### S-matrix for the scattering of zero energy states representing ordinary scattering events as a solution to the problems?

The properties of factorizing S-matrices are extremely beautiful but it seems that we have been trying to make sense of them in a wrong interpretational framework. The proper understanding of the situation might require a radically new interpretation relying on the idea that particle reactions correspond to creation of negative energy states from vacuum in TGD framework. This idea has not been used in any manner hitherto.

**1. Fundamental scattering as a scattering zero energy states representing particle reactions**

In TGD framework that initial and final states of particle reaction form zero energy states and it is scattering of these states representing particles reactions that we actually observe in TGD Universe.

- What is observed are not particles in the initial and final states but particle reactions identified as zero energy states consisting of positive and negative energy particles which we interpret as initial and final states of the particle reaction. This interpretation might well require an appropriate generalization of the notion of S-matrix and to modify the notion of unitarity which must be in some form be still there.
What comes in mind is the construction of factorizing S-matrix for zero energy states consisting of the particles of initial and finals states having arbitrary momenta. This scattering would describe scattering between zero energy states interpreted in terms of ordinary particle reactions.

- The basic property of factorizing S-matrices is that they affect only the internal degrees of freedom: therefore the scattering between zero energy states does not affect the initial and final momenta of positive and negative energy states so that the curses of factorizing S-matrices become blessings. The quantum jumps describing scattering makes it possible to experience these reactions consciously and affects only internal degrees of freedom which are not detected.
- One should be even ready to give up the cherished Lorentz invariance and color symmetries since Jones inclusions associated with the scattering experiment could mean symmetry breaking caused by the selection of subgroups SO(1,1)× SO(2) SO(1)× SO(3), and U(2)subset of SU(3). In the recent picture these choices affect even the geometry and topology of imbedding space and they reflect directly the effect of experimenter to the measurement situation, which simply cannot neglected as is usually done axiomatically. One could also speak about number theoretic breaking o symmetries induced by the requirement that fundamental commutative sub-manifolds of imbedding space are 2-dimensional.
- One can generalize the construction of factorizing S-matrices without difficulties to the case in which incoming and outgoing states are zero energy states representing particle reactions. Pass-by is possible also for negative and positive energy states if pass by is considered for the projections of rapidities on iπ-&eta
_{i}to real plane. The construction goes through as such for the other tensors. Also the crossing symmetry and other symmetries of factorizing S-matrices generalized in obvious manner.

**2. Generalization of unitarity conditions**

Consider now the unitarity conditions for the scattering between zero energy states observed as particle reactions. The aim is to derive ordinary unitarity conditions from the unitarity in the scattering of zero energy states as what might be regarded as thermal averages with motivation for averaging coming from the fact that the internal degrees of freedom affected by thescattering between zero energy states are not detected.

The unitarity conditions for the scattering of zero energy states read formally as

∑_{ m+ n-} S_{m+n-→ m + n-} S^{*} _{ r+s- → m+ n- } =δ_{m+,r+} δ _{n-,s-} .

Note that the summed final states are expressed using italic. The sum over the final zero energy states can be also written as a trace for the product of matrices labelled by incoming zero energy states.

Tr(S_{ m+n-} S^{*}_{r+s-}) =δ_{m+,r+}δ_{n-,s-} .

One can put s_{-}=n_{-} on both sides and perform the sum over n_{-} to get

∑ _{n-}Tr(S_{{m+n-} S^{*}_{r+n-}) =δ_{m+,r+}∑_{n-}δ_{n-,n-} .

For the factors of type II_{1} the sum

∑_{n-}δ_{n-,n-} is equal to the trace Tr(Id)=1 of the identity matrix so that one obtains

∑_{n-}Tr(S_{m+n-} S^{*}_{r+n-}) =δ_{m+,r+} .

One could also divide the left hand side by Tr(Id) to get average as one understands it usually.

The usual unitarity condition would read
∑_{n-}S_{m+n-→ m+n-} S^{*}_{r+n-→ r+n-} =δ_{m+,r+} .

This condition would be replaced with an average over the final states in the scattering described by a factorizing S-matrix.

The interpretation of the result would be as a thermal expectation value of the unitarity condition in the sense of hyper-finite factors of type II_{1}. This averaging is necessary if we do not have any control over the scattering between zero energy states: this scattering is just a means to become conscious about the existence of the state we usually interpret as change of state. What looks a very non-physical feature of factorizing S-matrices turns into a victory since the trace is only over final states which are characterized by the same collection of momenta and same particle number and uncertainties relate only to the internal degrees of freedom which we cannot measure and whose basic function is to make it possible to consciously perceive the particle reaction as a zero energy state.

**3. Should one accept spontaneous breaking of Lorentz symmetry?**

The proposed picture does not seem to provide any way to achieve unitarity by summing over all choices of (M^{2},E^{2}) and (S^{2},S^{1}) pairs since in a more general scattering situation the crucial transversal or longitudinal momentum exchange can occur only under very special situations. Hence a statistical averaging of the probabilities over Lorentz group seems to be the only manner to achieve Lorentz invariance.

An interesting question, is whether breaking of Lorentz symmetry is already encountered in the hadronic scattering in quark model description, which involves the reduction of Lorentz group to SO(1,1)× SO(2), and longitudinal and transverse momenta.

** 4. Summary**

To sum up, quantum classical correspondence combined with the number theoretical view about conformal invariance could fix highly uniquely the dependence of S-matrix on cm degrees of freedom and on net momenta and color quantum numbers associated with various lightcones whose tips define the arguments of n-point function. If one is ready to accept the new view about scattering event as a scattering between two zero energy states between representing initial * resp.* final states as positive * resp.* negative energy particles of the state one obtains physically highly non-trivial counterpart of S-matrix. One must also accept the generalization of the unitarity condition to what might be regarded as a thermal average stating explicitly the assumption that experimenter does not have control over the scattering between zero energy states whose basic function is to make the zero energy state representing ordinary scattering consciously observed. The properties of factorizing S-matrices are ideal for this purpose.

The price being paid is the breaking of the full Lorentz and color invariances, which at the level of Jones inclusion means a change of the geometry topology of imbedding space and space-time. This kind of breaking of course happens in realistic experimental situation. Lorentz invariance is obtained only in statistical sense.

The most fascinating aspect of the new interpretation is that the factorizing and physically almost trivial S-matrices of integrable 2-D systems generalize to S-matrices describing scattering between states with vanishing net quantum numbers could be imbedded into 4-D theory in such a manner that the resulting S-matrix could be physically completely non-trivial and perhaps even physically realistic.

The basic requirement on S-matric between zero energy states is its almost triviality. This motivates the hope that the tensor factoring of 2-D factorizing matrices could be extended also to configuration space degrees of freedom so that each complex configuration space dimension would contribute one factorizing S-matrix to the tensor product.

The chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is now (it could change!).

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