^{3}between Minkowskian and Euclidean regions X

^{3}of the space-time surface identified as light-like partonic orbits has turned out to be a difficult technical problem. By holomorphy as a realization of generalized holography, the 4-metric at X

^{3}degenerates to 2-D effective Euclidean metric apart from 2-D delta function singularities X

^{2}at which the holomorphy fails but the metric is 4-D.

One must treat both the bosonic and fermionic situations. There are two options for the treatment of the interface dynamics.

- The interface X
^{3}is regarded as an independent dynamic unit. The earlier approaches rely on this assumption. By the light-likeness of X^{3}, C-S-K action is the only possible option. The problem with U(1) gauge invariance disappears if C-S-K action is identified as a total divergence emerging from the instanton term for Kähler action.One can assign to the instanton term a corresponding contribution to the modified Dirac action at X

^{3}. It however seems that the instanton term associated with the 4-D modified Dirac action does not reduce to a total divergence allowing to localize it a X^{3}.In this approach, conservation laws require that the normal components of the canonical momentum currents from the Minkowskian and Euclidean sides add up to the divergence of the canonical momentum currents associated with the C-S-K action.

- Since the interface is not a genuine boundary, one can argue that one should treat the situation as 4-dimensional. This approach is adopted in this article. In the bosonic degrees of freedom, the C-S-K term is present also for this option could determine the bosonic dynamics of the boundary apart from a 2-D delta function type singularities coming from the violation of the minimal surface property and of the generalized holomorphy. At vertices involving fermion pair creation this violation would occur.

- It is essential that the 3-D light-like orbit X
^{3}is a 2-sided surface between Minkowskian and Euclidean domains. The variation of the C-S-K term emerging from a total divergence could determine the dynamics of the interface except possibly at the singularities X^{3}, where the interior contributions from the 2 sides give rise to a 2-D delta function term. - The contravariant metric diverges at X
^{3}since by holography one has g_{uv}=0 at X^{3}outside X^{2}. The condition J_{uv}= 0 could guarantee that the contribution of the Kähler action remains finite. The contribution from Kähler action to field equations could be even reduced to the divergence of the instanton term at X^{3}by what I have called electric-magnetic duality proposed years ago (see this). At X^{3}, the dynamics would be effectively reduced to 2-D Euclidean degrees of freedom outside X^{2}. Everything would be finite as far as Kähler action is considered. - Since the metric at X
^{3}is effectively 2-D, the induced gamma matrices are proportional to 2-D delta function and by J_{uv}=0 condition the contribution of the volume term to the modified gamma matrices dominates over the finite contribution of the Kähler action. This holds true outside the 2-D singularities X^{2}. In this sense the idea that only induced gamma matrices matter at the interfaces, makes sense.In order to obtain the counterpart of Einstein's equations the metric must be effectively 2-D also at X

^{2}so that det(g_{2})=0 is true although holomorphy fails. It seems that one must assume induced, rather than modified, gamma matrices (effectively reducing to the induced ones at X^{3}outside X^{2}) since for the latter option the gravitational vertex would vanish by the field equations.The situation is very delicate and I cannot claim that I understand it sufficiently. It seems that the edge of the partonic orbit due to the turning of the fermion line and involving hypercomplex conjugation is essential.

- For the modified Dirac equation to make sense, the vanishing of the covariant derivatives with respect to light-like coordinates seems necessary. One would have D
_{u}Ψ=0 and D_{v}Ψ=0 in X^{3}except at the 2-D singularities X^{2}, where the induced metric would have diagonal components g_{uu}and g_{vv}. This would give rise to the gauge boson vertices involving emission of fermion-antifermion pairs. - By the generalized holomorphy, the second fundamental form H
^{k}vanishes outside X^{2}. At X^{2}, H^{k}is proportional to a 2-D delta function and also the Kähler contribution can be of comparable size This should give the TGD counterpart of Einstein's equations and Newtonian equations of motion and to the graviton vertex.The orientations of the tangent spaces at the two sides are different. The induced metric at the Minkowskian side would become 4-D. At the Euclidean side it could be Euclidean and even metrically 2-D.

There is however an objection. The use of the induced gammas in the modified Dirac equation seems necessary although the non-vanishing of H^{k} seems to violate the hermiticity at the vertices. Can the turning of the fermion line and the exotic smooth structure allow to get rid of this problem?

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.

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