String world sheets, partonic 2-surfaces and vanishing of induced (classical) weak fields
Well-definedness of the em charge is the fundamental on spinor modes. Physical intuition suggests that also classical Z0 field should vanish - at least in scales longer than weak scale. Above the condition guaranteeing vanishing of em charge has been discussed at very general level. It has however turned out that one can understand situation by simply posing the simplest condition that one can imagine: the vanishing of classical W and possibly also Z0 fields inducing mixing of different charge states.
- Induced W fields mean that the modes of Kähler-Dirac equation do not in general have well-defined em charge. The problem disappears if the induced W gauge fields vanish. This does not yet guarantee that couplings to classical gauge fields are physical in long scales. Also classical Z0 field should vanish so that the couplings would be purely vectorial. Vectoriality might be true in long enough scales only. If W and Z0 fields vanish in all scales then electroweak forces are due to the exchanges of corresponding gauge bosons described as string like objects in TGD and represent non-trivial space-time geometry and topology at microscopic scale.
- The conditions solve also another long-standing interpretational problem. Color rotations induce rotations in electroweak-holonomy group so that the vanishing of all induced weak fields also guarantees that color rotations do not spoil the property of spinor modes to be eigenstates of em charge.
- The representation of the covariantly constant curvature tensor is given by
R01= e0 ∧ e1-e2∧ e3 , R23= e0∧ e1- e2∧ e3 ,
R02=e0∧ e2-e3 ∧ e1 , R31 = -e0∧
e2+e3∧ e1 ,
R03 = 4e0∧ e3+2e1∧ e2
, R12 = 2e0∧ e3+4e1∧ e2 .
R01=R23 and R03= R31 combine to form purely left handed classical W boson fields and Z0 field corresponds to Z0=2R03.
Kähler form is given by
J= 2(e0∧e3+e1∧ e2) .
- The vanishing of classical weak fields is guaranteed by the conditions
e0∧ e1-e2∧e3 =0 ,
e0∧ e2-e3 ∧e1 ,
4e0∧ e3+2e1∧e2 .
- There are many manners to satisfy these conditions. For instance, the condition e1= a× e0 and e2=-a×e3 with arbitrary a which can depend on position guarantees the vanishing of classical W fields. The CP2 projection of the tangent space of the region carrying the spinor mode must be 2-D.
Also classical Z0 vanishes if a2= 2 holds true. This guarantees that the couplings of induced gauge potential are purely vectorial. One can consider other alternaties. For instance, one could require that only classical Z0 field or induced Kähler form is non-vanishing and deduce similar condition.
- The vanishing of the weak part of induced gauge field implies that the CP2 projection of the region carrying spinor mode is 2-D. Therefore the condition that the modes of induced spinor field are restricted to 2-surfaces carrying no weak fields sheets guarantees well-definedness of em charge and vanishing of classical weak couplings. This condition does not imply string world sheets in the general case since the CP2 projection of the space-time sheet can be 2-D.
- Additional consistency condition to neutrality of string world sheets is that Kähler-Dirac gamma matrices have no components orthogonal to the 2-surface in question. Hence various fermionic would flow along string world sheet.
- If the Kähler-Dirac gamma matrices at string world sheet are expressible in terms of two non-vanishing gamma matrices parallel to string world sheet and sheet and thus define an integrable distribution of tangent vectors, this is achieved. What is important that modified gamma matrices can indeed span lower than 4-D space and often do so (massless extremals and vacuum extremals representative examples). Induced gamma matrices defined always 4-D space so that the restriction of the modes to string world sheets is not possible.
- String models suggest that string world sheets are minimal surfaces of space-time surface or of imbedding space but it might not be necessary to pose this condition separately.
- The vanishing conditions for induced weak fields allow also 4-D spinor modes if they are true for entire spatime surface. This is true if the space-time surface has 2-D projection. One can expect that the space-time surface has foliation by string world sheets and the general solution of K-D equation is continuous superposition of the 2-D modes in this case and discrete one in the generic case.
- If the CP2 projection of space-time surface is homologically non-trivial geodesic sphere S2, the field equations reduce to those in M4× S2 since the second fundamental form for S2 is vanishing. It is possible to have geodesic sphere for which induced gauge field has only em component?
- If the CP2 projection is complex manifold as it is for string like objects, the vanishing of weak fields might be also achieved.
- Does the phase of cosmic strings assumed to dominate primordial cosmology correspond to this phase with no classical weak fields? During radiation dominated phase 4-D string like objects would transform to string world sheets.Kind of dimensional transmutation would occur.
- Electroweak gauge potentials do not couple to νR at all. Therefore em neutrality condition is un-necessary if the induced gamma matrices do not mix right handed neutrino with left-handed one. This is guaranteed if M4 and CP2 parts of Kähler-Dirac operator annihilate separately right-handed neutrino spinor mode. Also νR modes can be interpreted as continuous superpositions of 2-D modes and this allows to define overlap integrals for them and induced spinor fields needed to define WCW gamma matrices and super-generators.
- For covariantly constant right-handed neutrino mode defining a generator of super-symmetries is certainly a solution of K-D. Whether more general solutions of K-D exist remains to be checked out.