Connection with parity breaking, massivation, and PCAC hypothesis

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 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

   L_L/R= J+/- I

   is non-vanishing. For c≠ 1, both J_L and J_R 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.

2. A more general condition would allow c to depend on space-time coordinates. The conservation of J forces conservation of I if the condition ∂_αcI^α=0 is true. This gives a non-trivial condition only in regions with 4-D CP₂ and M⁴ projections.
3. The twistor lift of TGD requires that also M⁴ has Kähler structure. Therefore J and I and corresponding Kähler gauge potential A have both M⁴ part and CP₂ parts and Kähler action K, J_K, J and I are sums of M⁴ and CP₂ parts:

   A_K= A(M⁴)+A(CP₂),
   J_K=J_K(M⁴)+J_K(CP₂) ,
   K = K(M⁴)+K(CP₂) ,
   J =J(M⁴)+J(CP₂) ,
   I= I(M⁴)+I(CP₂) .
   

   Only the divergence for the sum I of M⁴ and CP₂ parts of the instanton currents must vanish:

   ∂_αI^α=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).

4. 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.

   D_αI^α=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.

5. D_αI^α≠ 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.

6. The geometric model for massivation based on zitterbewegung of DE provides additional insight. 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.

   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 article Comparing the Berry phase model of super-conductivity with the TGD based model or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.