Jones inclusions and construction of Smatrix and U matrixTGD leads naturally to zero energy ontology which reduces to the positive energy ontology of the standard model only as a limiting case. In this framework one must distinguish between the Umatrix characterizing the unitary process associated with the quantum jump (and followed by state function reduction and state preparation) and the Smatrix defining timelike entanglement between positive and negative energy parts of the zero energy state and coding the rates for particle reactions which in TGD framework correspond to quantum measurements reducing timelike entanglement. 1. Smatrix In zero energy ontology Smatrix characterizes time like entanglement of zero energy states (this is possible only for HFFs for which Tr(SS^{+})=Tr(Id)=1 holds true). Smatrix would code for transition rates measured in particle physics experiments with particle reactions interpreted as quantum measurements reducing time like entanglement. In TGD inspired quantum measurement theory measurement resolution is characterized by Jones inclusion (the group G defines the measured quantum numbers), N subset M takes the role of complex numbers, and state function reduction leads to N ray in the space M/N regarded as N module and thus from a factor to a subfactor. The finite number theoretic braid having Galois group G as its symmetries is the spacetime correlate for both the finite measurement resolution and the effective reduction of HFF to that associated with a finitedimensional quantum Clifford algebra M/N. SU(2) inclusions would allow angular momentum and color quantum numbers in bosonic degrees of freedom and spin and electroweak quantum numbers in spinorial degrees of freedom. McKay correspondence would allow to assign to G also compact ADE type Lie group so that also Lie group type quantum numbers could be included in the repertoire. Galois group G would characterize subspaces of the configuration space ("world of classical worlds") number theoretically in a manner analogous to the rough characterization of physical states by using topological quantum numbers. Each braid associated with a given partonic 2surface would correspond to a particular G that the state would be characterized by a collection of groups G. G would act as symmetries of zero energy states and thus of Smatrix. Smatrix would reduce to a direct integral of Smatrices associated with various collections of Galois groups characterizing the number theoretical properties of partonic 2surfaces. It is not difficult to criticize this picture.
2. Umatrix In a welldefined sense U process seems to be the reversal of state function reduction. Hence the natural guess is that Umatrix means a quantum transition in which a factor becomes a subfactor whereas state function reduction would lead from a factor to a subfactor. Various arguments suggest that U matrix could be almost trivial and has as a basic building block the so called factorizing Smatrices for integrable quantum field theories in 2dimensional Minkowski space. For these Smatrices particle scattering would mean only a permutation of momenta in momentum space. If Smatrix is invariant under inclusion then U matrix should be in a welldefined sense almost trivial apart from a dispersion in zero modes leading to a superpositions of states characterized by different collections of Galois groups. 3. Relation to TGD inspired theory of consciousness Umatrix could be almost trivial with respect to the transitions which are diagonal with respect to the number field. What would however make U highly interesting is that it would predict the rates for the transitions representing a transformation of intention to action identified as a padictoreal transition. In this context almost triviality would translate to a precise correlation between intention and action. The general vision about the dynamics of quantum jumps suggests that the extension of a subfactor to a factor is followed by a reduction to a subfactor which is not necessarily the same. Breathing would be an excellent metaphor for the process. Breathing is also a metaphor for consciousness and life. Perhaps the essence of living systems distinguishing them from subsystems with a fixed state space could be cyclic breathing like process N→ M supset N → N_{1} subset M→ .. extending and reducing the state space of the subsystem by entanglement followed by deentanglement. One could even ask whether the unique role of breathing exercise in meditation practices relates directly to this basic dynamics of living systems and whether the effect of these practices is to increase the value of M:N and thus the order of Galois group G describing the algebraic complexity of "partonic" 2surfaces involved (they can have arbitrarily large sizes). The basic hypothesis of TGD inspired theory of cognition indeed is that cognitive evolution corresponds to the growth of the dimension of the algebraic extension of padic numbers involved. If one is willing to consider generalizations of the existing picture about quantum jump, one can imagine that unitary process can occur arbitrary number of times before it is followed by state function reduction. Unitary process and state function reduction could compete in this kind of situation. 4. Fractality of Smatrix and translational invariance in the lattice defined by subfactors Fractality realized as the invariance of the Smatrix in Jones inclusion means that the Smatrices of N and M relate by the projection P: M→N as S_{N}=PS_{M}P. S_{N} should be equivalent with S_{M} with a trivial relabelling of strands of infinite braid. Inclusion invariance would mean translational invariance of the Smatrix with respect to the index n labelling strands of braid defined by the projectors e_{i}. Translations would act only as a semigroup and Smatrix elements would depend on the difference mn only. Transitions can occur only for mn≥ 0, that is to the direction of increasing label of strand. The group G leaving N elementwise invariant would define the analog of a unit cell in lattice like condensed matter systems so that translational invariance would be obtained only for translations m→ m+ nk, where one has n≥ 0 and k is the number of M(2,C) factors defining the unit cell. As a matter fact, this picture might apply also to ordinary condensed matter systems. For more details see the chapter Was von Neumann Right After All?.
