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.
- 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.
- 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!
- 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.
- 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!
- 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.
- 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.
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).
For a summary of earlier postings see Links to the latest progress in TGD.