S-matrix as a functor and the groupoid like structure formed by S-matrices

In zero energy ontology S-matrix can be seen as a functor from the category of Feynman cobordisms to the category of operators. S-matrix can be identified as a "complex square root" of the positive energy density matrix S= ρ^1/2₊S₀, where S₀ is a unitary matrix and ρ₊ is the density matrix for positive energy part of the zero energy state. Obviously one has SS^*=ρ₊. S^*S=ρ_- gives the density matrix for negative energy part of zero energy state. Clearly, S-matrix can be seen as a matrix valued generalization of Schrödinger amplitude. Note that the "indices" of the S-matrices correspond to configuration space spinors (fermions and their bound states giving rise to gauge bosons and gravitons) and to configuration space degrees of freedom (world of classical worlds). For hyper-finite factor of II₁ it is not strictly speaking possible to speak about indices since the matrix elements are traces of the S-matrix multiplied by projection operators to infinite-dimensional subspaces from right and left.

The functor property of S-matrices implies that they form a multiplicative structure analogous but not identical to groupoid. Groupoid has associative product and there exist always right and left inverses and identity in the sense that ff^-1 and f^-1f are defined but not identical in general, and one has fgg^-1=f and f^-1fg= g.

The reason for the groupoid like property is that S-matrix is a map between state spaces associated with initial and final sets of partonic surfaces and these state spaces are different so that inverse must be replaced with right and left inverse. The defining conditions for the groupoid are however replaced with more general ones. Associativity holds also now but the role of inverse is taken by hermitian conjugate. Thus one has the conditions fgg^*=fρ_{g,+} and f^*fg= ρ_f,-g, and the conditions ff^*=ρ₊ and f^*f=ρ_- are satisfied. Here ρ_f+/- is density matrix associated with positive/negative energy parts of zero energy state. If the inverses of the density matrices exist, groupoid axioms hold true since f^-1_L=f^*ρ_f,+^-1 satisfies ff^-1_L= Id₊ and f_R^-1=ρ_f,-^-1f^* satisfies f^-1_Rf= Id_-.

There are good reasons to believe that also tensor product of its appropriate generalization to the analog of co-product makes sense with non-triviality characterizing the interaction between the systems of the tensor product. If so, the S-matrices would form very beautiful mathematical structure bringing in mind the corresponding structures for 2-tangles and N-tangles. Knowing how incredibly powerful the group like structures have been in physics one has good reasons to hope that groupoid like structure might help to deduce a lot of information about the quantum dynamics of TGD.

A word about nomenclature is in order. S has strong associations to unitarity and it might be appropriate to replace S with some other letter. The interpretation of S-matrix as a generalized Schrödinger amplitude would suggest Ψ-matrix. Since the interaction with Kea's M-theory blog (with M denoting Monad or Motif in this context) helped to realize the connection with density matrix, also M-matrix might work. S-matrix as a functor from the category of Feynman cobordisms in turn suggests C or F. Or could just Matrix denoted by M in formulas be enough? Certainly it would inspire feeling of awe but create associations with M-theory in the stringy sense of the word but wouldn't it be fair if stringy M-theory could leave at least some trace to physics;-)!

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