Thursday, February 25, 2016

Does M4 Kähler form imply new physics?

The introduction of M4 Kähler form strongly suggested by the twistor formulation of TGD could bring in new gravitational physics.

  1. As found, the twistorial formulation of TGD assigns to M4 a self dual Kähler form whose square gives Minkowski metric. It can (but need not if M4 twistor space is trivial as bundle) contribute to the 6-D twistor counterpart of Kähler action inducing M4 term to 4-D Kähler action vanishing for canonically imbedded M4.

  2. Self-dual Kähler form in empty Minkowski space satisfies automatically Maxwell equations and has by Minkowskian signature and self-duality a vanishing action density. Energy momentum tensor is proportional to the metric so that Einstein Maxwell equations are satisfied for a non-vanishing cosmological constant! M4 indeed allows a large number of self dual Kähler fields (I have christened them as Hamilton-Jacobi structures). These are probably the simplest solutions of Einstein-Maxwell equations that one can imagine!

  3. There however exist quite a many Hamilton-Jacobi structures. However, if this structure is to be assigned with a causal diamond (CD) it must satisfy additional conditions, say SO(3) symmetry and invariance under time translations assignable to CD. Alternatively, covariant constancy and SO(2)⊂ SO(3) symmetry might be required.

    1. In the case of causal diamond (CD) a spherically symmetric self-dual monopole Kähler form with non-vanishing components Jtr= εtrθφJθφ, Jθφ=cos(θ) carrying radial electric and magnetic fields with identical gravitational charges looks rather natural option. The time-like line connecting the tips of CD would carry a genuine self-dual monopole so that Dirac monopole would not be in question. The potential associated with J could be chosen to be Aμ ↔ (1/r,0,0,sin(θ). I have considered this kind of possibility earlier in context of TGD
      inspired model of anyons but gave up the idea.

      The moduli space for CDs with second tip fixed would be hyperbolic space H3=SO(3,1)/SO(3) or a space obtained by identifying points at the orbits of some discrete subgroup of SO(3,1) as suggested by number theoretic considerations. This induced Kähler field could make the blackholes with center at this line to behave like M4 magnetic monopoles if the M4 part of Kähler form is induced into the 6-D lift of Kähler action with extremely small coefficients of order of magnitude of cosmological constant. Cosmological constant and the possibility of CD monopoles would thus relate to each other.

    2. Covariant constancy is an alternative option. This would leave only the fields Jtz =Jxy=1 unique apart from Lorentz transformation: it would be attractive to assign this Kähler with given CD to define a preferred plane M2 required also by the number theoretic vision. Now however rotational invariance is broken to SO(1,1)× SO(2). SO(3,1)/SO(1,1)× SO(2) would define moduli for CDs. Magnetic and electric parts of Kähler form would be in z-direction and flux tubes would tend to be in this direction. One would have clearly a preferred direction and it is difficult to imagine how the gravitational field of blackhole could correlate with these fluxes unless one assigns to each flux tube its own CD.

  4. A further interpretational problem is that the classical coupling of M4 Kähler gauge potential to induced spinors is not small. Can one really tolerate this kind of coupling equivalent to a coupling to a self dual monopole field carrying electric and magnetic charges? One could of course consider the condition that the string world sheets carrying spinor modes are such that the induced M4 Kähler form vanishes and gauge potential become pure gauge. M4 projection would be 2-D Lagrange manifold whereas CP2 projection would carry vanishing induce W and possibly also Z0 field in order that em charge is well defined for the modes. These conditions would fix the string world sheets to a very high degree in terms of maps between this kind of 2-D sub-manifolds of M4 and CP2. Spinor dynamics would be determined by the avoidance of interaction!

    It must be emphasized that the imbedding space spinor modes characterizing the ground states of super-symplectic representations would not couple to the monopole field so that at this level Poincare invariance is not broken. The coupling would be only at the space-time level and force spinor modes to Lagrangian sub-manifolds.

  5. At the static limit of GRT and for gij≈ δij implying SO(3) symmetry there is very close analogy with Maxwell's equations and one can speak of gravi-electricity and gravi-magnetism with 4-D vector potential given by the components of g. The genuine U(1) gauge potential does not however relate to the gravimagnetism in GRT sense. Situation would be analogous to that for CP2, where one must add to the spinor connection U(1) term to obtain respectable spinor structure. Now the U(1) term would be added to trivial spinor connection of flat M4: its presence would be justified by twistor space Kähler structure. If the induced M4 Kähler form is present as a classical physical field it means genuinely new contribution to gravitational interaction and assignable to cosmological constant.

    I have talked much about gravitational flux or gravitons are carried along Kähler magnetic monopole flux tubes. This is quite respectable hypothesis. One can however ask whether the gravitational interaction could be mediated along flux tubes of M4 Kähler magnetic field carrying monopole flux. For the proposed SO(3) symmetric option the flux tubes would emanate radially from the origin and one could assign to each gravitating object CD. It is of course quite possible that Kähler magnetic flux tubes and gravitational flux tubes are one and same thing in astrophysical systems. Note however that Kähler magnetic monopole fluxes do not involve genuine monopole like M4 Kähler fluxes in SO(3) symmetric case.

For background see the chapter From Principles to giagrams of "Towards M-matrix" or the article From Principles to Diagrams.

For a summary of earlier postings see Links to the latest progress in TGD.

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