How to handle the interfaces between Minkowskian and Euclidean regions of space-time surface?

The treatment of  the dynamics at the   interfaces X³ between Minkowskian and Euclidean regions X³  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 X³  degenerates to 2-D effective Euclidean metric apart from 2-D delta function singularities  X² 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.

1.  The  interface X³ is  regarded as an independent dynamic unit.  The earlier approaches rely on this assumption.  By the light-likeness of X³, 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 X³.  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 X³.

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. In the 4-dimensional treatment  there are no analogs of the boundary conditions at the interface.

1. It is essential that the 3-D light-like orbit X³ 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 X³,  where the  interior contributions from the 2 sides  give rise to  a 2-D  delta function term.
2. The contravariant metric diverges at X³  since  by holography one has g_uv=0 at X³ outside X². 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 X³  by what I have called  electric-magnetic duality   proposed years ago (see this).  At X³, the dynamics would be effectively reduced to 2-D Euclidean degrees of freedom outside X². Everything would be finite as far as Kähler action is considered.
3. Since the metric at X³ 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 X². 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 X² so that det(g₂)=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 X³  outside X²) 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_uΨ=0 and D_vΨ=0 in X³ except at the 2-D singularities X², 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 X². At X²,  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. The following overview of the symmetry breaking through the generation of 2-D  singularities is suggestive.  Masslessess and holomorphy are violated via the  generation of the analog of Higgs expectation at the vertices. The use of the  induced gamma matrices violates supersymmetry  guaranteed by the use of the modified gamma matrices   but only at the vertices.

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.

Source: http://matpitka.blogspot.com/2024/05/how-to-handle-interfaces-between.html