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^αg^αβh^k_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.

2. Therefore L_V, which involves cosmological constant Λ, disappears from the scattering amplitudes by the field equations for L_B although it is implicitly present. The number theoretic evolution of the S_K+S_I 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. S_I would contribute to classical charges and its M⁴ part would contribute to the Poincare charges.

3. An objection against this proposal is that the divergence of the modified gamma matrices defined by the S_K+S_I need not be well-defined. It should be proportional to a lower-dimensional delta function located at the singularity.

   For 3-D light-like light-partonic orbits, the contravariant induced metric appearing in the trace of the second fundamental form has diverging components but it is not clear whether the trace of the second fundamental form can give rise to a 3-D delta function at this limit. Chern-Simons action at the light-like partonic orbit coming from the instanton term is well-defined and field equations should not give rise to a singularity except at partonic 2-surfaces, which have been identified as analogs of vertices at which the partonic 2-surface X² splits to two.

   At X² the trace of the second fundamental form can be well-defined and proportional to a 2-D delta function at X² since the 4-metric metric has one light-like direction at X² and has a vanishing determinant and is therefore is effectively 2-D (the light-like components of g_uv =g_vu of the 4-metric vanish). Therefore vertices would naturally correspond to partonic 2-surfaces, which split to two at the vertex. This is indeed the original proposal.

4. The divergence of g^μνh^k_ν vertex as the trace of the second fundamental form D_αh^k_β defined by the covariant derivatives of coordinate gradients, appears in the vertex. The second fundamental form is orthogonal to the space-time surface and can be written as

   g^μνD_ν∂_μh^k= P^k_lH^l , P^k_l = h^k_l- g^μνh^k_μh_lrh^r_ν ,

   H^k= g^αβ (∂_α+B^k_α)g^αβh^k_β , B^k_α= B^k_lmh^m_α .

   P^k_l projects to the normal space of the space-time surface. H^k is covariant derivative of h^k_α and B^k_α= B^k_lmh^m_α is the projection of the Riemann connection of H to the space-time surface.

5. This allows a very elegant physical interpretation. In linear Minkowski coordinates for M⁴, one has B^k_α=0 but the presence of the CP₂ contribution coming from the orthonormal projection implies that the covariant divergence is non-vanishing and proportional to the radius squared of 2. Vertex is proportional to the trace of the second fundamental form, whose CP₂ part is analogous to the Higgs field of the standard model. This field is vanishing in the interior by the minimal surface property in analogy with the generalized Equivalence Principle.

   The trace of the second fundamental form is a generalization of acceleration from 1-D case to 4-D situation so that the interaction vertices are lower-dimensional regions of the space-time surface which experience acceleration. The regions outside the vertices represent massless fields geometrically. At the singularities the Higgs-like field is non-vanishing so that there is mass present. The analog of Higgs vacuum expectation is non-vanishing only at the defects.

   It seems that a circle is closing. I started more than half a century ago from Newton's "F=ma" and now I discover it in the interaction vertex, which is the core of quantum field theories! I almost see Newton nodding and smiling and saying "What I said!".

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.

2. 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 L_C-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 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.

3. The real exponential exp(K) of the real Kähler function defined by S_K+S_V would be visible in the WCW vacuum functional and bring in an additional dependence on the α_K 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/α_K=1/k appears in the action.

1. For Option 1 only iS_V appears in S_D and iS_K+ iS_C-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 iS_C-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 S_K+S_V and S_I 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 D_αTα_kC-S-K.

2. 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/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 TGD Universe? or the chapter with the same title.

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.