The twistor lift of TGD led to the introduction of Kähler form also in M^{4} factor of imbedding space
M^{4}×CP_{2}. The moduli space of causal diamonds (CDs) introduced already early
allow to save Poincare invariance at the level of WCW. One of the very nice things is that the selfduality of J(M^{4}) leads to a new mechanism of breaking for P,CP, and T in long scales, where these breakings indeed take place. P corresponds to chirality selection in living matter, CP to matter antimatter asymmetry and T could correspond to preferred arrow of clock time. TGD allows both arrows but T breaking could make other arrow dominant. Also the hierarchy of Planck constant is expected to be important.
Can one say anything quantitative about these various breakings?
 J(M^{4}) is proportional to Newton's constant G in the natural scale of Minkowski coordinates defined by twistor sphere of T(M^{4}). Therefore CP breaking is expected to be proportional to l_{P}^{2}/R^{2} or to its square root l_{P}/R. The estimate for l_{P}/R is X== l_{P}/R≈ 2^{12}≈ 2.5× 10^{4}.
The determinant of CKM matrix is equal to phase factor by unitarity (UU^{†}=1) and its imaginary part characterizes CP breaking. The imaginary part of the determinant should be proportional to the Jarlskog invariant J= +/ Im(V_{us}V_{cb}V*_{ub} V*_{cs}) characterizing CP breaking of CKM matrix (see this).
The recent experimental estimate is J≈ 3.0× 10^{5}. J/X≈ 0 .1 so that there is and order of magnitude deviation. Earlier experimental estimate used in padic mass calculations was by almost order of magnitude larger consistent with the value of X. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does nto distinguish between different meson so that it is better to talk about orders of magnitude only.
The parameter used to characterize matter antimatter asymmetry (see this) is the ratio R=[n(Bn(B*)]/n(γ)) ≈ 9× 10^{11} of the difference of baryon and antibaryon densities to photon density in cosmological scales. One has X^{3} ≈ 1.4 × 10^{11}, which is order of magnitude smaller than R.
 What is interesting that P is badly broken in long length scales as also CP. The same could be true for T. Could this relate to the thermodynamical arrow of time? In ZEO state function reductions to the opposite boundary change the direction of clock time. Most physicist believe that the arrow of thermodynamical time and thus also clock time is always the same. There is evidence that in living matter both arrows are possible. For instance, Fantappie has introduced the notion of syntropy as time reversed entropy. This suggests that thermodynamical arrow of time could correspond to the dominance of the second arrow of time and be due to selfduality of J(M^{4}) leading to breaking of T. For instance, the clock time spend in time reversed phase could be considerably shorter than in the dominant phase. A quantitative estimate for the ratio of these times might be given some power of the the ratio X=l_{P}/R.
For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.
