_{2}would suggest. Poincare invariance would be still achieved by allowing the moduli space for CDs and extension of WCW by this moduli space necessitated also by quantum measurement theory and theory of consciousness (flow and arrow of time).

Why J(M^{4}) would be needed? First of all, it provides the breaking of Poincare invariance involve with quantum measurement. The time direction defining rest system and quantization axis of spin indeed imply symmetry breaking. Secondly, it might allow to understand CP breaking in kaon-anti-kaon system and other similar systems from first principles since M^{4}-electric dipole momentum would contribute to mass splitting between particle. This small asymmetry could lead to matter-antimatter asymmetry. Matter and antimatter would have different values of h_{eff} and dark relative to each other. The following arguments already given in previous posting (the reposting is motivated by the enormous importance of these mysteries for modern physics) support this view quantitatively.

CP breaking in hadronic systems is one of the poorly understood aspects of fundamental physics and relates closely to the mysterious matter-antimatter asymmetry. The constant electric part of self dual J(M^{4}) implies CP breaking. I have earlier considered the possibility that Kähler electric fields could cause this breaking but this breaking would be local. Second possibility is that matter and antimatter correspond to different values of h_{eff} and are dark relative to each other.

Could J(M^{4}) explain the observed CP breaking as appearing already at the level of imbedding space M^{4}× CP_{2} and could this breaking explain hadronic CP breaking and matter anti-matter asymmetry? Could M^{4} part of Kähler electric field induce different h_{eff}/h=n for particles and antiparticles?

To answer these questions one can study Dirac equation at imbedding space level coupled to the gauge potential A(M^{4}) for J(M^{4}).

- The coupling of Kähler form to leptons is 3 times larger than to to quarks as in the case of A(CP
_{2}). This would give coupling k=1 for quarks an k=3 for leptons. k corresponds to fermion number which is opposite for fermions and antifermions having therefore opposite values of k at the respective space-time sheets.

- The potential satisfies ∂
_{μ}A^{μ}(M^{4})=0. Let the non-vanishing components of the Kähler gauge potential be (A_{0},A_{z})=ε (x,+/- y). The sign fact ε+/- 1 corresponds to self dual and antiself-dual options, let us assume self-duality as in the case of CP_{2}Kähler form. Scalar d'Alembertian reads as (∂^{μ}∂_{μ}+ A^{μ}A_{μ})Ψ= -m^{2}Ψ.

- Assuming momentum eigenstate in time and z-direction (plane M
^{2}), one obtains by separation of variables (H_{1}+H_{2})Ψ= (E^{-}m^{2}-k_{z}^{2})Ψ. H_{x}= -∂_{x}^{2}+k^{2}x^{2}and H_{y}= -∂_{y}^{2}+k^{2}y^{2}) are oscillator Hamiltonians. The spectrum is of H_{x}+H_{y}is given by k_{T}^{2}= (n_{1}+n_{2}+1)2^{1/2}|k| and one obtains E^{2}=m^{2}+k_{z}^{2}+k_{T}^{2}. This contribution is CP invariant and same for fermions and anti-fermions. The special feature is the presence of zero point transversal momentum. It is not possible to have a particle, which would be completely at rest. One can also say that m^{2}is increased 2^{1/2}|k| hbar^{2}/L^{2}, L= 1 m if standard convention for metric is used. For other conventions the numerical value of CP_{2}radius is scale by L/L_{new}. L must correspond to some physical scale assignable to particle: secondary p-adic length scale is the natural identification.

- Spinor d'Alembertian contains also dipole moment term kX=J
^{muν}Σ_{μν}giving a contribution, which depends on the sign of k: E^{2}=m^{2}+k_{z}^{2}+k_{T}^{2}+ kX. The term is sum of magnetic and electric dipole moment terms. The coupling k changes sign in CP operation and be of opposite sign for fermions and anti-fermions. One has a breaking of CP for given spin state. The dependence of X on spin state gives a test for the theory and also for the predicted CP breaking.

- Scaling covariance allows in principle all values L. To estimate the size of the effect one must fix the length scale L. CP
_{2}size has only different value using L as unit and in flat background it does not matter. L should correspond to the size scale of the CD associated with particle. The secondary p-adic length scale of fermion defining also the size scale of its magnetic body is a natural guess so that Δ E^{2}/E^{2}≈ 2Δ E/E≈ Δ m/m ∼ 2/p^{1/2}, p≈ 2^{k}would hold true. This mass splitting is very small. For weak bosons having k=89 the mass splitting would be of order 3× 10^{-4}eV. For small values of p at ultrahigh energies the scale of CP breaking is larger, which conforms with the idea that matter-antimatter-asymmetry has emerged in very early cosmology.

The recent experiment found that the mass difference Δ m/m for proton and antiproton satisfies Δ m <69× 10

^{-12}m ≈ 6.9× 10^{-2}eV (see this) so that this gives no constraints. Kaon-antikaon mass difference is estimated to be about 3.5× 10^{-6}eV (see this). This would correspond to a p-adic length scale k=96. Top quark is mainly responsible for the mixing of neutral kaon and its antiparticle in the model of based on loops involving decay to virtual quark pairs. The estimate from p-adic mass calculations for top quark mass scale is k=94 so that the order of magnitude estimate has correct of order of magnitude (being by factor 4 too large). This is an encouraging sign.

How the mass splitting of neutral kaons would result? In quark model kaon and antikaon can be regarded as sdbbar and dsbar pairs. The net spins vanishes but the mass splitting due to electric moment dipole moment term X is non-vanishing due to the different sign of coupling k. The sign of the mass splitting is also opposite for kaon and antikaon.

- One can also consider the modified Dirac equation for canonically imbedded M
^{4}which is simplest preferred extremal. The coupling to J(M^{4}) to modified Dirac equation in space-time interior with gamma matrices replaced with modified gamma matrices are obtained as contractions of canonical momentum currents with M^{4}gamma matrices. Completely analogous phenomenon happens for CP_{2}type extremals. T^{αβ}=0 so that the modified gamma comes from J^{αβ}J^{k}_{l}∂_{β}m^{l}γ_{k}. These give just ordinary gamma matrices so that the two Dirac equations are identical.

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

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https://arxiv.org/pdf/1111.4427v1.pdf

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