### Quantum quantum mechanics and quantum S-matrix

The description of finite measurement resolution in terms of Jones inclusion N subset M seems to boil down to a simple rule. Replace ordinary quantum mechanics in complex number field C with that in N to obtain "quantum quantum mechanics". This means that the notions of unitarity, hermiticity, Hilbert space ray, etc.. are replaced with their N counterparts.

The full S-matrix in M should be reducible to a finite-dimensional quantum S-matrix in the state space generated by quantum Clifford algebra M/N which can be regarded as a finite-dimensional matrix algebra with non-commuting N-valued matrix elements. This suggests that full S-matrix can be expressed as S-matrix with N-valued elements satisfying N-unitarity conditions.

Physical intuition also suggests that the transition probabilities defined by quantum S-matrix must be commuting hermitian operators whose collective spectrum defines a large class of transition probabilities satisfying probability conservation. It is obvious that these conditions pose very powerful additional restrictions on the S-matrix.

Since the probabilities act as operators on states generated by operators of N, they contain information about the state in N degrees of freedom. Hence the spectrum would reflect a sensitivity to the context defined by the state in N degrees of freedom.

Quantum S-matrix defines N-valued entanglement coefficients between quantum states with N-valued coefficients. How this affects the situation? The non-commutativity of quantum spinors has a natural interpretation in terms of fuzzy state function reduction meaning that quantum spinor corresponds effectively to a statistical ensemble which cannot correspond to pure state. Does this mean that predictions for transition probabilities must be averaged over the ensemble defined by "quantum quantum states"?

For a brief summary of quantum TGD see the article TGD: an Overall View.

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