This leads to an M-matrix which by simple argument is unique for Jones inclusions with M:N<4. In the more general case one obtains an M-matrix with non-trivial density matrix, whose dimension correspond to the number of irreducibles in the representation of N in M induced by inclusion and to which different summands of M-matrix correspond to. The presence of type I factors complicates of course this picture but does not bring anything mathematically new. This picture is extremely attractive physically but should be made more precise. In the following an attempt in this direction is made.
1. How to define the inclusion of N physically?
The overall picture looks beautiful but it is not clear how one could define the inclusion N subset M precisely. One must distinguish between two cases corresponding to the unitary U-matrix representing unitary process associated with the quantum jump and defined between zero energy states and M-matrix defining the time-like entanglement between positive and negative energy states.
- In the case of U-matrix both N and M corresponds to zero energy states. The time scale of the zero energy state created by N should be shorter than that for the state defined naturally as the temporal distance t+- between the tips of the light-cones M4+/- associated with the state and defining diamond like structure.
- In the case of M-matrix one has zero energy subalgebra of algebra creating positive or negative energy states in time scale t+-. In this case the time scale for zero energy states is smaller than t+-/2. The defining conditions for the Connes tensor product are analogous to crossing symmetry but with the restriction that the crossed operators create zero energy states.
- In standard QFT picture the action of the element of N multiplies the positive or negative energy parts of the state with an operator creating a zero energy state.
- At the space-time level one can assign positive/negative energy states to the incoming/outcoing 3-D lines of generalized Feynman diagrams (recall that in vertices the 3-D light-like surfaces meet along their ends). At the parton level the addition of a zero energy state would be simply addition of a collection of light-like partonic 3-surfaces describing a zero energy state in a time scale shorter than that associated with incoming/outgoing positive/negative energy space-time sheet. The points of the discretized number theoretic braid would naturally contain the insertions of the second quantized induced spinor field in the description of M-matrix element in terms of N-point function.
- At first look this operation looks completely trivial but this is not the case. The point is that the 3-D lines of zero energy diagram and those of the original positive/negative energy diagram must be assigned to single connected 4-D space-time surface. Note that even the minima of the generalized eigenvalue &lamdba; (to which Higgs vacuum expectation is proportional) are not same as for the original positive energy state and free zero energy state since the minimization is affected by the constraint that the resulting space-time sheet is connected.
- What happens if one allows several disconnected space-time sheets in the initial state? Could/should one assign the zero energy state to a particular incoming space-time sheet? If so, what space-time sheet of the final state should one attach the *-conjugate of this zero energy state? Or should one allow a non-unique assignment and interpret the result in terms of different phases? If one generalizes the connectedness condition to the connectedness of the entire space-time surface characterizing zero energy state one would bet rid of the question but can still wonder how unique the assignment of the 4-D space-time surface to a given collection of light-like 3-surfaces is.
Second problem relates to the realization of Hermitian conjugation N → N* at the space-time level. Intuitively it seems clear that the conjugation must involve M4 time reflection with respect to some origin of M4 time mapping partonic 3-surfaces to their time mirror images and performing T-operation for induced spinor fields acting at the points of discretized number theoretic braids.
Suppose that incoming and outgoing states correspond to light-cones M4+ and M4- with tips at points m0=0 and m0=t+-. This does not require that the preferred sub-manifolds M2 and S2II are same for positive/negative energy states and inserted zero energy states. In this case the point (m0=t+-/2,mk=0) would be the natural reflection point and the operation mapping the action of N to the action of N* would be unique.
Can one allow several light cones in the initial and final states or should one restrict M-matrix to single diamond like structure defined by the two light-cones? The most reasonable option seems to be an assignment of a diamond shape pair of light-cones to each zero energy component of the state. The temporal distance t+- between the tips of the light-cones would assign a precise time scale assigned to the zero state. The zero energy states inserted to a state characterized by a time scale t+- would correspond to time scales t<t+-/2 so that a hierarchy in powers of 2 would emerge naturally. Note that the choice of quantization axes (manifolds M2 and S2II) could be different at different levels of hierarchy.
This picture would apply naturally also in the case of U-matrix and make the cutoff hierarchy discrete in accordance with p-adic length scale hypothesis bringing in also quantization of the time scales t+-. In the case of U-matrix N would contain besides the zero energy algebra of M-matrix also the subalgebra for which the positive and negative energy parts reside at different sides of the center of the diamond.
3. How to generalize the notion of observable?
The almost-uniqueness of M-matrix seems too good to be true and in this kind of situation it is best to try to find an argument killing the hypothesis. The first test is whether the ordinary quantum measurement theory with Hermitian operators identified as observables generalizes.
The basic implication is that M should commute with Hermitian operators of N assuming that they exist in some sense. All Hermitian elements of N could be regarded not only as observables but also as conserved charges defining symmetries of M which would be thus maximal. The geometric counterpart for this would be the fact that configuration space is a union of symmetric spaces having maximal isometry group. Super-conformal symmetries of M-matrix would be in question.
The task is to define what Hermiticity means in this kind of situation. The super-positions N+N* and products N*N defined in an appropriate sense should Hermitian operators. One can define what the products MN* and NM mean. There are also two Hermitian conjugations involved: M conjugation and N conjugation.
- Consider first Hermitian conjugation in M. The operators of N creating zero energy states on the positive energy side and N* acting on the negative energy side are not Hermitian in the hermitian conjugation of M. If one defines MN*== N*M and NM== MN, the operators N+N* and N*N indeed commute with M by the basic condition. One could label the states created by M by eigenvalues of a maximally commuting sub-algebra of N. Clearly, the operators acting on positive and negative energy state spaces should be interpreted in terms of a polarization N=N++N- such that N+/- acts on positive/negative energy states.
- In the Hermitian conjugation of N which does not move the operator from positive energy state to negative energy state there certainly exist Hermitian operators and they correspond to zero energy states invariant under exchange of the incoming and outgoing states but in time scale t+-/2. These operators are not Hermitian in M. The commutativity of M with these operators follows also from the basic conditions.
It thus seems that the conjecture survives the first test.
4. Fractal hierarchy of state function reductions
In accordance with fractality, the conditions for the Connes tensor product at a given time scale imply the conditions at shorter time scales. On the other hand, in shorter time scales the inclusion would be deeper and would give rise to a larger reducibility of the representation of N in M. Formally, as N approaches to a trivial algebra, one would have a square root of density matrix and trivial S-matrix in accordance with the idea about asymptotic freedom.
M-matrix would give rise to a matrix of probabilities via the expression P(P+→ P-) = Tr[P+M+P-M], where P+ and P- are projectors to positive and negative energy energy N-rays. The projectors give rise to the averaging over the initial and final states inside N ray. The reduction could continue step by step to shorter length scales so that one would obtain a sequence of inclusions. If the U-process of next quantum jump can return the M-matrix associated with M or some larger HFF, U process would be kind of reversal for state function reduction.
Analytic thinking proceeding from vision to details; human life cycle proceeding from dreams and wild actions to the age when most decisions relate to the routine daily activities; the progress of science from macroscopic to microscopic scales in the spirit of strong reductionism; even biological decay processes: all these have an intriguing resemblance to the fractal state function reduction process proceeding to to shorter and shorter time scales. Since this means increasing thermality of M-matrix, U process as a reversal of state function reduction might break the second law of thermodynamics.
The conservative option would be that only the transformation of intentions to action by U process giving rise to new zero energy states can bring in something new and is responsible for evolution. The non-conservative option is that the biological death is the U-process of the next quantum jump leading to a new life cycle. Breathing would become a universal metaphor for what happens in quantum Universe. The 4-D body would be lived again and again.
For more details see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".