### How to construct scattering amplitudes?

Lubos Motl ) told about two new hep-th papers, by Pate, Raclariu, and Strominger (see this) and by Nandan, Schreiber, Volovich, Zlotnikov (see this) related to a new approach to scattering amplitudes based on the replacement of the quantum numbers associated with Poincare group labelling particles appearing in the scattering amplitudes with quantum numbers associated with the representations of Lorentz group.

Why I got interested was that in zero energy ontology (ZEO) the key object is causal diamond (CD) defined as intersection of future and past directed M^{4} light-cones with points replaced with CP_{2}. Space-time surfaces are inside CD and have ends at its light-like boundaries. The Lorentz symmetries associated with the boundaries of CD could be more natural than Poincare symmetry, which would emerge in the integration over the positions of CDs of external particles arriving to the opposite light-like boundaries of the big CD defining the scattering region where preferred extremal describing the scattering event resides.

I did my best to understand the articles and - of course relate these ideas to TGD, where the construction of scattering amplitudes is the basic challenge. My technical skills are too limited for to meet this challenge at the level of explicit formulas but I can try to understand the physics and mathematics brought in by TGD.

While playing with more or less crazy and short-lived ideas inspired by the reading of the articles I finally realized that there is perhaps no point in starting from quantum field theories. TGD is not quantum field theory and I must start from TGD itself.

In TGD framework the picture inspired by adelic physics is roughly following.

- Cognitive representations realizing number theoretic universality of adelic physics consist of points of imbedding space with coordinates in the extension of rationals. The number of points is typically finite. Cognitive representation should contain as subset the points associated with n-point functions, which are essentially correlation functions.

Fundamental fermions are building bricks of elementary particles, and a good guess is that fundamental fermions correspond to singular points for which the action of subgroup of Galois group of extension is trivial so that several points collapse together.

- One must sum over the orbits of a subgroup S
_{D}of symplectic group of light-cone boundary acting as isometries of both boundaries of CD. S_{D}consists of isometries with parameters in the extension of rationals defining the adele. All orbits needed to represent the pairs of initial and final 3-surfaces at the boundaries of CD allowed by the action principle must be realized so that single orbit very probably is not enough.

- Correlations code for the quantum dynamics. In quantum field theories quantum fluctuations of fields at distinct points of space-time correlate and give rise to n-point functions expressible in terms of propagators and vertices: massless fields and conformal fields define the basic example. Operator algebra or path integral describes them mathematically.

In TGD correlations between imbedding space points belonging to the space-time surface result from classical deterministic dynamics: the points of 3-surface at opposite boundaries of CD are not independent.

This dynamics is non-linear geometric analog for the dynamics of massless fields: space-time sheets as preferred extremals are indeed minimal surfaces with string world sheets appearing as singularities. Minimal surface property is forced by the volume action implied by the twistor lift and having interpretation in terms of cosmological constant. The correlation between points at the same boundary of CD are expected to be independent since these 3-surfaces chosen rather freely as analogs of boundary values for fields.

Fermionic dynamics governed by modified Dirac action is dictated completely by super-symplectic and super-conformal symmetries. Second quantization of fermions at space-time level is necessary to realized WCW spinor structure: WCW gamma matrices are linear combinations of fermionic oscillator operators.

- This suggests that the attempts to guess the conformal field theory producing the correlation functions makes things much more complex than they actually are. It should be possible to understand how these correlations emerge from the classical dynamics of space-time surfaces.

- Sum over the symplectic orbits of cognitive representations must be involved as also vacuum expectation values in the fermionic sector for fermionic fields which must appear in vertices for external particles. At the level of cognitive representations anti-commutators for oscillator operators involve Kronecker deltas so that one has discretized variant of second quantization.

- This could be achieved by expanding the restriction Ψ
^{A}_{|X3}of the imbedding space harmonic Ψ^{A}restricted to 3-surface at end of space-time surface as sum of modes Ψ_{n}of the induced spinor field. This would be counterpart for the induction procedure. One can assign to singular points bilinear of type Ψbar^{A}_{|X3}D^{↔}Ψ. D denotes modified Dirac operator. This gives to propagators connecting fermions vertices at the opposite ends of space-time surface.

- A more concrete picture must rely on a concrete model for elementary particles. Elementary particles have as building bricks pair of wormhole contacts with fermion lines at the light-like orbits of the throats at which the signature of the metric changes from Minkowskian to Euclidian. Particle is necessarily a pair of two wormhole contacts and flux tube connects them at both space-time sheets and forms a closed flux tube carrying monopole flux.

All particles consist of fundamental fermions and anti-fermions: for instance gauge bosons involve fermion and anti-fermion responsible for the quantum numbers at the opposite throats of second wormhole contact. Second wormhole contact involves neutrino pair neutralizing electroweak isospin in scales longer than the size of the flux tube structure.

- The topological counterpart of 3-vertex appearing in Feynman diagram corresponds to a replication of this kind of 3-surface highly analogous to bio-replication. In replication vertex, there is no singularity of 3-surface analogous to that appearing in the vertices of stringy diagrams but space-time surface is singular just like 1-D manifold is singular for at vertex of Feynman diagram.

These singular replicating 3-surfaces and the partonic 2-surfaces give rise to the counterparts of interaction vertices. Fermionic 4-vertex is impossible and fermion lines can only be re-shared between outgoing partonic orbits. This is however not enough as will be found. It will be found that also the creation of fermion pair as effective turning of fermion lines entering along "upper" wormhole throat and turning back at Euclidian wormhole throat and continuing along the orbit of "lower" wormhole throat must be possible.

- The appearance of a sub-CD assignable to the partonic 2-surface is possible but does not solve the problem considered. There would be incoming fermion line at lower boundary and 1 fermion line and fermion and anti-fermion line associated with the boson at the "upper" boundary of CD. There would be non-locality in the scale of the partonic 2-surface and sub-CD meaning that the lines can end to vacuum. Now one would encounter the same difficulty but only in shorter scale.

- Could one say that fermion line turns backwards in time? A line turning back could be described as an annihilation of fermion pair to vacuum carrying classical gauge field, which is standard process. In QFT picture this would be achieved if a bilinear Ψ DΨ is allowed in the vertex where annihilation takes place. Not in TGD: fermionic action vanishes identically by field equations expressing essentially the conservation of fermion current and various super currents obtained as contractions fermion field with modes.

Could fermion-antifermion pair creation occur at singular points associated with partonic surfaces representing the turning of fermion line backwards in time. This looks still too singular.

Rather, the turning backwards in time should mean that a fermion line arriving from future along the orbit of "upper" throat (say) goes through Euclidian wormhole throat and continues along the orbit of "lower" throat back to future than making discontinuous turn-around. Euclidian regions of space-time surface representing one key distinction between GRT and TGD would thus be absolutely essential for the generalized scattering diagrams. An exchange of momentum with classical field would be Feynman diagrammatic manner to say this.

New oscillator operator pairs emerge at the partonic vertices and would correspond to the above described turn-around for fermion line at wormhole contact. Fermion pairs present at the "lower" boundary of CD could also disappear.

- The anti-commutation relations fermions are modified due to the presence of vacuum gauge fields so that the anti-commutator of fermionic creation operators a
^{†}_{m}and anti-fermionic creation operators b^{†}_{n}is non-vanishing. A proper formulation of the fermionic anti-commutation relations at the ends of space-time surface is needed. One can imagine that although standard anti-commutation relations at the lower end of space-time surface hold true, the time evolution of Ψ in the presence of vacuum gauge potentials implies that the vacuum expectations < vac| a^{†}_{m}b^{†}_{n}|vac > are non-vanishing. This would require that for instance b^{†}_{n}and a_{n}are mixed.

- Is the first guess enough? It is not as becomes clear after a thought about the continuum limit. The WCW degrees of freedom are described at continuum limit in terms of super-symplectic algebra (SSA) acting on ground state are neglected. Imbedding space spinor modes characterizee only the ground staes of these representations. These degrees of freedom are essential already in elementary particle physics.

Sub-algebra SSA

_{m}of SSA with conformal weights coming as m-multiples of those of SSA and its commutator with SSA annihilate the physical states, and one obtains a hierarchy. How to describe these states in the discretization? The natural possibility are the representations of S_{D}such that (S_{D})_{m}and the subgroup generated by the commutator algebra are represented trivially. One has non-trivial S_{D}_{m}representations at both ends of WCW such that the action of S_{D}on the tensor product acts trivially.

There are also fermionic degrees of freedom. The challenge is to identify among other things WCW gamma matrices as fermionic super charges and it would be nice if all charges were Noether charges. The simplest guess is that the algebra generated by fermionic Noether charges Q^{A}for symplectic transformations h^{k}→ h^{k}+j^{Ak}assumed to induce isometries of WCW and Noether supercharges Q_{n}and their conjugates for the shifts Ψ→ Ψ+ε u_{n}, where u_{n}is a solution of the modified Dirac equation, is enough.

The commutators Γ

^{A}_{n}=[Q^{A},Q_{n}] are super-charges labelled by A and n. One would like to identify them as gamma matrices of WCW. The problem is that they are labelled by (A,n) and isometry generators are labelled by A only. There should be one-one correspondence. Do all supercharges Γ^{A}_{n}except Γ^{A}_{0}corresponding to u_{0}=constant annihilate the physical state so that one would have 1-1 correspondence. This would correspond to what happens quite generally in super-conformal algebras.

The generators of this fermionic algebra could be used to generate more general states. One should also construct the discretized versions of the generators as sums over points of the cognitive representation at the ends of space-time surface. Note that this requires tangent space data.

- What about the conservation of four-momentum and other conservation laws? This can be handled by quantum classical correspondence (QCC). The momentum and color labels defined by fermionic quantum numbers in Cartan algebra can be assumed to be equal to the corresponding classical Noether charges for particle-like space-time surfaces entering to CD. The technical problem is that if one knows only the discretization - even with tangent space data - one does not know the values of these charges! It might be that M
^{8}-H correspondence in which M^{8}side fixes space-time surfaces as roots for real or imaginary parts of octonionic polynomials from the data at discrete set of points is needed.

- ZEO means deviations from ordinary description. S
_{D}invariance of zero energy state forces sum over the 4-surfaces of the orbit with identical coefficients. Symplectic invariance implies time-like entanglement. One can describe this in terms of hermitian square root Ψ of density matrix satisfying Ψ †Ψ =ρ. The coefficients of different orbits need not be same and allows description in terms of dynamical density matrix. If there is Yangian symmetry also this entanglement is analogous to the entanglement due to statistics.

Surprisingly - and somewhat disappointingly after decades of attempts to understand unitarity in TGD - unitarity is trivial in ZEO since state basis is defined essentially by the rows of matrices and orthogonality conditions their orthogonality and therefore unitarity. More concretely, for single state at the passive end state function normalization to unity defined by inner product as sum over 3-surfaces at active end would give conservation of probability. Orthogonality of the state basis with inner product as sum over surfaces passive boundary gives orthogonality for the coefficients defining rows of a matrix and therefore unitarity. In the case that single orbit or even several of them defines the states one obtains the same result.

What then guarantees the orthogonality of zero energy states? In ordinary quantum mechanics the property of being eigenstates of some hermitian operator guarantees orthogonality. In TGD zero energy states would be solutions of the analog of massless Dirac equation in WCW consisting of pairs of 3-surfaces with members at the ends of preferred extremals inside CD. This generalizes Super Virosoro conditions of superconformal theories and would provide the orthonormal state basis.

_{1}+n

_{2}points at the boundaries of CD, and summation over the orbits of the symplectic group. All these mathematical objects would emerge from classical dynamics. The sum over the orbits for chosen spinor harmonics would produce n-point functions, vertices and propagators. It is difficult to imagine anything simpler and quantum classical correspondence would be complete.

See the article Scattering amplitudes and orbits of cognitive representations under subgroup of symplectic group respecting the extension of rationals or the chapter The Recent View about Twistorialization in TGD Framework.

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