Zero energy ontology replaces S-matrix with M-matrix and groups M-matrices to rows of U-matrix. S-matrix appears as factor in the decomposition of M-matrix to a product of hermitian square root of density matrix and unitary S-matrix interpreted in standard sense.
Note that one cannot drop the S-matrix factor from M-matrix since M-matrix is neither unitary nor hermitian and the dropping of S would make it hermitian. The analog of the decomposition of M-matrix to the decomposition of Schrödinger amplitude to a product of its modulus and of phase is obvious.
The interpretation is in terms of square root of thermodynamics. This interpretation should give the analogs of the Feynman rules ordinary quantum theory producing unitary matrix when one has pure quantum states so that density matrix is projector in 1-D sub-space of state space (for hyper-finite factors of type II1 something more complex is required).
This is the case. M-matrices are in this case just the projections of S-matrix to 1-D subspaces defined by the rows of S-matrix. The state basis is naturally such that the positive energy states at the lower boundary of CD have well-defined quantum numbers and superposition of zero energy states does not contain different quantum numbers for the positive energy states. The state at the upper boundary of CD is the state resulting in the interaction of the particles of the initial state. Unitary of the resulting U-matrix reduces to that for S-matrix.
A more general situation allows square roots of density matrices which are diagonalizable hermitian matrices satisfying the orthogonality condition that the traces
The matrices span the Lie algebra of infinite-dimensional unitary group. The hermitian square roots of M-matrices would reduce to the Lie algebra of infinite-D unitary group. This does not hold true for zero energy states.
If one however assumes that S commutes with the algebra spanned by the square roots of density matrices and allows powers of S one obtains a larger algebra complely analogous to Kac-Moody algebra in the sense that powers of S takes the role of powers of exp(i nφ) in Kac-Moody algebra generators. The commutativity of S and density matrices means that the square roots of density matrices span symmetry algebra of S. The Hermitian sub-Lie-algebra commuting with S is large: for SU(N) it would correspond to SU(N-1)× U(1) so that the symmetry algebra is huge in infinite-D case.
A possible interpretation for the sub-space spanned by M-matrices proportional to Sn is in terms of the hierarchy of CDs. If one assumes that the size scales of CDs come as octaves 2m of a fundamental scale then one would have m=n. Second possibility is that scales of CDs come as integer multiples of the CP2 scale: in this case the interpretation of n would be as this integer: this interpretation conforms with the intuitive picture about S as TGD counterpart of time evolution operator. This interpretation could also make sense for the M-matrice associated with the hierarchy of dark matter for which the scales of CDs indeed come as integers multiples of the basic scale.
If the square roots of density matrices are required to have only non-negative eigenvalues -as I have carelessly proposed in some contexts,- only projection operators are possible for Cartan algebra so that only pure states are possible. If one allows both signs one can have more interesting density matrices and this is the only manner to obtain square root of thermodynamics. Note that the standard representation for the Cartan algebra of finite-dimensional Lie group corresponds to non-pure state. For ρ=Id one obtains M=S defining the ordinary S-matrix. The orthogonality of this zero energy state with respect to other ones requires
stating that SU(N=∞) Lie algebra element is in question.
The reduction of the construction of U to that of S is an enormous simplification and reduces to the problem of finding the TGD counterpart of S-matrix. Note that the finiteness of the norm of SS†=Id requires that hyper-finite factor of type II1 is in question with the definining property that the infinite-dimensional unit matrix has unit norm. This means that state function reduction is always possible only into an infinite-dimensional subspace only.
The natural guess is that the Lie algebra generated by zero energy states is just the generalization of the Yangian symmetry algebra of N=4 SUSY postulated to be a symmetry algebra of TGD (see this). The characteristic feature of the Yangian algebra is the multi-locality of its generators. The generators of the zero energy algebra are zero energy states and indeed form a hierarchy of multi-local objects defined by partonic 2-surfaces at upper and lower light-like boundaries of causal diamonds. Zero energy states themselves would define the symmetry algebra of the theory and the construction of quantum TGD also at the level of dynamics -not only quantum states in sense of positive energy ontology- would reduce to the construction of infinite-dimensional Lie-algebra! It is hard to imagine anything simpler!
For background see the chapter Construction of Quantum Theory: M-matrix.