Time-like entanglement coefficients as a square root of density matrix?

I have been pondering the problem of understanding the conditions to be posed on the most general quantum state in zero energy ontology. I started from the identification of time-like entanglement coefficients as the unitary S-matrix working in case of hyperfinite factors of type II₁. Imbedding space degrees of freedom very naturally bring in a factor of type I so that that thermal S-matrices are very natural quantum states and obtained by replacing time parameter assignable to S-matrix with complex valued parameter. All quantum states do not however correspond to thermal states and one can wonder what might be the most general identification of the quantum state in zero energy ontology.

Since density matrix formalism defines a very general formulation of quantum theory and since the quantum states in zero energy ontology are analogous to operators, the idea that time-like entanglement coefficients in some sense define a square root of density matrix is rather natural. This would give the defining conditions

ρ⁺= SS⁺ ,ρ^-= S⁺S , Tr(ρ^+/-)=1 .

ρ^+/- would define density matrix for positive/negative energy states. In the case HFFs of type II₁ one obtains unitary S-matrix and also the analogs of pure quantum states are possible for factors of type I. The numbers p⁺_m,n=|S_m,n²|/ρ⁺_m,m and p^-_m,n=|S_n,m²|/ρ^-_m,m give the counterparts of the usual scattering probabilities.

A physically well-motivated hypothesis would be that S has expression S= ρ^1/2 S₀ such that S₀ is a universal unitary S-matrix, and ρ^1/2 is square root of a state dependent density matrix. Note that in general S is not diagonalizable in the algebraic extension involved so that it is not possible to reduce the scattering to a mere phase change by a suitable choice of state basis.

What makes this kind of hypothesis aesthetically attractive is the unification of two fundamental matrices of quantum theory to single one. This unification is completely analogous to the combination of modulus squared and phase of complex number to a single complex number: complex valued Schrödinger amplitude is replaced with operator valued one.

For more details about the recent situation concerning the understanding of S-matrix see the revised chapter Construction of Quantum Theory: S-Matrix of "Towards S-matrix".