Does measurement resolution fix the allowed M-matrices through Connes tensor product?

Kea mentioned in her blog finite dimensional matrices which form an algebra analogous to algebra associated with quantum groups defined by primitive roots of unity.

This brought in my mind hyperfinite factors of type II₁ and the inclusion N subset M defining quantum measurement theory with a finite measurement resolution characterized by N and with complex rays of state space replaced with N rays. What this really means is far from clear.

1. Naively one expects that matrices whose elements are elements of N give a representation for M. Now however unit operator has unit trace and one cannot visualize the situation in terms of matrices in case of M and N.

2. The state space with N resolution would be formally M/N consisting of N rays. For M/N one has finite-D matrices with non-commuting elements of N. In this case quantum matrix elements should be multiplets of selected elements of N, not all possible elements of N. One cannot therefore think in terms of the tensor product of N with M/N regarded as a finite-D matrix algebra.

3. What does this mean? Obviously one must pose a condition implying that N action commutes with matrix action just like C: this poses conditions on the matrices that one can allow. Connes tensor product does just this.

The starting point is the Jones inclusion sequence

N subset M subset M \otimes_N M subset...

M \otimes_N M is Connes tensor product which can be seen as elements of the ordinary tensor product commuting with N action so that N indeed acts like complex numbers in M. M/N is in this picture represented with M in which operators defined by Connes tensor products of elements of M. The replacement M→ M/N corresponds to the replacement of the tensor product of elements of M defining matrices with Connes tensor product.

One can try to generalize this picture to zero energy ontology.

1. M \otimes_N M would be generalized by M₊ \otimes_N M_-. Here M₊ would create positive energy states and second M_- negative energy states and N would create zero energy states in some shorter time scale resolution: this would be the precise meaning of finite measurement resolution.

2. Connes entanglement with respect to N would define a non-trivial(!) and unique recipe for constructing M-matrices as generalization of S-matrices expressible as products of square root of density matrix and unitary S-matrix but it is not how clear how many M-matrices this allows. In any case M-matrices would depend on the triplet (N,M₊,M_-) and this would correspond to p-adic length scale evolution giving replacing coupling constant evolution in TGD framework. Thermodynamics would enter the fundamental quantum theory via the square root of density matrix. Nice example of precognition that I choose the letter M;-)! Or should I replace M with {\cal M};-)?

3. Addition: The defining condition for the variant of the Connes tensor product proposed here has the following equivalent forms

   MN= N^*M ,

   N=M^-1N^*M ,

   N^*=MNM^-1 .

   If M₁ and M₂ are two M-matrices satisfying the conditions then the matrix M₁₂=M₁M₂^-1 satisfies the following equivalent conditions

   N=M₁₂NM₁₂^-1 ,

   [N,M₁₂]=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 M₁₂=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= a₁M₁+a₂M₂+..., 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 L₀ 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.

4. Zero energy ontology is a key element of this picture and the most compelling argument for zero energy ontology is the possibility of describing coherent states of Cooper pairs without giving up fermion number, charge, etc. conservation and automatic emerges of length scale dependent notion of quantum numbers (quantum numbers identified as those associated with positive energy factor)

Last time I wrote about the reduction of interactions to Connes tensor product for year and half ago (see the earlier posting Connes tensor product as universal definition of interactions,... and this) but without reference to zero energy ontology as fundamental manner to define measurement resolution with respect time and assuming unitarity.

To sum up, interactions would be an outcome of a finite measurement resolution and at the never-achievable limit of infinite measurement resolution the theory would be free: this would be the counterpart of asymptotic freedom! For more details see the chapter Construction of Quantum Theory: S-matrix