Prime Hilbert spaces and infinite primes
Kea told in her blog about a result of quantum information science which seems to provide an additional reason why for p-adic physics.
Suppose that one has N-dimensional Hilbert space which allows N+1 mutually unbiased basis. This means that the moduli squared for the inner product of any two states belonging to different basis equals to 1/N. If one knows all transition amplitudes from a given state to all states of all N+1 mutually unbiased basis, one can fully reconstruct the state. For N=pn dimensional N+1 unbiased basis can be found and the article of Durt gives an explicit construction of these basis by applying the properties of finite fields. Thus state spaces with pn elements - which indeed emerge naturally in p-adic framework - would be optimal for quantum tomography. For instance, the discretization of one-dimensional line with length of pn units would give rise to pn-D Hilbert space of wave functions.
The observation motivates the introduction of prime Hilbert space as as a Hilbert space possessing dimension which is prime and it would seem that this kind of number theoretical structure for the category of Hilbert spaces is natural from the point of view of quantum information theory. One might ask whether the tensor product of mutually unbiased bases in the general case could be constructed as a tensor product for the bases for prime power factors. This can be done but since the bases cannot have common elements the number of unbiased basis obtained in this manner is equal to M+1, where M is the smallest prime power factor of N. It is not known whether additional unbiased bases exists.
1. Hierarchy of prime Hilbert spaces characterized by infinite primes
The notion of prime Hilbert space provides a new interpretation for infinite primes, which are in 1-1 correspondence with the states of a supersymmetric arithmetic QFT. The earlier interpretation was that the hierarchy of infinite primes corresponds to a hierarchy of quantum states. Infinite primes could also label a hierarchy of infinite-D prime Hilbert spaces with product and sum for infinite primes representing unfaitfully tensor product and direct sum.
- At the lowest level of hierarchy one could interpret infinite primes as homomorphisms of Hilbert spaces to generalized integers (tensor product and direct sum mapped to product and sum) obtained as direct sum of infinite-D Hilbert space and finite-D Hilbert space. (In)finite-D Hilbert space is (in)finite tensor product of prime power factors. The map of N-dimensional Hilbert space to the set of N-orthogonal states resulting in state function reduction maps it to N-element set and integer N. Hence one can interpret the homomorphism as giving rise to a kind of shadow on the wall of Plato's cave projecting (shadow quite literally!) the Hilbert space to generalized integer representing the shadow. In category theoretical setting one could perhaps see generalize integers as shadows of the hierarchy of Hilbert spaces.
- The interpretation as a decomposition of the universe to a subsystem plus environment does not seem to work since in this case one would have tensor product. Perhaps the decomposition could be to degrees of freedom to those which are above and below measurement resolution. Perhaps one should try to interpret physically the process of transferring degrees of freedom from tensor product to direct sum.
- The construction of these Hilbert spaces would reduce to that of infinite primes. The analog of the fermionic sea would be infinite-D Hilbert space which is tensor product of all prime Hilbert spaces Hp with given prime factor appearing only once in the tensor product. One can "add n bosons" to this state by replacing of any tensor factor Hp with its n+1:th tensor power. One can "add fermions" to this state by deleting some prime factors Hp from the tensor product and adding their tensor product as a finite-direct summand. One can also "add n bosons" to this factor.
- At the next level of hierarchy one would form infinite tensor product of all infinite-D prime Hilbert spaces obtained in this manner and repeat the construction. This can be continued ad infinitum and the construction corresponds to abstraction hierarchy or a hierarchy of statements about statements or a hierarchy of n:th order logics. Or a hierarchy of space-time sheets of many-sheeted space-time. Or a hierarchy of particles in which certain many-particle states at the previous level of hierarchy become particles at the new level (bosons and fermions). There are many interpretations.
- Note that at the lowest level this construction can be applied also to Riemann Zeta function. ζ would represent fermionic vacuum and the addition of fermions would correspond to a removal of a product of corresponding factors ζp from ζ and addition of them to the resulting truncated ζ function. The addition of bosons would correspond to multiplication by a power of appropriate ζp. At zeros of ζ the modified zeta functions reduce to their fermionic parts. The analog of ζ function at the next level of hierarchy would be product of all these modified ζ functions and probably fails to exist as a smooth function since the product would typically converge to either zero or infinity.
2. Hilbert spaces assignable to infinite integers and rationals make also sense
- Also infinite integers make sense since one can form tensor products and direct sums of infinite primes and of corresponding Hilbert spaces. Also infinite rationals exist and this raises the question what kind of state spaces inverses of infinite integers mean.
- Zero energy ontology suggests that infinite integers correspond to positive energy states and their inverses to negative energy states. Zero energy states would be always infinite rationals with real norm which equals to real unit.
- The existence of these units would give for a given real number an infinite rich number theoretic anatomy so that single space-time point might be able to represent quantum states of the entire universe in its anatomy (number theoretical Brahman=Atman).
Also the world of classical worlds (light-like 3-surfaces of the imbedding space) might be imbeddable to this anatomy so that basically one would have just space-time surfaces in 8-D space and configuration space would have representation in terms of space-time based on generalized notion of number. Note that infinitesimals around a given number would be replaced with infinite number of number-theoretically non-equivalent real units multiplying it.
3. Should one generalize the notion of von Neumann algebra?
Especially interesting are the implications of the notion of prime Hilbert space concerning the notion of von Neumann algebra -in particular the notion of hyper-finite factors of type II1 playing a key role in TGD framework. Does the prime decomposition bring in additional structure? Hyper-finite factors of type II1 are canonically represented as infinite tensor power of 2×2 matrix algebra having a representation as infinite-dimensional fermionic Fock oscillator algebra and allowing a natural interpretation in terms of spinors for the world of classical worlds having a representation as infinite-dimensional fermionic Fock space.
Infinite primes would correspond to something different: a tensor product of all p×p matrix algebras from which some factors are deleted and added back as direct summands. Besides this some factors are replaced with their tensor powers.
Should one refine the notion of von Neumann algebra so that one can distinguish between these algebras as physically non-equivalent? Is the full algebra tensor product of this kind of generalized hyper-finite factor and hyper-finite factor of type II1 corresponding to the vibrational degrees of freedom of 3-surface and fermionic degrees of freedom? Could p-adic length scale hypothesis - stating that the physically favored primes are near powers of 2 - relate somehow to the naturality of the inclusions of generalized von Neumann algebras to HFF of type II1?
For background see that chapter TGD as a Generalized Number Theory III: Infinite Primes.