### How uniquely Connes tensor product defines the M-matrix?

I told already earlier about a highly unique identification of M-matrix (product of square root of density matrix and unitary S-matrix) defining time-like entanglement coefficients between positive and negative energy parts of zero energy states and characterizing quantum dynamics in zero energy ontology.

The identification is based on the notion of measurement resolution represented as Jones inclusion. The zero energy states created by included algebra N creating zero energy states for which the time interval between positive and negative energy parts of the state is shorter than that for M are below measurement resolution. Hence complex rays are replaced with N rays and M-matrix must "*-commute" with N: in other words MN=N^{*}M in element-wise manner. One obtains a hierarchy of M-matrixes chacterized by inclusions, which are not necessary Jones inclusions.

I conjectured that this leads to a highly unique M-matrix. This is indeed the case. The defining condition for the variant of the Connes tensor product proposed here has the following equivalent forms

MN= N^{*}M ,

N=M^{-1}N^{*}M ,

N^{*}=MNM^{-1} .

If M_{1} and M_{2} are two M-matrices satisfying the conditions then the matrix M_{12}=M_{1}M_{2}^{-1} satisfies the following equivalent conditions

N=M_{12}NM_{12}^{-1} ,

[N,M_{12}]=0 .

Jones inclusions with M:N≤4 are irreducible which means that the operators commuting with N consist of complex multiples of identity. Hence one must have M_{12}=1 so that M-matrix is unique in this case. For M:N>4 the complex dimension of commutator algebra of N is 2 so that M-matrix depends should depend on single complex parameter. The dimension of the commutator algebra associated with the inclusion gives the number of parameters appearing in the M-matrix in the general case.

When the commutator has complex dimension d >1 , the representation of N in M is reducible: the matrix analogy is the representation of elements of N as direct sums of d representation matrices. M-matrix is a direct sum of form M= a_{1}M_{1}+a_{2}M_{2}+..., where M_{i} are unique. The condition ∑_{i};|a_{i}|^{2}=1 is satisfied and*-commutativity holds in each summand separately.

** Questions**: Could M_{i} define unique universal unitary S-matrices in their own blocks? Could the direct sum define a counterpart of a statistical ensemble? Could irreducible inclusions correspond to pure states and reducible inclusions to mixed states? Could different values of energy in thermodynamics and of the scaling generator L_{0} in p-adic thermodynamics define direct summands of the inclusion? The values of conserved quantum numbers for the positive energy part of the state indeed naturally define this kind of direct direct summands.

For more details see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".

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