### Comments about M-matrix and Connes tensor product

I have proposed that the identification of M-matrix as Connes tensor product defined by finite measurement resolution could lead to a universal definition of dynamics. This hypothesis is fascinating but - mainly due to my poor understanding of HFFS of type II

_{1}- has remained just an interesting hypothesis. In the following I represent a formulation of this idea which is more precise than the earlier formulations and take the role of skeptic and reconsider also hyper-finite factors of type III

_{1}appearing in quantum field theories. I also consider the possibility that M-matrices could relate to a quantum variant of so called 2-vector space formulated by John Baez and collaborators.

** 1. Finite measurement resolution and M-matrix**

Consider first the formulation of finite measurement resolution in terms of inclusions of HFFs of type II_{1}.

- Finite measurement resolution means that M-matrix elements are defined by integrating- that is taking a trace- over the degrees of freedom defined by the sub-factor N subset M in the sense that states created by N from a given state do not differ from it in given measurement resolution. As a consequence,N takes the role of complex numbers in the ordinary quantum theory.
- This requires that the tensor product decomposition
M= (M/N)⊗ N .
exists in some sense. The factor space M/N could be seen as an analog of or even identical with a tensor product of quantum spinors with different quantum phases q=exp(i2π/n), n > 3, and would have a fractal quantum dimension fixed by the inclusion. Quantum spinors are finite-dimensional.

- Consider M-matrix elements M
_{(r,n),(r1,n1)}, where r labels the states of M/N. Transition probabilities with a finite measurement resolution are defined by taking trace over N meaning summation over both n and n_{1}. One obtainsp(r

_{1}→ r_{2})=∑_{n,n1}M_{(r,n), (r1,n1)}M^{+}_{(r1,n1),(r,n)}. The subscript '+' refers to Hermitian conjugation here.
- Suppose one multiplies the state by element n of N. Since one has a tensor product structure, the trace over N separates out so that one obtains a factor Tr(n
^{+}n) giving a multiplicative constant just as is obtained by multiplying the states appearing in the ordinary S-matrix by a constant. Note that the measurement resolution could be different for initial and final states: in this case the smaller algebra would serve as measurement algebra. - The condition that N acts like complex numbers has not been used yet and the above picture makes sense even without this condition. For the ordinary S-matrix the action of complex number on final state affects S-matrix in the same manner as the action of its conjugate on initial state. Generalizing, in positive energy ontology S-matrix elements would be anti-linear with respect to (say) initial quantum states as inner product like and the action of element n on the final state would be equal to the action of n
^{+}on the initial state. In zero energy ontology M-matrix is bilinear in positive and negative energy parts of the zero energy state. The condition that N acts like complex numbers means that the action of an element of N on the final final state in M-matrix induces the same effect as the action of n (-rather than that of n^{+}) on the initial state. This condition is rather powerful and expected to determine the M-matrix highly uniquely. - This picture does not require gauge invariance with respect to N. One can however consider also a stronger condition stating that the action of unitary elements of N act as gauge symmetries. In zero energy ontology this would mean that symmetries represent elements of infinite-dimensional orthogonal group.
- Connes tensor product makes sense also in construction of many-particle states and gives rise to irreducible entanglement not visible in measurement resolution.

** 2. How unique the Connes tensor product really is?**

I have used to state in rather light hearted manner that Connes tensor product is highly unique but what the situation really is? Is my belief just a folk wisdom that I have gather somewhere? Let us try to think ourselves. How unique the M-matrix could be?

- The simplest solution to the conditions could be written as a
*formal*tensor product M=M_{1}⊗ M_{2}, where M_{1}is M-matrix in M/N and M_{2}projection operator P_{N}in to N. The appearance of P_{N}N would say that in N degrees of freedom entanglement coefficients are identical and basic condition would be satisfied since P_{N}would commute with elements of N. If one replaces P_{N}with a more general operator, it must also commute with all elements of N, which means that it must represent direct sum of HFFs of type II_{1}since the definition property of factors is that only unit matrix commutes with all operators of the factor. - This representation would suggest that M
_{1}can be chosen completely freely as an analog of a complex square root of Hermitian density matrix. The skeptic would conclude that Connes tensor product does not say anything about the physically interesting part of M-matrix! - This formal representation very probably does not make sense as such since M
_{1}is tensor product of quantum spinor spaces and tensor product property might reveal itself only as a factorization of trace into a product of traces over M/N and N parts of the tensor product like structure.

This argument must be of course take as childish babbling of a poor physicist knowing nothing about the delicacies of this branch of mathematics. It is quite possible that the delicacies of the Connes tensor product bring in uniqueness.

** 3. Is the resulting M-matrix realistic?**

Are there hopes that the resulting M-matrix is realistic?

- The resulting M-matrix defined in quantum Hilbert space has fractal quantum dimension. Since the unitary representations of quantum groups are finite-dimensional, one can expect that finite-dimensionality results. Therefore skeptic would ask whether the use of mere HFFs of type II
_{1}is enough and whether the outcome is just a topological S-matrix of braid theories. This would be dissapointing. One can however easily add factor of type I as a tensor product factor to give HFF of type II giving also the structure of ordinary wave mechanics. - The physical picture is that the orbits of fermions and antifermions at light-like 3-surfaces defined braids and the topological M-matrix can be assigned with the legs of generalized Feynman diagrams. The remaining configuration space degrees of freedom could correspond to factors of type I.
- In finite measurement resolution space-time surface decomposes to strings connecting the braid strands at different light-like 3-surfaces and this stringy interaction should guarantee that the theory does not reduce to a topological QFT. This interaction is between braids so that something new is needed. Does this something new allow a description in terms of finite measurement resolution too or is factor of type I enough? Or could the almost TQFT property mean that N does not act anymore like complex numbers so that Connes tensor product is deformed?
- One can also wonder what is the role of the dynamics in the parameters labeling the copies of HFF. One interpretaton is in terms of zero modes for the Kähler metric of world of classical worlds and therefore as non-quantum fluctuating degrees of freedom. One can also consider interpretation of superposition over these parameters in terms of thermodynamical degrees of freedom.

**4. Should one replace HFFs of type II _{1} with HFFs of type III_{1}?**

These skeptic arguments encourage a serious reconsideration of an alternative approach for achieving uniqueness by extending the HFF of type II_{1} with HFF of type III_{1} (I have considered this generalization already earlier). The dream would be that any M-matrix with a finite measurement resolution is obtained from a universal M-matrix with infinite measurement resolution existing in some sense by tracing over HFF N of type III_{1} and multiplying by the projector to N. Tomita-Takesaki theorem raises the hope that this universal M-matrix indeed exists and is unique apart from the inner automorphisms of HFF of type III_{1} and complex parameter t whose real and imaginary parts have interpretation as time and temperature.

- In quantum field theories in Minkowski space hyper-finite factors of type III
_{1}appear and it would not be surprising that they would appear also in quantum TGD. These factors can be expressed as so called crossed products of hyper-finite factors of type hyper-finite II_{∞}factor and real numbers, whereas the latter correspond to a tensor product of hyper-finite II_{1}factor and I_{∞}factor (for the basic wisdom about von Neumann algebras see this). - One can assign to any M-matrix in factor M an M-matrix associated with any quantum space M/N by tracing over N. This could make sense also for HFFs of type III
_{1}. In this case the trace Tr(n^{*}n) over N would be always infinite so that multiplication by operator of n of N would be analogous to a multiplication of the moduli square of S-matrix elements with an infinite scale factor. Maybe this relates to infinite multiplicative renormalization factors appearing in quantum field theories. - Tomita-Takesaki theorem assumes a von Neuman algebra acting on complex Hilbert space and the existence of a cyclic state Ω of Hilbert space (vacuum state in physics). Furthermore, the existence of a faithful state ω(x) = (xΩ,Ω) satisfying by definition ω(x
^{*}x) > 0 is assumed. Faithful state is analogous to a thermodynamical ensemble. For any anti-linear operator S one has the polar decomposition S=JΔ^{1/2}, where Δ = S^{*}S > 0 is positive self-adjoint operator and J is anti-unitary involution with maps the algebra acting in Hilbert space to the algebra commutating with it by x→ JxJ. ω→ σ^{ω}_{t}= AdΔ^{it}is canonical "evolution" associated with ω and maps von Neumann algebra to itself. This automorphism is unique apart of inner automorphism and thus characterizes the von Neumann algebra itself. So called KMS condition holds true: ω(yx) =ω(σ_{t}^{ω}(x)y), t→ i, in the sense of analytic continuation. This condition is easy to understand by checking it for matrix algebras. - The facts that the automorphism defined by Δ characterizes the algebra itself and the automorphism is non-trivial for factors of type III led Connes and Rovelli to propose that thermodynamics represents a deeper level of physics than Hamiltonian (unique apart from inner automorphism) defining it and one could define Lorentz invariance thermodynamical time using thermodynamics for von Neumann algebras of type III.
- In zero energy ontology where M-matrix corresponds to a generalized thermodynamical state a variant of this idea suggests itself. The automorphism for hyper-finite factor of type III
_{1}would define M-matrix depending on complex parameter t apart from inner automorphism of the factor. The real part of the parameter t would correspond to the Lorentz invariant temporal distance between the tips of CD quantized in powers of 2. Imaginary part of t would correspond to the inverse temperature.

** 5. The notion of 2-vector space and entanglement with finite measurement resolution**

John Baez and collaborators are playing with very formal looking formal structures by replacing vectors with vector spaces. Direct sum and tensor product serve as basic arithmetic operations for vector spaces and one can define category of n-tuples of vectors spaces with morphisms defined by linear maps between vectors spaces of the tuple. n-tuples allow also elementwise product and sum. The obtain results which make them happy. For instance, the category of linear representations of give group forms 2-vector spaces since direct sums and tensor products of representations as well as n-tuples make sense. The 2-vector space however looks more or less trivial from the point of physics.

The situation could become more interesting in quantum measurement theory with finite measurement resolution described in terms of inclusions of hyperfinite factors of type II_{1}. The reason is that Connes tensor product replaces ordinary tensor product and brings in interactions via irreducible entanglement as a representation of finite measurement resolution. The category in question could give Connes tensor products of quantum state spaces and describing interactions. For instance, one could multiply M-matrices via Connes tensor product to obtain category of M-matrices having also the structure of 2-operator algebra.

- The included algebra represents measurement resolution and this means that the infinite-D sub-Hilbert spaces obtained by the action of this algebra replace the rays. Sub-factor takes the role of complex numbers in generalized QM so that one would obtain non-commutative quantum mechanics. For instance, quantum entanglement for two systems of this kind would not be between rays but between infinite-D subspaces corresponding to sub-factors. One could build a generalization of QM by replacing rays with sub-spaces. It seems that quantum group concept does more or less this: the states in representations of quantum groups could be seen as infinite-dimensional Hilbert spaces.
- One could speak about both operator algebras and corresponding state spaces modulo finite measurement resolution as quantum operator algebras and quantum state spaces with fractal dimension defined as factor space like entities obtained from HFF by dividing with the action of included HFF. Possible values of the fractal dimension are fixed completely for Jones inclusions. Maybe these quantum state spaces could define the notions of quantum 2-Hilbert space and 2-operator algebra via direct sum and tensor production operations. Fractal dimensions would make the situation interesting both mathematically and physically.

Suppose one takes the fractal factor spaces as the basic structures and keeps the information about inclusion.

- Direct sums for quantum vectors spaces would be just ordinary direct sums with HFF containing included algebras replaced with direct sum of included HFFs.
- The tensor products for quantum state spaces and quantum operator algebras are not anymore trivial. The condition that measurement algebras act effectively like complex numbers would require Connes tensor product involving irreducible entanglement between elements belonging to the two HFFs. This would have direct physical relevance since this entanglement cannot be reduced in state function reduction.
- The sequences of super-conformal symmetry breakings identifiable in terms of inclusions of super-conformal algebras and corresponding HFFs could have a natural description using the 2-Hilbert spaces and quantum 2-operator algebras.

For background see the chapter Construction of Quantum Theory: M-matrix of "Towards M-matrix".

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