^{4}, call it J. J is covariantly constant self-dual 2-form, whose square is the negative of the metric. There is a moduli space for these Kähler forms parametrized by the direction of the constant and parallel magnetic and electric fields defined by J. J partially characterizes the causal diamond (CD): hence the notation J(CD) and can be interpreted as a geometric correlate for fixing quantization axis of energy (rest system) and spin.

Kähler form defines classical U(1) gauge field and there are excellent reasons to expect that it gives rise to U(1) quanta coupling to the difference of B-L of baryon and lepton numbers. There is coupling strength α_{1} associated with this interaction. The first guess that it could be just Kähler coupling strength leads to unphysical predictions: α_{1} must be much smaller. Here I do not yet completely understand the situation. One can however check whether the simplest guess is consistent with the empirical inputs from CP breaking of mesons and antimatter asymmetry. This turns out to be the case.

One must specify the value of α_{1} and the scaling factor transforming J(CD) having dimension length squared as tensor square root of metric to dimensionless U(1) gauge field F= J(CD)/S. This leads to a series of questions.

How to fix the scaling parameter S?

- The scaling parameter relating J(CD) and F is fixed by flux quantization implying that the flux of J(CD) is the area of sphere S
^{2}for the twistor space M^{4}× S^{2}. The gauge field is obtained as F=J/S, where S= 4π R^{2}(S^{2}) is the area of S^{2}.

- Note that in Minkowski coordinates the length dimension is by convention shifted from the metric to linear Minkowski coordinates so that the magnetic field B
_{1}has dimension of inverse length squared and corresponds to J(CD)/SL^{2}, where L is naturally be taken to the size scale of CD defining the unit length in Minkowski coordinates. The U(1) magnetic flux would the signed area using L^{2}as a unit.

^{2}) relates to Planck length l

_{P}? l

_{P}is either the radius l

_{P}=R of the twistor sphere S

^{2}of the twistor space T=M

^{4}× S

^{2}or the circumference l

_{P}= 2π R(S

^{2}) of the geodesic of S

^{2}. Circumference is a more natural identification since it can be measured in Riemann geometry whereas the operational definition of the radius requires imbedding to Euclidian 3-space.

How can one fix the value of U(1) coupling strength α_{1}? As a guideline one can use CP breaking in K and B meson systems and the parameter characterizing matter-antimatter symmetry.

- The recent experimental estimate for so called Jarlskog parameter characterizing the CP breaking in kaon system is J≈ 3.0× 10
^{-5}. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does not distinguish between different meson so that it is better to talk about orders of magnitude only.

- Matter-antimatter asymmetry is characterized by the number r=n
_{B}/n_{γ}∼ 10^{-10}telling the ratio of the baryon density after annihilation to the original density. There is about one baryon 10 billion photons of CMB left in the recent Universe.

_{1}.

- Since the action is obtained by dimensional reduction from the 6-D Kähler action, one could argue α
_{1}= α_{K}. This proposal leads to unphysical predictions in atomic physics since neutron-electron U(1) interaction scales up binding energies dramatically.

U(1) part of action can be however regarded a small perturbation characterized by the parameter ε= R

^{2}(S^{2})/R^{2}(CP_{2}), the ratio of the areas of twistor spheres of T(M^{4}) and T(CP_{2}). One can however argue that since the relative magnitude of U(1) term and ordinary Kähler action is given by ε, one has α_{1}=ε× α_{K}so that the coupling constant evolution for α_{1}and α_{K}would be identical.

- ε indeed serves in the role of coupling constant strength at classical level. α
_{K}disappears from classical field equations at the space-time level and appears only in the conditions for the super-symplectic algebra but ε appears in field equations since the Kähler forms of J resp. CP_{2}Kähler form is proportional to R^{2}(S^{2}) resp. R^{2}(CP_{2}) times the corresponding U(1) gauge field. R(S^{2}) appears in the definition of 2-bein for R^{2}(S^{2}) and therefore in the modified gamma matrices and modified Dirac equation. Therefore ε^{1/2}=R(S^{2})/R(CP_{2}) appears in modified Dirac equation as required by CP breaking manifesting itself in CKM matrix.

NTU for the field equations in the regions, where the volume term and Kähler action couple to each other demands that ε and ε

^{1/2}are rational numbers, hopefully as simple as possible. Otherwise there is no hope about extremals with parameters of the polynomials appearing in the solution in an arbitrary extension of rationals and NTU is lost. Transcendental values of ε are definitely excluded. The most stringent condition ε=1 is also unphysical. ε= 2^{2r}is favoured number theoretically.

- p-Adic mass calculations predict electron mass as

m

_{e}= hbar/R(CP_{2})(5+Y)^{1/2}.

Expressing m

_{e}in terms of Planck mass m_{P}and assuming Y=0 (Y∈ (0,1)) gives an estimate for l_{P}/R(CP_{2}) as

l

_{P}R(CP_{2}) ≈ 2.0× 10^{-4}.

- From l
_{P}= 2π R(S^{2}) one obtains estimate for ε, α_{1}, g_{1}=(4πα_{1})^{1/2}assuming

α_{K}≈ α≈ 1/137 in electron length scale.

ε = 2

^{-30}≈ 1.0× 10^{-9},

α

_{1}=εα_{K}≈ 6.8× 10^{-12},

g

_{1}= (4πα_{1}^{1/2}≈ 9.24 × 10^{-6}.

_{P}= R(S

^{2}) and l

_{P}=2π R(S

^{2}). Only the length of the geodesic of S

^{2}has meaning in the Riemann geometry of S

^{2}whereas the radius of S

^{2}has operational meaning only if S

^{2}is imbedded to E

^{3}. Hence l

_{P}= 2π R(S

^{2}) is more plausible option.

For ε=2^{-30} the value of l_{P}^{2}/R^{2}(CP_{2}) is l_{P}^{2}/R^{2}(CP_{2})=(2π)^{2} × R^{2}(S^{2})/R^{2}(CP_{2}) ≈ 3.7× 10^{-8}. l_{P}/R(S^{2}) would be a transcendental number but since it would not be a fundamental constant but appear only at the QFT-GRT limit of TGD, this would not be a problem.

One can make order of magnitude estimates for the Jarlskog parameter J and the fraction r= n(B)/n(γ).
Here it is not however clear whether one should use ε or α_{1} as the basis of the estimate

- The estimate based on ε gives

J∼ ε^{1/2}≈ 3.2× 10^{-5},r∼ ε ≈ 1.0× 10

^{-9}.

The estimate for J happens to be very near to the recent experimental value J≈ 3.0× 10

^{-5}. The estimate for r is by order of magnitude smaller than the empirical value.

- The estimate based on α
_{1}gives

J∼ g_{1}≈ 0.92× 10^{-5},

r∼ α

_{1}≈ .68× 10^{-11}.

The estimate for J is excellent but the estimate for r by more than order of magnitude smaller than the empirical value. One explanation is that α

_{K}has discrete coupling constant evolution and increases in short scales and could have been considerably larger in the scale characterizing the situation in which matter-antimatter asymmetry was generated.

^{2}(S

^{2})/R

^{2}(CP

_{2}). One can however argue that since the relative magnitude of U(1) term and ordinary Kähler action is given by ε, one has α

_{1}=ε× α

_{K}.

Quantal U(1) force must be also consistent with atomic physics. The value of the parameter α_{1} consistent with the size of CP breaking of K mesons and with matter antimatter asymmetry is α_{1}= εα_{K} = 2^{-30}α_{K}.

- Electrons and baryons would have attractive interaction, which effectively transforms the em charge Z of atom Z
_{eff}= rZ, r=1+(N/Z)ε_{1}, ε_{1}=α_{1}/α=ε × α_{K}/α≈ ε for α_{K}≈ α predicted to hold true in electron length scale. The parameter

s=(1 + (N/Z)ε)

^{2}-1= 2(N/Z)ε +(N/Z)^{2}ε^{2}

would characterize the isotope dependent relative shift of the binding energy scale.

The comparison of the binding energies of hydrogen isotopes could provide a stringent bounds of the value of α

_{1}. For l_{P}= 2π R(S^{2}) option one would have α_{1}=2^{-30}α_{K}≈ .68× 10^{-11}and s≈ 1.4× 10^{-10}. s is by order of magnitude smaller than α^{4}≈ 2.9× 10^{-9}corrections from QED (see this). The predicted differences between the binding energy scales of isotopes of hydrogen might allow to test the proposal.

- B=N would be neutralized by the neutrinos of the cosmic background. Could this occur even at the level of single atom or does one have a plasma like state? The ground state binding energy of neutrino atoms would be α
_{1}^{2}m_{ν}/2 ∼ 10^{-24}eV for m_{ν}=.1 eV! This is many many orders of magnitude below the thermal energy of cosmic neutrino background estimated to be about 1.95× 10^{-4}eV (see this). The Bohr radius would be hbar/(α_{1}m_{ν}) ∼ 10^{6}meters and same order of magnitude as Earth radius. Matter should be U(1) plasma. U(1) superconductor would be second option.

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

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