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.

- 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). - 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? - 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. - 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.

** 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.

- 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

^{α}g^{αβ}h^{k}_{β}γ_{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. - 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 M^{4}part would contribute to the Poincare charges. - 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 X

^{2}splits to two.At X

^{2}the trace of the second fundamental form can be well-defined and proportional to a 2-D delta function at X^{2}since the 4-metric metric has one light-like direction at X^{2}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. - The divergence of g
^{μν}h^{k}_{ν}vertex as the trace of the second fundamental form D_{α}h^{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 asg

^{μν}D_{ν}∂_{μ}h^{k}= P^{k}_{l}H^{l}, P^{k}_{l}= h^{k}_{l}- g^{μν}h^{k}_{μ}h_{lr}h^{r}_{ν},H

^{k}= g^{αβ}(∂_{α}+B^{k}_{α})g^{αβ}h^{k}_{β}, B^{k}_{α}= B^{k}_{lm}h^{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}_{lm}h^{m}_{α}is the projection of the Riemann connection of H to the space-time surface. - This allows a very elegant physical interpretation. In linear Minkowski coordinates for M
^{4}, one has B^{k}_{α}=0 but the presence of the CP_{2}contribution coming from the orthonormal projection implies that the covariant divergence is non-vanishing and proportional to the radius squared of2. Vertex is proportional to the trace of the second fundamental form, whose CP _{2}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.

- 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. - 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. - 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.

_{K}=1/k appears in the action.

- 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. - 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.

- 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
_{α}T^{αkC-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 M
^{4}spinors. This looks natural since CP_{2}is compact.The volume term included in the definition of the induced gamma matrices must be normalized by 1/L

_{p}^{4}. 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_{p}^{4}= 3Λ/8π G. This guarantees the expected rather slow coupling constant evolution induced by that of α_{K}diverging in short scales.

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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