The latest vision about the role of hyperfinite factors in TGD

I realized of the importance of von Neumann algebras known as hyper-finite factors for more than half decade ago. The mathematics involved is extremely heavy technically and for a physicist at my age and with my brain the only reasonable goal is to understand this notion conceptually and see whether it relates naturally to own visions. Fermionic Fock space finding geometrization in quantum TGD is indeed a canonical representation for HFFs of II₁ having very close relations to quantum groups, topological quantum field theories, statistical mechanics, etc.. so that there are excellent motivations for taking HFFs of various types seriously.

It is clear that at least the hyper-finite factors of type II₁ assignable to WCW (world of classical worlds) spinors must have a profound role in TGD. Whether also HFFS of type III₁ appearing in relativistic quantum field theories emerge when WCW spinors are replaced with spinor fields in WCW is not completely clear. I have proposed several ideas about the role of hyper-finite factors in TGD framework. In particular, inclusions of factors and Connes tensor product provide an excellent candidate for defining the notion of measurement resolution.

The perspective provided by zero energy ontology, the recent advances in the understanding of M-matrix at QFT limit using the notion of bosonic emergence, as well as the more mature view about what these mysterious factors are, motivate a fresh discussion of the subject. There are several question to be considered but before posing these questions it is good to give some references.

1. Basic notions related to factors

It would take too much text to explain the basic ideas and facts of factors so that I give links to references that I have used with comments.

1. There is Wikipedia article about von Neumann algebras. Hyper-finite factors of type II₁, II_∞ and type III₁ are of special importance from TGD point of view. HFFs of type III₁ are encountered also in quantum field theories.

2. The slides of Longo with title Operator Algebras and Index Theorems in Quantum Field Theory should be useful. In particular, it explains Tomita-Takesaki formula, which turned out to have very nice interpretation in zero energy ontology. The ordinary fermionic Fock space does not satisfy the conditions of the theorem (existence of cyclic and separable state) but the space formed by zero energy states naturally does so. Positive and negative energy parts of zero energy states for which vacua are annihilated by creation resp. annihilation operators corresponds to the decomposition of the space of bounded operators of Hilbert space to a "vee" product of factors and its commutant generalizing the tensor product in the description of mutually non-interacting systems.

3. The article Von Neumann algebra automorphisms and time- thermodynamics relation in general covariant quantum theories of Connes and Rovelli should give ideas about physical interpretation. The insights given by article allowed to realize that the hope about the identification of the modular automorphism Δ^it as M-matrix for a complex value of t is not realistic.

4. The article The role of Type III Factors in Quantum Field Theory by Yngvason gives a good idea about their role in relativistic QFT and helped to realize that WCW local Clifford algebra and WCW spinor fields from fixed causal diamond define naturally HFF of type II_∞ since isotropies correspond to group SO(3) leaving the tips of CD invariant. The extension to the union of WCWs associated with Lorentz boosts of CD would give naturally rise to HFF of type III₁.

5. Crossed product is an important concept which I have not applied earlier in TGD framework. Essentially semi-direct product of algebra with group is in question and under some assumptions gives rise to factor. Now the algebra is that of WCW spinor fields in CD and group Lorentz group transforming CDs. Crossed product construction leads to an understanding of factors in quantum TGD and it seems that the relevant factors in TGD framework are HFFs of type III₁. The minimal choices for the group in question is Lorentz group and its non-compactness implies HFF of type III₁ as a result. Translations acting on either tip of CD induce Lorentz transformation so that they induce also as modular automorphisms.

6. KMS The notion of KMS state is important and gives a connection between thermodynamics and factors.

7. A strongly TGD colored brief summary about basics of von Neumann algebras and the recent view about M-matrix contra HFFS can be found in the section "The latest vision about the role of HFFs in TGD" of the chapter Construction of Quantum Theory: M-matrix of "Towards S-matrix".

2. Conceptual problems

It is safest to start from the conceptual problems and take a role of skeptic.

1. Under what conditions the assumptions of Tomita-Takesaki formula stating the existence of modular automorphism and isomorphy of the factor and its commutant hold true? What is the physical interpretation of the formula M^¢=JMJ relating factor and its commutant in TGD framework? The answer turned out to be that positive and zero energy parts of the zero energy state relate in this manner and the transformation involves also time reflection with respect to the center of CD in rest frame of CD inducing transformation of light-like 3-surfaces and space-time surfaces to the time-mirrored counterparts.

2. Is the identification M=Δ^it sensible is quantum TGD and zero energy ontology, where M-matrix is "complex square root" of exponent of Hamiltonian defining thermodynamical state and the notion of unitary time evolution is given up? The notion of state ω leading to Δ is essentially thermodynamical and one can wonder whether one should take also a "complex square root" of ω to get M-matrix giving rise to a genuine quantum theory.

3. TGD based quantum measurement theory involves both quantum fluctuating degrees of freedom assignable to light-like 3-surfaces and zero modes identifiable as classical degrees of freedom assignable to interior of the space-time sheet. Zero modes have also fermionic counterparts. State preparation should generate entanglement between the quantal and classical states. What this means at the level of von Neumann algebras?

4. What is the TGD counterpart for causal disjointness. At space-time level different space-time sheets could correspond to such regions whereas at imbedding space level causally disjoint CDs would represent such regions.

3. Technical problems

There are also more technical questions.

1. What is the von Neumann algebra needed in TGD framework? Does one have a a direct integral over factors (at least direct integral over zero modes)? Which factors appear in it? Can one construct the factor as a crossed product of some group G with direct physical interpretation (say Lorentz group affecting CD) and of naturally appearing factor A? Is A a HFF of type II_∞ assignable to a fixed CD? What is the natural Hilbert space H in which A acts?

2. What are the geometric transformations induced modular automorphisms of II_∞ inducing the scaling down of the trace? Is the action of G induced by the boosts in Lorentz group. Could also translations and scalings induce the action? What is the factor associated with the union of Poincare transforms of CD? log(Δ) is Hermitian algebraically: what does the non-unitarity of exp(log(Δ)it) mean physically?

3. Could ω correspond to a vacuum which in conformal degrees of freedom depends on the choice of the sphere S² defining the radial coordinate playing the role of complex variable in the case of the radial conformal algebra. Does ^*-operation in M correspond to Hermitian conjugation for fermionic oscillator operators and change of sign of super conformal weights?

The exponent of the modified Dirac action gives rise to the exponent of Kähler function as Dirac determinant and fermionic inner product defined by fermionic Feynman rules. It is implausible that this exponent could as such correspond to ω or Δ^it having conceptual roots in thermodynamics rather than QFT. If one assumes that the exponent of the modified Dirac action defines a "complex square root" of ω the situation changes. This raises technical questions relating to the notion of square root of ω.

1. Does the complex square root of ω have a polar decomposition to a product of positive definite matrix (square root of the density matrix) and unitary matrix and does ω^1/2 correspond to the modulus in the decomposition? Does the square root of Δ have similar decomposition with modulus equal equal to Δ^1/2 in standard picture so that modular automorphism, which is inherent property of von Neumann algebra, would not be affected?

2. Δ^it or rather its generalization is defined modulo a unitary operator defined by some Hamiltonian and is therefore highly non-unique as such. This non-uniqueness applies also to |Δ|. Could this non-uniqueness correspond to the thermodynamical degrees of freedom?

4. Cautious conclusions

The cautious conclusions are following.

1. The notion of state as it appears in the theory of factors is not enough for the purposes of quantum TGD. The reason is that state in this sense is essentially the counterpart of thermodynamical state.

2. The construction of M-matrix might be understood in the framework of factors if one replaces state with its "complex square root" natural if quantum theory is regarded as a "complex square root" of thermodynamics. The replacement of exponent of Hamiltonian with imaginary exponent of action is the counterpart for this generalization in QFT framework.

3. The identification of Δ^it as M-matrix is not consistent with zero energy ontology and causal diamond as basic building block of WCW. It might be that the varying thermodynamical part of M-matrix -say that giving rise to p-adic thermodynamics- involves Δ^t.

4. Also the idea that Connes tensor product could fix M-matrix is too optimistic but an elegant formulation in terms of partial trace for the notion of M-matrix modulo measurement resolution exists and Connes tensor product allows interpretation as entanglement between sub-spaces consisting of states not distinguishable in the measurement resolution used. The partial trace also gives rise to non-pure states naturally. M-matrix as a complex square root of ω could be determined uniquely from the condition of universality and the requirement that it decomposes to a tensor product of M-matrices associated with factors space M/N and N for any reasonable choices of N.

The challenge is to show that the identification of M-matrix as a "complex square root" of state ω is consistent with the definition of M-matrix using only the modified Dirac action for second quantized induced spinor fields identifiable as square root" of Kähler action emerges. The key idea is the notion of bosonic emergence used as a guideline in the construction of QFT limit of TGD and meaning that bosonic propagators and vertices emerge radiatively from fermionic loops.

For details see either the section "The latest vision about the role of HFFs in TGD" of the chapter Construction of Quantum Theory: M-matrix of the book "Towards S-matrix" or of the chapter Was von Neumann Right After All of the book "Overall View about TGD".