Precise definition of the notion of unitarity for Connes tensor product

Connes tensor product for free fields provides an extremely promising manner to define S-matrix and I have worked out the master formula in a considerable detail. The subfactor N subset of M in Jones represents the degrees of freedom which are not measured. Hence the infinite number of degrees of freedom for M reduces to a finite number of degrees of freedom associated with the quantum Clifford algebra N/M and corresponding quantum spinor space.

The previous physical picture helps to characterize the notion of unitarity precisely for the S-matrix defined by Connes tensor product. For simplicity restrict the consideration to configuration space spin degrees of freedom.

1. Tr(Id)=1 condition implies that it is not possible to define S-matrix in the usual sense since the probabilities for individual scattering events would vanish. Connes tensor product means that in quantum measurement particles are described using finite-dimensional quantum state spaces M/N defined by the inclusion. For standard inclusions they would correspond to single Clifford algebra factor C(8). This integration over the unobserved degrees of freedom is nothing but the analog for the transitions from super-string model to effective field theory description and defines the TGD counterpart for the renormalization process.

2. The intuitive mathematical interpretation of the Connes tensor product is that N takes the role of the coefficient field of the state space instead of complex numbers. Therefore S-matrix must be replaced with N-valued S-matrix in the tensor product of finite-dimensional state spaces. The notion of N unitarity makes sense since matrix inversion is defined as S_ij→ S_ji^† and does not require division (note that i and j label states of M/N). Also the generalization of the hermiticity makes sense: the eigenvalues of a matrix with N-hermitian elements are N Hermitian matrices so that single eigenvalue is abstracted to entire spectrum of eigenvalues. Kind of quantum representation for conceptualization process is in question and might have direct relevance to TGD inspired theory of consciousness. The exponentiation of a matrix with N Hermitian elements gives unitary matrix.

3. The projective equivalence of quantum states generalizes: two states differing by a multiplication by N unitary matrix represent the same ray in the state space. By adjusting the N unitary phases of the states suitably it might be possible to reduce S-matrix elements to ordinary complex vacuum expectation values for the states created by using elements of quantum Clifford algebra M/N, which would mean the reduction of the theory to TGD variant of conformal field theory or effective quantum field theory.

4. The probabilities P_ij for the general transitions would be given by

   P_ij=N_ijN_ij^† ,

   and are in general N-valued unless one requires

   P_ij=p_ije_N ,

   where e_N is projector to N. N_ij is therefore proportional to N-unitary matrix. S-matrix is trivial in N degrees of freedom which conforms with the interpretation that N degrees of freedom remain entangled in the scattering process.

5. If S-matrix is non-trivial in N degrees of freedom, these degrees of freedom must be treated statistically by summing over probabilities for the initial states. The only mathematical expression that one can imagine for the scattering probabilities is given by

   p_ij=Tr(N_ijN_ij ^†)_N .

   The trace over N degrees of freedom means that one has probability distribution for the initial states in N degrees of freedom such that each state appears with the same probability which indeed was von Neumann's guiding idea. By the conservation of energy and momentum in the scattering this assumption reduces to the basic assumption of thermodynamics.

6. An interesting question is whether also momentum degrees of freedom should be treated as a factor of type II₁ although they do not correspond directly to configuration space spin degrees of freedom. This would allow to get rid of mathematically unattractive squares of delta functions in the scattering probabilities.

For details see the chapter Was von Neumann Right After All of "TGD: an Overview".