What modified Dirac action is and how it determines scattering amplitudes?

Holography=generalized holomorphy property means that minimal surface field equations are true outside singularities for any general coordinate invariant action constructible in terms of the induced geometry. However, the twistor lift of TGD suggests that 6-D Kähler action is the fundamental action. It reduces to 4-D Kähler action plus volume term in the dimensional reduction guaranteeing that the 6-surface can be regarded as a generalization of twistor space having space-time surface as a base-space and 2-sphere.

One can express the induced spinor field obtained as a restriction of the second quantized H spinor field to the space-time surface and it satisfies modified Dirac equation (see this).

Modified Dirac action L_D is defined for the induced spinor fields.

1. It is fixed by the condition of hermiticity stating that the canonical momentum currents appearing in it have a vanishing divergence. If the modified gamma matrices Γ^α are defined by an action S_B defining the space-time surface itself, they are indeed divergenceless by field equations. This implies a generalization of conformal symmetry to the 4-D situation (see this) and the modes of the modified Dirac equation define super-symplectic and generalized conformal charges defining the gamma matrices of WCW (see this).
2. Generalized holomorphy implies that S_B could be chosen to correspond to modified gamma matrices defined by the sum of L_K+L_V or even by L_V defining induced gamma matrices. Which option is more plausible?
3. An attractive guiding physical idea is that the singularities are not actually singularities if exotic diffeo structure is inducted. Field equations hold true but with S_K+S_V. The singularities would cancel. One would avoid problems with the conservation laws by using exotic diffeo structure.
4. At the short distance limit for which α_K is expected to diverge as a U(1) coupling, the action reduces to S_V and the defects would be absent. Only closed cosmic strings and monopole flux tubes would be present but wormhole contacts and string world sheets identifiable as defects are absent: this would be the situation in the primordial cosmology (see this). Only dark energy as classical energy of the cosmic strings and monopole flux tubes would be present and there would be no elementary particles and elementary particle scattering at this limit.

One can consider several options assuming that the singularities are not actually present for the exotic diffeo structures.

Option 1: The first option relies on the assumption that the exponential of the modified Dirac action is imaginary and analogous to the phase defined by the action in QFTs. This is enough in TGD since fermions are the only fundamental particles and bosonic action is a purely classical notion.

1. Volume action is in a very special role in that it represents both the classical dynamics of particles as 3-D surfaces as analogs of geodesic lines, the classical geometrized dynamics of massless fields, and generalizes the Laplace equations of complex analysis.

This motivates the proposal that only induced the gamma matrices Γ^αgαβhk_βγ_k (no contribution from L_K) corresponding to S_V appear in L_D and the bosonic action S_B=S_K+S_V+S_I, where S_I is real, is defined by the twistor lift of TGD. The field equations are satisfied also at the singularities so that the contributions from S_K+S_I and S_V cancel each other at the singularity in accordance with the idea that an exotic diffeo structure is in question. Both S_K and S_I contributions would have an imaginary phase.

The number theoretical parameters of the bosonic action determined by the hierarchy of extensions of rationals would parametrize different exotic diffeo structures and make themselves visible in the quantum dynamics in this way. S_I would contribute to classical charges and its M⁴ part would contribute to the Poincare charges.

Option 2: Modified gamma matrices are defined by S_K+S_V +iS_I and the real part of the singularity vanishes. The imaginary part cannot vanish simultaneously.

1. The exponent of Kähler function defines a real vacuum functional and K is determined by S_K+S_V whereas the action exponential of QFTs of QFTs defines a phase. In topological QFTs, the contribution of the instanton term S_D,I is naturally purely imaginary and could define “imaginary part of the Kähler function K, which does not contribute to the Kähler metric of WCW.

One can argue that this must be the case also for S_D. Hence the contribution of S_K+S_V to S_D would be real and differ by a multiplication with i from that in QFTs whereas the contribution of iS_I would be imaginary. One must admit that this is not quite logical. Also the contribution to the Noether charges would be imaginary. This does not look physically plausible.

For the representations of Kac-Moody algebras the coefficient of Chern-Simons action is k/4π and allows an interpretation as quantization of α_K as α_K= 1/k. Scattering vertices would be universal and determined by an almost topological field theory. Almost comes from the fact that the exponent of S_B defines the vacuum functional.

The volume term included in the definition of the induced gamma matrices must be normalized by 1/L_p⁴. L_p is a p-adic length scale and is roughly of order of a biological scale L_p≈ 10^-4 meters if the scale dependent cosmological constant Λ corresponds to the inverse squared for the horizon radius. One has 1/L_p⁴= 3Λ/8π G. This guarantees the expected rather slow coupling constant evolution induced by that of α_K diverging in short scales. See the article What gravitons are and could one detect them in the TGD Universe?) and the chapter the chapter About the Relationships Between Electroweak and Strong Interactions and Quantum Gravity in the TGD Universe.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.