MIP*=RE: What it could possibly mean?

I receive a very interesting link to a popular article explaining a recently discovered deep result in mathematics having implications also in physics. The article by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen has a rather concise title “ MIP*=RE” . In the following I try to express the impressions of physcist about the result.

1. RE (recursively enumerable languages) denotes all problems solvable by computer. P denotes the problems solvable in polynomial time. NP does not refer to non-polynomial time but to “non-deterministic polynomial acceptable problems” – I hope this helps the reader! It is not known whether P = NP is true.

2. IP problems that can be solved in collaboration of interrogator and prover who tries to convince interrogator that her proof is convincing with high enough probability. MIP involves several provers treated like crimicals trying to prove that they are innocent and being not allowed to communicate. MIP* is the class of solvable problems in which provers are allowed to entangle.

The finding, which is characterizedas shocking, is that all problems solvable by a Turing computer belong to this class: RE= MIP*. All problems solvable by computer would reduces to problems in MIP*. Quantum computation could indeed give something genuinely new to the classical computation.

Connes embedding problem and the notion of finite measurement/cognitive resolution

Alain Connes formulate Connes embedding problem. The question is whether infinite matrices forming factor of type II₁ can be always approximated by finite-D matrices that is imbedded in a hyperfinite factor of type II₁ (HFF). Factors of type II> and their hyperfinite counterpart are special class von Neumann algebras possibly relevant for quantum theory this is the case by definition.

Here comes the first connection with TGD. There are intuitive physicist’s arguments demonstrating that in TGD the operator algebras involved with TGD are HFFs provides a description of finite measurement resolution. The inclusion of HFFs defines the notion of resolution: included factor represents the degrees of freedom not seen in the resolution used (see this and this).

Second TGD based approach to finite resolution is purely number theoretic and involves adelic physics as a fusion real physics with various p-adic physics as correlates of cognition. Cognitive representations are purely number theoretic ad unique discretizations of space-time surfaces defined by a given extension of rationals forming an evolutionary hierarchy. These to views should be closely related: in particular the hierarchy of extensions of rationals should define a hierarchy of inclusions of HFFs.

Hyperfinite factors involve new structures like quantum groups and quantum algebras reflecting the presence of additional symmetries: actually the “world of classical worlds” (WCW) as the space of space-time surfaces as maximal group of isometries and this group has a fractal hierarchy of isomorphic groups imply inclusion hierarchies of HFFs. By analogs of gauge conditions this infinite-D group reduces to a hierarchy of effectively finite-D groups. For quantum groups the infinite number of irreps of compact group effectively reduces to a finite number of them.

Does the result mean that the reduction of most general quantum theory to TGD like theory relying on HFFs is not possible? This would not be surprising taking into account gigantic symmetries responsible for the cancellation of infinities in TGD framework and the very existence of WCW geometry. The suggestive conclusion is that for hyperfinite factors the analog of MIP*=RE cannot hold true: TGD Universe would not allow to solve all problems solvable by Turing Computer.

Here however a word of warning is in question.

1. The ordinary computational paradigm is formulated for Turing machines manipulating natural numbers by recursive algorithms. Programs would essentially represent a recursive function n→ f(n). What happens to this paradigm when extensions of rationals define cognitive representations as unique space-time discretizations with algebraic numbers as the limit giving rise to a dense in the set of reals. Can one naively apply the results deduced for the Turing machine.

Extensions of rationals form discrete structures but it is not at all clear whether the Turing machine paradigm formulated for natural numbers generalizes straight-forwardly for algebraic numbers. Also all p-adic number fields are included in the Adele associated with a given extension of rationals besides reals. Adelic theorem suggests a wild generalization: all these extensions of all these p-adic numbers fields restricted to extension of rationals describe real physics. Real physics is understable by cognition if there is enough time.

Tsirelson problem (see this) is another problem mentioned in the popular article as a physically interesting application. The problem relates to the mathematical description of quantum measurement.

Three systems are considered. There are two systems O₁ and O₂ representing observers and the third representing the measured system M. The measurement reducing the entanglement between M and O₁ or O₂ can regarded as producing correspondence between state of M and O₁ or O2, and one can think that O₁ or O₂ measures only obserservables in its own state space as a kind of image of M.

There are two manners to see the situation. The provers correspond not now to the observers and the two situations correspond to provers without and with entanglement.

1. One assumes that the state spaces H1 resp. H2 of O₁ resp. O₂ form a tensor product and that O₁ and O₂ choose their observables X₁ resp. X₂ independently (the provers cannot communicate). By repeating measurements correlation functions for the measurement outcomes by O₁ and O₂ are obtained as also probabilities for the various outcomes of the joint measurements interpreted as measurements of X₁ ⊗ X₂.
2. One can also think that the two systems form a single system O so that O₁ and O₂ can entangle. This corresponds to the situation in which entanglement between the provers is allowed. Now X₁ and X₂ cannot be independent but must be commuting observables. Also now one can construct a correlation function for the products X₁ X₂ of the commuting observers X₁ and X₂.

Are these manners to see the situation equivalent? Tsirelson demonstrated that this is the case for finite-dimensional Hilbert spaces, which can be indeed decomposed toa tensor product of factors associated with O₁ and O₂. For the infinite-dimensional case the situation remained open. According to the article, the new result implies that this is not the case. For hyperfinite factors the situation can be also approximated with a finite-D Hilbert space so that the situations are equivalent.

In TGD this would be the case if the hypothesis about HFF property is true.This does not mean that the entanglement between observers would not bring in anything new. Entanglement could make possible much faster problem solving and could allow to transform MIS-hard problems to problems solvable in polynomial time (I hope that this sentence makes sense!).

The HFF property however relates to a finite measurement resolution involving discretization at space-time level: does only cognitively finitely representable physics have the HFF property or is it a property of the full physics of continuum TGD? Is the continuum physics captured completely by the hierarchy of number theoretic discretizations. What is important is that continuum physics involves transcendentals and in mathematics this brings in analytic formulas and partial differential equations. At least at the level of mathematical consciousness the emergence of the notion of continuum means a gigantic step.

The quantization of induced spinors in TGD looks different in discrete and continuum cases. Discrete case is very simple since anticommutators give discrete Kronecker deltas. In the continuum case one has delta functions possibly causing infinite vacuum energy like divergences in conserved Noether charges (Dirac sea). In here I have proposed how these problems could be avoided by avoiding anticommutators giving delta-function. The solution is based on zero energy ontology and TGD based view about space-time.

This also relates in an interesting manner to consciousness. Quantum entanglement makes in the TGD framework possible telepathic sharing of mental images represented by sub-selves of self. For the series of discretizations of physics by HFFs and cognitive representations, this seems to mean nothing new. I have not yet decided whether I should feel disappointed or not. If HFF property is not true in the continuum picture then quantum entanglement between conscious entities could mean a new tool of conscious problem solving, which could be called transcendence.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.