Some TGD inspired questions and thoughts about hyperfinite factors of type II1

I have had a very interesting discussion with Baba Ilya Iyo Azza about von Neumann algebras. I have a background of physicists and have suffered a lot of frustration in trying to understand hyperfinite factors of type II₁ (HFFs) by trying to read mathematicians' articles.

I cannot understand without a physical interpretation and associations to my own big vision TGD. Yesterday I again stared at the basic definitions, ideas and concepts trying to build a physical interpretation. I try to summarize what I possibly understood.

1. One starts from the algebra of bounded operators in Hilbert space B(H). von Neumann algebra is a subalgebra of B(H). Already here an analog of inclusion is involved (see this). There are also inclusions between von Neumann algebras.

   What could the inclusion of von Neumann algebra to B(H) as subalgebra mean physically?

2. In the TGD framework, I can find several analogies. Space-time is a 4-surface in H=M⁴× CP₂: analog of inclusion reducing degrees of freedom. Space-time is not only an extremal of an action, but also satisfies holography so that this 4-surface is almost uniquely defined by a 3-surface. I talk about preferred extremals (PEs).

   Clearly, there is an analogy with von Neumann algebras, in particular HFFs with extremely nice mathematical properties, as a subalgebra of B(H) and quantum classical correspondence suggests that this analogy is not accidental.

The notion of the commutant M' of M is essential. Also M' defines HFF.What could be its physical interpretation?

1. In TGD, one has indeed an excellent candidate for the commutant. Supersymplectic symmetry algebra (SSA) of δ M⁴₊× CP₂ (δ M⁴₊ denotes the boundary of a future directed light-cone) is proposed to act as isometries of the "world of classical worlds" (WCW) consisting of space-time surfaces as PEs (very, very roughly).

   Symplectic symmetries are generated by Hamiltonians, which are products of Hamiltonians associated with δ M⁴₊ (metrically sphere S²) and CP₂. Symplectic symmetries are conjectured to act as isometries of WCW and gamma matrices of WCW extend symplectic symmetries to super-symplectic ones.

   Hamiltonians and their super-counterparts generate the super-symplectic algebra (SSA) and quantum states are created by using them. SSA is a candidate for HFF. Call it M. What about M ?

2. The symplectic symmetries leave invariant the induced Kähler forms of CP₂ and contact form of δ M⁴₊ (assignable to the analog of Kähler structure in M⁴).
3. The wave functions in WCW depending of magnetic fluxes defined by these Kähler forms over 2-surfaces are physically observables which commute SSA and with M. These fluxes are in a central role in the classical view about TGD and define what might perhaps be regarded as a dual description necessary to interpret quantum measurements.

   Could M' correspond or at least include the WCW wave functions (actually the scalar parts multiplying WCW spinor fields with WCW spinor for given 4-surface a fermionic Fock state) depending on these fluxes only? I have previously talked of these degrees of freedom as zero modes commuting with quantum degrees of freedom and of quantum classical correspondence.

4. Note that there are also number theoretic degrees of freedom, which naturally appear from the number theoretic M⁸ description mapped to H-description: Galois groups and their representations, etc...

There are further algebraic notions involved. The article of John Baez (see this) describes these notions nicely.

1. The condition M'' = M is a defining algebraic condition for von Neumann algebras. What does this mean? Or what could its failure mean? Could M'' be larger than M? It would seem that this condition is achieved by replacing M with M''.

   M''=M codes algebraically the notion of weak continuity, which is motivated by the idea that functions of operators obtained by replacing classical observable by its quantum counterpart are also observables. This requires the notion of continuity. Every sequence of operators must approach an operator belonging to the von Neumann algebra and this can be required in a weak sense, that is for matrix elements of the operators.

2. There is also the notion of hermitian conjugation defined by an antiunitary operator J: a^†= JAJ. This operator is absolutely essential in quantum theory and in the TGD framework it is geometrized in terms of the Kä form of WCW. The idea is that entire quantum theory, rather than only gravitation or gravitation and gauge interactions should be geometrized. Left multiplication by JaJ corresponds to right multiplication by a.
3. The notion of factor as a building brick of more complex structures is also central and analogous to the notion of simple group or prime. It corresponds to a von Neumann algebra, which is simple in the sense that it has a trivial center consisting of multiples of unit operators. The algebra is direct sum or integral over different factors.
4. A highly non-intuitive and non-trivial axiom relating to HFFs is that the trace of the unit operator equals to 1. The intuitive idea is that the density matrix for an infinite-D system identified as a unit operator gives as its trace total probability equal to one. These factors emerge naturally for free fermions. For factors of type I associated with three bosons, the trace equals n in the n-D case and ∞ in the infinite-D case.

   The factors of type I_∞ are tensor products of factors of type I and HFFs and could describe free bosons and fermions.

   In quantum field theory (QFT), factors of type III appear and in this case the notion of trace becomes useless. These factors are pathological and in QFT they lead to divergence difficulties. The physical reason is the idea about point-like particles, leading in scattering amplitudes to powers of delta functions having no mathematical meaning. In the TGD framework, the generalization of a point-like particle to 3-surface saves from these difficulties and leads to factors of type I and HFFs.

   Measurement resolution implies unique number theoretical discretization and further simplifies the situation in the TGD framework. In particular, "hyperfinite" expresses the fact that the approximation of a factor with its finite-D cutoff is an excellent approximation.

One cannot avoid philosophical considerations related to the interpretations of quantum measurement theory. The standard interpretations are known to lead to problems in the case of HFFs.

1. An important aspect related to the probabilistic interpretation is that physical states are characterized by a density matrix so that quantum theory reduces to probability theory, which would become in some sense non-commutative for von Neumann algebras.

   The problem is that no pure normal states as counterparts of quantum states do not exist for HFFs. Furthermore, the phenomenon of interference central in quantum theory does not have a direct description. One can of course argue that in practice the system studied is entangled with the environment and that this forces the description in terms of a density matrix even when pure states are realized at the fundamental level.

2. TGD strongly suggests the generalization of the state as density matrix to a "complex square root" of density matrix proportional to exponent of a real valued Kähler function of WCW identified as Kähler action for the space-time region as a preferred extremal and a phase factor defined by the analog of of action exponential. The quantum state would be proportional to an exponent of Kähler function of WCW identified as Kähler action for space-time surface as a preferred extrema.
3. There are also problems with the interpretations of quantum theory, which actually strongly suggest that something is badly wrong with the standard ontology.

   This requires a generalization of quantum measurement theory (see this and this) based on zero energy ontology (ZEO) and Negentropy Maximization Principle (NMP) \cite{allb/nmpc}. The key motivation is that ZEO is implied by an almost exact holography forced by general coordinate invariance for space-times as 4-surface. That holography and, as a consequence, classical determinism are not quite exact, has important implications for the understanding of the space-time correlates of cognition and intentionality in the TGD framework.

   In the TGD framework, the basic postulate is that quantum measurement as a reduction of entanglement can in principle occur for any entangled system pair.

Consider now the standard construction leading to a hierarchy of HFFs and their inclusions.

1. One starts from an inclusion M⊂ N of HFFs. I will later consider what these algebras could be in the TGD framework.
2. One introduces the spaces L²(M) resp. L²(N) of square integrable functions in M resp. N.

   From the physics point of view, bringing in L² is something extremely non-trivial. Space is replaced with wave functions in space: this corresponds to what is done in wave mechanics, that is quantization! One quantizes in M, particles as points of M are replaced by wave functions in M, one might say.

3. At the next step one introduces the projection operator e as a projection from L²(N) to L²(M): this is like projecting wave functions in N to wave functions in M. I wish I could really understand the physical meaning of this. The induction procedure for second quantized spinor fields in H to the space-time surface by restriction is completely analogous to this procedure.

   After that one generates a HFF as an algebra generated by e and L²(N): call it < L²(N), e>. One has now 3 HFFs and their inclusions: M₀== M, M₁== N, and < L²(N), e>== M₂.

   An interesting question is whether this process could generalize to the level of induced spinor fields?

4. Even this is not enough! One constructs L²(M₂)== M₃ including M₂. One can continue this indefinitely. Physically this means a repeated quantization.

   One could ask whether one could build a hierarchy M₀, L²(M₀),..., L²(L²...(M₀))..): why is this not done?

   The hierarchy of projectors e_i to M_i defines what is called Temperley-Lieb algebra involving quantum phase q=exp(iπ/n) as a parameter. This algebra resembles that of S_∞ but differs from it in that one has projectors instead of group elements. For the braid group e_i²=1 is replaced with a sum of terms proportional to e_i and unit matrix: mixture of projector and permutation is in question.

5. There is a fascinating connection in TGD and theory of consciousness. The construction of what I call infinite primes (see this) is structurally like repeated second quantization of a supersymmetric arithmetic quantum field theory involving fermions and bosons labelled by the primes of a given level I conjectured that it corresponds physically to quantum theory in the manysheeted space-time.

   Also an interpretation in terms of a hierarchy of statements about statements about .... bringing in mind hierarchy of logics comes to mind. Cognition involves generation of reflective levels and this could have the quantization in the proposed sense as a quantum physical correlate.

Connes tensor product is natural for modules having algebra as coefficients. For instance, matrix multiplication has an interpretation as Connes tensor product reduct tensor product of matrices to a matrix product. The number of degrees of freedom is reduced.

1. Inclusion of Galois group algebra of extension to its extension could define Connes tensor product. Composite polynomial instead of product of polynomials: this would describe interaction physically: the degree of composite is product of degrees of factors and the same holds true for the product of polynomials. This rule for the dimensions holds also for the tensor product. Composite structure implies correlations and formation of bound states so that the number of degrees of freedom is reduced.
2. Also the inclusion SSA_n+1 to SSA_n should define Connes tensor product. Note that the inclusions are in different directions. Could it be that these two inclusion sequences correspond to the sequences assignable to M and M'?

   What about the already mentioned "classical" degrees of freedom associated with the fluxes of the induced Kähler form? Should one include the additional degrees of freedom to M' or are they dual to the number theoretic degrees of freedom assignable to Galois groups. How does the M⁸-H duality, relating number theoretic and geometric descriptions in analogy with Langlands duality, relate to this?

I do not have an intuitive grasp about category theory. In any case, one would have a so-called 2-category (see this). M and N correspond to 0-morphisms (objects). One can multiply L²(M) and L²(N) by M or N. The bimodules _ML²(M)_M, _NL²(N)_N correspond to 1-morphisms which are units whereas the bimodules _MM_N, and _NM_M correspond to generating 1-morphisms mapping M into N. Bimodule map corresponds to 2-morphisms. Connes tensor product defines a tensor functor.

Extended ADE Dynkin diagrams for ADE Lie groups, which correspond to finite subgroups of SU(2) by McKay correspondence, characterize inclusions of HFFs. For a subset of ADE groups not containing E₇ and D_2n+1, there are inclusions, which correspond to Dynkin diagrams of finite subgroups of the quantum group SU(2)_q. What is interesting that E₆ (tetrahedron) and E₈ (icosahedron/dodecahedron) appear in the TGD based model of bioharmony and genetic code but not E₇ (see this).

1. Why finite subgroups of SU(2) (or SU(2)_q) should appear as characterizers of the inclusions in the tunnel hierarchies with the same value of the quantum dimension M_n+1:M_n of quantum algebra.

   In the TGD interpretation M_n+1 reduces to a tensor product of M_n and quantum group, when M_n represents reduced measurement resolution and quantum group the added degrees of freedom. Quantum groups would represent the reduced degrees of freedom. This has a number theoretical counterpart in terms of finite measurement resolution obtained when an extension....of rationals is reduced to a smaller extension. The braided relative Galois group would represent the new degrees of freedom.

2. The identification of HFF as tensor product of GL(2,c) or GL(n,C) and the identification as analog of McKay graph for the irreps of a closed subgroup defines an invariant characterizing the fusion rules involved with the reduction of the tensor product is involved but I do not really understand this. What comes to mind is that all the essential features of tensor products of HFFs reduce to tensor products of finite subgroups of SU(2) or of SU(2)_q.
3. In the TGD framework, SU(2) could correspond to a covering group of quaternionic automorphisms and number theoretic discretization (cognitive representations) would naturally lead to discrete and finite subgroups of SU(2).

What could HFFs correspond to in TGD?

1. Braid group B(G) of group (say Galois group as subgroup of S_n) and its group algebra would correspond to B(G) and L²(B(G)).
2. Braided Galois group and its group algebra could correspond to B(G) and L²(B(G)). Composite polynomials define hierarchies of Galois groups such that the included Galois group is a normal subgroup. This kind of hierarchy could define an increasing sequence of inclusions of braided Galois groups.
3. Elements of SSA are labelled by non-negative integers. One can construct a hierarchy of subalgebras SSA_n , such that elements with large conformal weight annihilate the physical state and also their commutators with SSA_n do this. SSA_n+1 is included by SSA_n and one has a kind of reversed sequence of inclusions.
4. Braided Galois groups and a hierarchy of SSA_n could correspond to commuting algebras M and M'.
5. Question: Does the standard construction bring in something totally new to these hierarchies or is the resulting structure equivalent with that given by the standard construction?

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

Articles related to TGD.