Connection of Beltrami flows with PCAC hypothesis, massivation,  and CP violation

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Connection of Beltrami flows with PCAC hypothesis, massivation,  and CP violation

Conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis (PCAC) are essential elements of old-fashioned hadron physics and hold true also in the standard model.

  1. The simplest ansatz, which realizes the Beltrami hypothesis, states that the vectorial Kähler current J equals apart from sign c=+/- 1 to instanton current I, which is axial current:

    J=+/- I .

    The condition states that only the left or right handed current chiral defined as

    JL/R= J+/- I

    is non-vanishing. For c≠ 1, both JL and JR are non-vanishing. Since both right- and left-handed weak currents exist, c≠ 1 seems to be a plausible option.

    By quantum classical correspondence, these currents serve as space-time correlates for the left- and right-handed fermion currents of the standard model. Note however that induced gamma matrices differ from those of M4: for instance, they are not covariantly constant but defines a current with divergence which vanishes by field equations.

    A more general condition would allow c to depend on space-time coordinates. The conservation of J forces conservation of I if the condition ∂αcIα=0 is true. This gives a non-trivial condition only in regions with 4-D CP2 and M4 projections.

  2. The twistor lift of TGD requires that also M4 has Kähler structure. Therefore J and I and corresponding Kähler gauge potential AK have both M4 part and CP2 parts and Kähler action K, AK, JK, J and I are sums of M4 and CP2 parts:

    AK= A(M4)+A(CP2),
    JK=JK(M4)+JK(CP2) ,
    K = K(M4)+K(CP2) ,
    J =J(M4)+J(CP2) ,
    I= I(M4)+I(CP2) .

    Only the divergence of I must vanish:

    αIα=0 .

    A possible interpretation is in terms of the 8-D variant of twistorialization by twistor lift requiring masslessness in an 8-D sense.

    PCAC states that the divergence of the axial current is non-vanishing. This is not in conflict with the conservation of the total instanton current I. PCAC corresponds to the non-conservation I(CP2), whose non-conservation is compensated by that of I(M4).

  3. For regions with at most 3-D M4- and CP2 projections, the M4- and CP2 instanton currents have identically vanishing divergence. In these regions the conservation of I is not lost if c has both signs. c could be also position dependent and even differ for I(M4) and I(CP2) in these regions.

    DαIα=0 is true for the known extremals. For the simplest CP2 type extremals and for extremals with 2-D CP2 projection, I itself vanishes. Therefore parity violation is not possible in these regions. This would suggest that these regions correspond to a massless phase.

  4. DαIα≠ 0 is possible only if both M4 and CP2 projections are 4-D. This phase is interpreted as a chaotic phase and by the non-conservation of electroweak axial currents could correspond to a massive phase.

    CP2 type extremals have 4-D projection and for them Kähler current and instanton current vanish identically so that also they correspond to massless phase (M4 projection is light-like). Could CP2 type extremals allow deformations with 4-D M4 projection (DEs)?

    The wormhole throat between space-time region with Minkowskian signature of the induced metric and CP2 type extremal (wormhole contact) with Euclidian signature is light-like and the 4-metric is effectively 3-D. It is not clear whether this allows 4-D M4 projection in the interior of DE.

The geometric model for massivation based on zitterbewegung of DE provides additional insight.
  1. M8-H duality allows to assign a light-like curve also to DE. For space-time surfaces determined by polynomials (cosmological constant Λ>0), this curve consists of pieces which are light-like geodesics.

    Also real analytic functions (Λ=0) can be considered and they would allow a continuous light-like curve, whose definition boils down to Virasoro conditions. In both cases, the zigzag motion with light-velocity would give rise to velocity v<c in long length scales having interpretation in terms of massivation.

  2. The interaction with J(M4) would be essential for the generation of momentum due to the M4 Chern-Simons term assigned with the 3-D light-like partonic orbit. M4 Chern-Simons term can be interpreted as a boundary term due to the non-vanishing divergence of I(M4) so that a connection with two views about massivation is obtained. Does the Chern-Simons term come from the Euclidean or Minkowskian region?
I have proposed two models for the generation of matter-antimatter asymmetry. In both models, CP breaking by M4 Kähler form is essential. Classical electric field induces CP breaking. CP takes self-dual (E,B) to anti-self-dual (-E,B) and self-duality of J(M4) does not allow CP as a symmetry.
  1. In the first model the electric part of J(M4) would induce a small CP breaking inside cosmic strings thickened to flux tubes inducing in turn small matter-antimatter asymmetry outside cosmic strings. After annihilation this would leave only matter outside the cosmic strings.
  2. In the simplest variant of TGD only quarks are fundamental particles and leptons are their local composites in CP2 scale. Both quarks and antiquarks are possible but antiquarks would combine leptons as almost local 3-quark composites and presumably realized CP2 type extremals with the 3 antiquarks associated with the partonic orbit. I should vanish identically for the DEs representing quarks and leptons but not for antiquarks and antileptons.

    Could the number of DEs with vanishing I be smaller for antiquarks than for quarks by CP breaking and could this induce leptonization of antiquarks and favor baryons instead of antileptons? Could matter-antimatter asymmetry be induced by the interior of DE alone or by its interaction with the Minkowskian space-time region outside DE.

In the standard model also charged weak currents are allowed. Does TGD allow their space-time counterparts? CP2 allows quaternionic structure in the sense that the conformally invariant Weyl tensor has besides W3=J(CP2) also charged components W+/-, which are however not covariantly constant. One can assign to W+/- analogs of Kähler currents as covariant divergences and also the analogs of instanton currents. These currents could realize a classical space-time analog of current algebra.

See the chapter Comparing the Berry phase model of superconductivity with the TGD based model or the article with the same title.