the article Why Mersenne Primes Are So Special? or the chapter Unified Number Theoretical Vision.

### Boolean algebras, Stone spaces, and p-adic physics

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.

See the chapter Infinite Primes and Motives or the article Boolean algebras, Stone spaces, and TGD.

### Langlands Program and TGD: Years Later

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?

See the chapter Langlands correspondence and TGD: years later or the article with the same title.

### p-Adicizable discrete variants of classical Lie groups and coset spaces in TGD framework

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 and TGD Inspired Theory of Consciousness

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.

### Why Mersenne primes are so special?

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.

1. p-Adic physics leads to the notion of negentropic entanglement (NE) for which number theoretic entanglement entropy is negative and thus measures genuine, possibly conscious information assignable to entanglement (ordinary entanglement entropy measures the lack of information about the state of either entangled system). NMP favors the generation of NE. NE can be however transferred from system to another (stolen using less diplomatically correct expression!), and this kind of transfer is associated with metabolism. This kind of transfer would be the most fundamental crime: biology would be basically criminal activity! Religious thinker might talk about original sin.

In living matter NE would make possible information storage. In fact, TGD inspired theory of consciousness constructed as a generalization of quantum measurement theory in Zero Energy Ontology (ZEO) identifies the permanent self of living system (replaced with a more negentropic one in biological death, which is also a reincarnation) as the boundary of CD, which is not affected in subsequent state function reductions and carries NE. The changing part of self - sensory input and cognition - can be assigned with opposite changing boundary of CD.

2. Also number theoretic stability can be considered. Suppose that one can assign to the system some extension of algebraic numbers characterizing the WCW coordinates ("world of classical worlds") parametrizing the space-time surface (by strong form of holography (SH) the string world sheets and partonic 2-surfaces continuable to 4-D preferred extremal) associated with it.

This extension of rationals and corresponding algebraic extensions of p-adic numbers would define the number fields defining the coefficient fields of Hilbert spaces (it might be necessary to assume that the coefficients belong to the extension of rationals also in p-adic sector although they can be regarded as p-adic numbers). Assume that you have an entangled system with entanglement coefficients in this number field. Suppose you want to diagonalize the corresponding density matrix. The eigenvalues belong in general case to a larger algebraic extension since they correspond to roots of a characteristic polynomials assignable to the density matrix. Could one say, that this kind of entanglement is stable (at least to some degree) against state function reduction since it means going to an eigenstate which does not belong to the extension used? Reader can decide!

3. Hilbert spaces are like natural numbers with respect to direct sum and tensor product. The dimension of the tensor product is product mn of the dimensions of the tensor factors. Hilbert space with dimension n can be decomposed to a tensor product of prime Hilbert spaces with dimensions which are prime factors of n. In TGD Universe state function reduction is a dynamical process, which implies that the states in state spaces with prime valued dimension are stable against state function reduction since one cannot even speak about tensor product decomposition, entanglement, or reduction of entanglement. These state spaces are quantum indecomposable and would be thus ideal for the storage of quantum information!

Interestingly, the system consisting of k qubits have Hilbert space dimension D=2k and is thus maximally unstable against decomposition to D=2-dimensional tensor factors! In TGD Universe NE might save the situation. Could one imagine a situation in which Hilbert space with dimension Mk=2k-1 stores the information stably? When information is processed this state space would be mapped isometrically to 2k-dimensional state space making possible quantum computations using qubits. The outcome of state function reduction halting the computation would be mapped isometrically back to Mk-D space. Note that isometric maps generalizing unitary transformations are an essential element in the proposal for the tensor net realization of holography and error correcting codes (see this).

Can one imagine any concrete realization for this idea? This question will be considered in the sequel.

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.

1. The set with k elements has 2k subsets. One of them is empty set and cannot be physically realized. Here the reader might of course argue that if they are realized as empty boxes, one can realize them. If empty set has no physical realization, the wave functions in the set of non-empty subsets with 2k-1 elements define 2k-1-dimensional Hilbert space. If 2k-1 is Mersenne prime, this state state space is stable against state function reductions since one cannot even speak about entanglement!

To make quantum computation possible one must map this state space to 2k dimensional state space by isometric imbedding. This is possible by just adding a new element to the set and considering only wave functions in the set of subsets containing this new element. Now also the empty set is mapped to a set containing only this new element and thus belongs to the state space. One has 2k dimensions and quantum computations are possible. When the computation halts, one just removes this new element from the system, and the data are stored stably!

2. Second realization relies on k bits represented as spins such that 2k-1 is Mersenne prime. Suppose that the ground state is spontaneously magnetized state with k+l parallel spins, with the l spins in the direction of spontaneous magnetization and stabilizing it. l>1 is probably needed to stabilize the direction of magnetization: l ≤ k suggests itself as the first guess. Here thermodynamics and a model for spin-spin interaction would give a better estimate.

The state with the k spins in direction opposite to that for l spins would be analogous to empty set. Spontaneous magnetization disappears, when a sufficient number of spins is in direction opposite to that of magnetization. Suppose that k corresponds to a critical number of spins in the sense that spontaneous magnetization occurs for this number of parallel spins. Quantum superpositions of 2k-1 states for k spins would be stable against state function reduction also now.

The transformation of the data to a processable form would require an addition of m≥ 1 spins in the direction of the magnetization to guarantee that the state with all k spins in direction opposite to the spontaneous magnetization does not induce spontaneous magnetization in opposite direction. Note that these additional stabilizing spins are classical and their direction could be kept fixed by a repeated state function reduction (Zeno effect). One would clearly have a critical system.

3. Third realization is suggested by TGD inspired view about Boolean consciousness. Boolean logic is represented by the Fock state basis of many-fermion states. Each fermion mode defines one bit: fermion in given mode is present or not. One obtains 2k states. These states have different fermion numbers and in ordinary positive energy ontology their realization is not possible.

In ZEO situation changes. Fermionic zero energy states are superpositions of pairs of states at opposite boundaries of CD such that the total quantum numbers are opposite. This applies to fermion number too. This allows to have time-like entanglement in which one has superposition of states for which fermion numbers at given boundary are different. This kind of states might be realized for super-conductors to which one at least formally assigns coherent state of Cooper pairs having ill-defined fermion number.

Now the non-realizable state would correspond to fermion vacuum analogous to empty set. Reader can of course argue that the bosonic degrees of freedom assignable to the space-time surface are still present. I defend this idea by saying that the purely bosonic state might be unstable or maybe even non-realizable as vacuum state and remind that also bosons in TGD framework consist of pairs of fundamental fermions.

If this state is effectively decoupled from the rest of the Universe, one has 2k-1-dimensional state space and states are stable against state function reduction. Information processing becomes possible by adding some positive energy fermions and corresponding negative energy fermions at the opposite boundaries of CD. Note that the added fermions do not have time-like quantum entanglement and do not change spin direction during time evolution.

The proposal is that Boolean consciousness is realized in this manner and zero energy states represents quantum Boolean thoughts as superposition of pairs (b1⊗ b2) of positive and negative energy states and having identification as Boolean statements b1→ b2. The mechanism would allow both storage of thoughts as memories and their processing by introducing the additional fermion.

So: why Mersenne primes would be so special?

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).

See the chapter Unified Number Theoretic Vision or the article Why Mersenne primes are so special?.