Virtual states correspond to Galois non-singlets with momentum which is algebraic integer and real states to Galois singlets with momentum which is ordinary integer. The scattering amplitudes are shown to have basic properties as in QFT. Scattering amplitudes associated with mere re-combinations of quark states to different Galois singlets as in the initial states. Quarks move as free particles.This corresponds to OZI rule and conforms with the assumption that all particles are Galois composites of quarks

One can also ask whether the counterpart of S-matrix has on mass shell virtual states as singularities. This turns out to be the case. Also the analogs of non-planar amplitudes are allowed.

**Explicit expressions for scattering probabilities**

The proposed identification of scattering probabilities as P(A→B)= g^{ABbar}g_{ABbar} in terms of components of the Kähler metric of the fermionic state space.

Contravariant component g^{ABbar} of the metric is obtained as a geometric series ∑_{n&ge 0} T^{n} from from the deviation T_{ABbar}= g_{ABbar}-δ_{ABbar} of the covariant metric g_{ABbar} from δ_{ABbar}.

g this is not a diagonal matrix. It is convenient to introduce the notation Z^{A}, A=1,...,n, Z^{Abar}=Z^{n+k}, k=n+1,...,2n. So that the
g_{BbarC} corresponds to g_{k+n,l}= δ_{k,l}+T_{k,l}.

and one has

g^{ABbar} to g^{k,l+n}= δ_{k,l}+T^{1}_{k,l}.

The condition g^{ABbar}g_{BbarC}= δ^{A}_{C} gives

g^{k,l+n}g_{l+n,m}= δ^{k}_{m} .

giving

∑_{l}(δ_{k,l}+T^{1}_{k,l})(δ_{l,m}+T_{l,m})

= δ_{k,m} + (T^{1}+ T + T^{1}T)_{km} = δ_{k,m} ,

which resembles the corresponding condition guaranteeing unitarity. The condition gives

T_{1}= -T/(1+T)>=- ∑_{n>1} ((-1)^{n}T^{n} .

The expression for P_{A→B} reads as

P(A→B)=g^{ABbar}g_{BbarA}

=[1-T/(1+T)+T^{†} -(T/(1+T))_{AB}T^{†}]_{AB} .

It is instructive to compare the situation with unitary S-matrix S=1+T. Unitarity condition SS^{†}= 1 gives

T^{†}=-T/(1+T) ,

and

P(A→B)=δ_{AB}+ T_{AB}+T^{†}_{AB}+ T^{†}_{AB}T_{AB}= [δ_{AB}-(T/(1+T))_{AB}+T_{AB} -(T/(1+T))_{AB}T_{AB} .

The formula is the same as in the case of Kähler metric.

**Do the notions of virtual state, singularity and resonance have counterparts?**

Is the proposal physically acceptable? Does the approach allow to formulate the notions of virtual state, singularity and resonance, which are central for the standard approach?

- The notion of virtual state plays a key role in the standard approach. On-mass-shell internal lines correspond to singularities of S-matrix and in a twistor approach for
*N*=4 SUSY, they seem to be enough to generate the full scattering amplitudes.If only off-mass-shell scattering amplitudes between on- mass-shell states are allowed, one can argue that only the singularities are allowed, which is not enough.

- Resonance should correspond to the factorization of S-matrix at resonance, when the intermediate virtual state reduces to an on-mass-shell state. Can the approach based on Kähler metric allow this kind of factorization if the building brick of the scattering amplitudes as the deviation of the covariant Kähler metric from the unit matrix δ
_{ABbar}is the basic building bricks and defined between on mass shell states?Note that in the dual resonance model, the scattering amplitude is some over contribution of resonances and I have proposed that a proper generalization of this picture could make sense in the TGD framework.

- Galois singlets with integer valued momentum components are the natural identification for on-mass-shell states. Galois non-singlet would be off-mass-shell state naturally having complex quark masses and momentum components as algebraic integers.
Virtual states could be arbitrary states without any restriction to the components of quark momentum except that they are in the extension of rationals and the condition coming from momentum conservation, which forces intermediate states to be Galois singlets or products of them.

Therefore momentum conservation allows virtual states as on mass shell states, that is intermediate states, which are Galois singlets but consist of Galois non-singlets identified as off-mass-shell lines. The construction of bound states formed from Galois non-singlets would indeed take place in this way.

- The expansion of the contravariant part of the scattering matrix T
^{1}= T/(1+T) appearing in the probabilityP

_{A→B}=g^{ABbar}g_{ABbar}=[1-T/(1+T)]

_{AB}+T^{†}_{AB}-[T/(1+T)]_{AB}T^{†}]_{AB}.would give a series of analogs of diagrams in which Galois singlets of intermediate states are deformed to non-singlets states.

- Singularities and resonances would correspond to the reduction of an intermediate state to a product of Galois singlets.

**What about the planarity condition in TGD?**

The simplest proposal inspired by the experience with the twistor amplitudes is that only planar polygon diagrams are possible since otherwise the area momenta are not well-defined. In the TGD framework, there is no obvious reason for not allowing diagrams involving permutations of external momenta with positive energies * resp*. negative energies since the area momenta x_{i+1}= ∑_{k=1}^{i} p_{k} are well-defined irrespective of the order. The only manner to uniquely order the Galois singlets as incoming states is with respect to their mass squared values given by integers.

**Generalized OZI rule**

In TGD, only quarks are fundamental particles and all elementary particles and actually all physical states in the fermionic sector are composites of them. This implies that quark and antiquark numbers are separately conserved in the scattering diagrams and the particle reaction only means the-arrangement of the quarks to a new set of Galois singlets.

At the level of quarks, the scattering would be completely trivial, which looks strange. One would obtain a product of quark propagators connecting the points at mass shells with opposite energies plus entanglement coefficients arranging them at positive and negative energy light-cones to groups which are Galois singlets.

This is completely analogous to the OZI role. In QCD it is of course violated by generation of gluons decaying to quark pairs. In TGD, gauge bosons are also quark pairs so that there is no problem of principle.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

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