About the construction of vertices

The understanding about the construction of S-matrix has increased considerably and I can now attack seriously the challenge of writing down the generalized Feynman rules.

1. Generalized Feynman diagrams

Let us first summarize the general picture.

1. Feynman diagrams are replaced with their higher dimensional variants with lines replaced with lightlike 3-surfaces identifiable as partonic orbits and with vertices replaced with partonic 2-surfaces along which lines meet. Lightlike 3-surfaces corresponding of maxima of Kähler function define generalized Feynman diagrams. There is no summation over the diagrams and each reaction corresponds to single minimal diagram. Quantum dynamics is 2-dimensional in the sense that vertices are defined by partonic 2-surfaces and 3-dimensional in the sense that different maxima of Kähler function defining points of spin glass energy landscape give rise to additional degeneracy essentially due to the presence of light-like direction.

2. S-matrix reduces to a unitary S-matrix depending parametrically on points of M⁴ defining arguments of N-point function in QFT approach. The momentum representation of S-matrix is obtained by taking a Fourier transform of this and is also unitary.

3. S-matrix is a generalization of braiding S-matrix in the sense that one assigns to the incoming/outgoing and internal lines a unitary braiding matrix. To the vertices, where braids replicate, one assigns a unitary isomorphism between tensor product of hyper-finite II₁ factors (HFFs) associated with incoming resp. outgoing lines. A crucial element in the construction is that these tensor products are themselves HFFs of type II₁.

4. Since also bosons are fermion-antifermion states located at partonic 2-surfaces, the construction of vertices reduces basically to that in the fermionic Fock space associated with the vertex and the space of small deformations of the generalized Feynman diagram around the maximum of Kähler function. The discrete set of points defining number theoretic strand define the basic unitary S-matrix and these points carry various quantum numbers. The natural assumption is that one can use at the vertex same fermionic basis for all incoming and outgoing lines and that unitary braiding S-matrix associated with lines induces a unitary transformation of basis. Its presence in internal lines gives rise to propagators as one integrates over the positions for tips of future and past lightcones containing at their light-like boundaries incoming and outgoing partons.

One can proceed by making simple guesses about the unitary isomorphism associated with the vertex.

1. The simplest guess would be that vertices involve only simple Fock space inner product. This would be like old fashioned quark model in which the quarks of incoming hadrons are re-arranged to from outgoing hadrons without pair creation or gluon emission. This trial does not work since it would not allow bosons which can be regarded as fermion-antifermion pair with either of them having non-physical helicity. This observation however serves as a valuable guideline.

2. An alternative guess is based on the observation that partonic 2-surface with punctures defined by number theoretical braids is analogous to closed bosonic string emitting particles. This would suggest that unitary S-matrix could be assigned with some conformal field theory or possibly string model. At least for non-specialist in conformal field theories this approach looks too abstract.

2. Vertices from free field theory defined by the modified Dirac operator

Something more concrete is required and to proceed one can try to apply the mathematical constraints from the basic definition of TGD.

1. The vertices should come out naturally from the modified Dirac action which contains the classical coupling of the gauge potentials (induced spinor connection) to fermions. Hence the modified Dirac action defining the analog of free field theory should appear as a basic building block in the definition of the inner product. Perturbation theory with respect to the induced gauge potential would conform with standard QFT but does not make sense. There is simply no decomposition of the modified Dirac operator D to "free" part and interaction term.

2. The vacuum expectation for the exponent of the modified Dirac action gives vacuum functional identified as exponent of Kähler function. When one sandwiches the exponent of Dirac action between many-fermion states, one obtains an inner product analogous to that in free field theory Feynman rules. How however the states are are not annihilated by D but are its generalized eigenstates with eigenvalues λ depending on p-adic prime by an overall scaling factor log(p) responsible for the coupling constant evolution. The generalized eigenvalue equation reads as DΨ= λ t^kΓ_kΨ, where t^k is lightlike vector t^k defining the tangent vector of partonic 3-surface or its M⁴ dual fixed once rest system and quantization axis of angular momentum has been fixed (it is not yet quite clear which option is correct). The notion of generalized eigenmode allows also to define Dirac determinant without giving up the separate conservation of H-chiralities (B and L). The generalized eigenstates are analogs of solutions of massless wave equation in the sense that the square of D annihilates them. Between states created by a monomial of fermionic oscillator operators the inner product reduces to a product of propagators.

3. A strict correspondence with free field theory would require that the incoming and outgoing states correspond to zero modes with λ=0 whereas internal lines as off mass shell states would correspond to non-vanishing eigenvalues λ . This assumption is however un-necessary since the four-momentum dependence comes only through the Fourier transform and one can regard all generalized eigenmodes as counterparts of massless modes. The restriction might be also inconsistent with unitarity.

4. For generalized eigenstates of D the modified Dirac propagator 1/D reduces to o^kΓ_k/λ. o^k is the light-like M⁴ dual of the lightlike vector t^k and λ is the generalized eigenvalue of D proportional to log(p). The propagator can be non-vanishing between vacuum and a boson consisting of fermion with physical helicity and antifermion with non-physical helicity so that non-trivial boson emission vertices are possible. At first it would seem that the inverse of the generalized eigenvalue λ contributes to the p-adic coupling constant evolution an overall 1/log(p) proportionality factor. However, since the inner product of un-normalized "bare" boson states (just fermion pair) is proportional to 1/log(p), the normalization of bosonic states cancels this factor so that algebraic number results. Thus fermionic contributions to the vertices are extremely simple since only the matrix o^kΓ_k remains. The conclusion made already earlier is that the p-adic coupling constant evolution must be due to the time evolution along parton lines dictated by the modified Dirac operator.

5. The fermionic contribution to the vertex says nothings about gauge couplings. All gauge coupling strengths must be proportional to the RG invariant Kähler coupling strength α_K, which can emerge only from the functional integral over small fluctuations around maximum of Kähler function K when the operator inverse of the covariant configuration space Kähler metric defining propagator is contracted between bosonic vector fields generating Kac-Moody and super-canonical symmetries in terms of which the deformation of the partonic 3-surface can be expressed. Obviously the configuration space spinor fields representing bosonic states must vanish at the maximum of K: otherwise coupling strength is of order unity. Geometrically this means that the maxima of Kähler function correspond to fixed points of these isometries.

3. Number theoretical constraints

The condition that S-matrix elements are algebraic numbers is an additional powerful guideline.

1. The most straightforward manner to guarantee that S-matrix elements are algebraic numbers is that vertex factors and propagators are separately algebraic numbers. log(p)-factors are obviously problematic number theoretically but normalization of the Fock space inner products cancels these factors. Thus coupling constant evolution can come only from the unitary time evolution with respect to the light-like coordinate of propagator lines dictated by the modified Dirac operator. Fermionic oscillator operators suffer a non-trivial unitary transformation depending on the p-adic prime p since (expressing it schematically) e^iHt is replaced by p^iHt.

2. The fundamental number theoretic conjecture is that the numbers p^s_n, where s_n=1/2+iy_n correspond to non-trivial zeros of Riemann zeta (or of more general zetas possibly involved), are algebraic numbers. If this is the case, then also the products and sums with rational coefficients involving finite number of nontrivial zeros of zeta are algebraic numbers and define a commutative algebra. The effect of the unitary time evolution operator should be expressible as an element of this algebra. Also larger algebraic extensions can be considered.

3. A simplified picture is provided by the dynamics of free number theoretic Hamiltonian for which eigenstates are labelled by primes and energy eigenvalues are given by E_p= log(p). Time evolution gives rise to phase factors exp(iE_pt)=p^it which are algebraic numbers in given extension of rationals for some quantized values of light-like coordinate t. If the conjectures about zeros of zeta hold true this is achieved if t is a linear combination of imaginary parts of zeros of zeta with integer coefficients: t= ∑_n k(n) y_n.

For more details see the chapter Hyper-Finite Factors and Construction of S-Matrix of the book "Towards S-matrix".

P.S. The following saying of the week from Tommaso Dorigo's blog somehow resonates with my inner feelings.

I make a living as a lawyer, and I spend a lot of time in the world of physics. I have encountered a lot of sleazy scumbags in the world of lawyers, but none of them are as bad as the bad guys in physics, and, although the bad guys in law do a lot of damage, I really think that the bad guys in physics do more damage to human civilization.