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

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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 LD 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 Γα are defined by an action SB 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 SB could be chosen to correspond to modified gamma matrices defined by the sum of LK+LV or even by LV 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 SK+SV. 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 αK is expected to diverge as a U(1) coupling, the action reduces to SV 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 Γαgαβhkβγk (no contribution from LK) corresponding to SV appear in LD and the bosonic action SB=SK+SV+SI, where SI is real, is defined by the twistor lift of TGD. The field equations are satisfied also at the singularities so that the contributions from SK+SI and SV cancel each other at the singularity in accordance with the idea that an exotic diffeo structure is in question. Both SK and SI contributions would have an imaginary phase.

  • Therefore LV, which involves cosmological constant Λ, disappears from the scattering amplitudes by the field equations for LB although it is implicitly present. The number theoretic evolution of the SK+SI makes itself visible in the scattering vertices. Outside the singularities both terms vanish separately but at singularities this is not the case. Only lower-D singularities contribute to the scattering amplitudes.
  • 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. SI would contribute to classical charges and its M4 part would contribute to the Poincare charges.

  • An objection against this option is that the divergence of the modified gamma matrices defined by the SK+SI need not be well-defined. It should be proportional to a delta function located at the singularity. For 3-D light-like surfaces the induced metric is singular and it is not clear whether the divergence at singularity gives 3-D delta function as a limit.
  • Option 2: Modified gamma matrices are defined by SK+SV +iSI 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 SK+SV whereas the action exponential of QFTs of QFTs defines a phase. In topological QFTs, the contribution of the instanton term SD,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 SD. Hence the contribution of SK+SV to SD would be real and differ by a multiplication with i from that in QFTs whereas the contribution of iSI 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.

  • One cannot require the vanishing of both the real part and imaginary part of the divergence of the modified gamma matrices at the singularity. The contribution of LC-S-K at the singularity would be non-vanishing and determine scattering amplitudes and imply their universality.
  • For the representations of Kac-Moody algebras the coefficient of Chern-Simons action is k/4π and allows an interpretation as quantization of αK as αK= 1/k. Scattering vertices would be universal and determined by an almost topological field theory. Almost comes from the fact that the exponent of SB defines the vacuum functional.

  • The real exponential exp(K) of the real Kähler function defined by SK+SV would be visible in the WCW vacuum functional and bring in an additional dependence on the αK and cosmological constant Λ, whose number theoretic evolution would fix the evolution of the other coupling strengths. Note that the induced spinor connection corresponds in gauge theories to gauge potentials for which the gauge coupling is absorbed as a multiplicative factor. There are therefore two options. For both cases 1/αK=1/k appears in the action.
    1. For Option 1 only iSV appears in SD and iSK+ iSC-S-K determines the scattering amplitudes for option 2). Exponent of the modified Dirac action defines the analog of the imaginary action exponential of QFTs.
    2. For Option 2 for which the entire action defines the modified gamma matrices the iSC-S-K defines the scattering amplitudes and one has an analog of topological QFT. This picture would conform with an old proposal that in some sense TGD is a topological quantum field theory. One can however argue that the treatment of SK+SV and SI in different ways does not conform with QFT treatment and also the Noether charges are a problem.

    Some technical remarks are in order.

    1. The spinor connection does not disappear from the dynamics at the singularities. It is transformed to components of projected Riemann connection of H appearing in the divergence DαTαkC-S-K.
  • The modified Dirac action must be dimensionless so that the scaling dimension of the induced spinors should be d=-3/2 and therefore same as the scaling dimension of M4 spinors. This looks natural since CP2 is compact.
  • The volume term included in the definition of the induced gamma matrices must be normalized by 1/Lp4. Lp is a p-adic length scale and is roughly of order of a biological scale Lp≈ 10-4 meters if the scale dependent cosmological constant Λ corresponds to the inverse squared for the horizon radius. One has 1/Lp4= 3Λ/8π G. This guarantees the expected rather slow coupling constant evolution induced by that of αK 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.


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