Under what conditions electric charge is conserved for modified Dirac equation?

One might think that talking about the conservation of electric charge at 21st century is a waste of time. In TGD framework this is certainly not the case.

1. In quantum field theories there are two manners to define em charge: as electric flux over 2-D surface sufficiently far from the source region or in the case of spinor field quantum mechanically as combination of fermion number and vectorial isospin. The latter definition is quantum mechanically more appropriate.
   
   
2. There is however a problem. In standard approach to gauge theory Dirac equation in presence of charged classical gauge fields does not conserve electric charge: electron is transformed to neutrino and vice versa. Quantization solves the problem since the non-conservation can be interpreted in terms of emission of gauge bosons. In TGD framework this does not work since one does not have path integral quantization anymore. Preferred extremals carry classical gauge fields and the question whether em charge is conserved arises. Heuristic picture suggests that em charge must be conserved. This condition might be actually one of the conditions defining what it is to be a preferred extremal. It is not however trivial whether this kind of additional condition can be posed.
   

What does the conservation of em charge imply in the case of the modified Dirac equation? The obvious guess that the em charged part of the modified Dirac operator must annihilate the solutions, turns out to be correct as the following argument demonstrates.

1. Em charge as coupling matrix can be defined as a linear combination Q= aI+bI₃, I₃=J_klΣ^kl, where I is unit matrix and I₃ vectorial isospin matrix, J_kl is the Kähler form of CP₂, Σ^kl denotes sigma matrices, and a and b are numerical constants different for quarks and leptons. Q is covariantly constant in M⁴× CP₂ and its covariant derivatives at space-time surface are also well-defined and vanish.
   
   
2. The modes of the modified Dirac equation should be eigen modes of Q. This is the case if the modified Dirac operator D commutes with Q. The covariant constancy of Q can be used to derive the condition
   

   [D,Q] Ψ= D₁Ψ=0 , D= Γ^μD_μ ,
   

   D₁=[D,Q]=Γ^μ₁D_μ , Γ^μ₁= [ Γ^μ,Q] .
   

   
   Note that here Γ^μ denotes modified gamma matrices rather than ordinary induced gamma matrices defined by contractions of energy momentum tensor with induced gamma matrices.
   Covariant constancy of J is absolutely essential: without it the resulting conditions would not be so simple.
   

   It is easy to find that also [D₁,Q]Ψ=0 and its higher iterates [D_n,Q]Ψ=0, D_n= [D_n-1,Q] must be true. The solutions of the modified Dirac equation would have an additional symmetry.
   
   

3. The commutator D₁=[D,Q] reduces to a sum of terms involving the commutators of the vectorial isospin I₃=J_klΣ^kl with the CP₂ part of the gamma matrices:
   

   D₁=[Q‚ D]= [I₃,Γ_r] ∂_μs^r T^αμ .
   

   In standard complex coordinates in which U(2) acts linearly the complexified gamma matrices can be chosen to be eigenstates of vectorial isospin. Only the charged flat space complexified gamma matrices Γ^A denoted by Γ⁺ and Γ^- possessing charges +1 and -1 contribute to the right hand side. Therefore the additional Dirac equation D₁Ψ=0 states
   

   D₁Ψ=[Q‚ D]Ψ= I₃(A)e_Ar Γ^A ∂_μs^r T^αμD_αΨ
   

   = (e_+r Γ⁺-e_-rΓ^-)∂_μs^r T^αμD_αΨ =0 .
   

   The next condition is
   

   D₂Ψ=[Q‚ D₁]Ψ=(e_+r Γ⁺+e_-rΓ^-) ∂_μs^r T^αμD_αΨ =0 .
   

   The remaining conditions give nothing new.
   
   

4. These equations imply two separate equations for the two charged gamma matrices
   

   D₊Ψ = Γ⁺T₊^αD_αΨ=0
   

   D_-Ψ = Γ^-T_-^αD_αΨ=0
   

   T_+/-^α= e_+/-r ∂_μs^r T^αμ .
   

   These conditions state what one might have expected: the charged part of the modified Dirac operator annihilates separately the solutions. The reason is that the classical W fields are proportional to e_r+/-.
   

   The above equations can be generalized to define a decomposition of the energy momentum tensor to charged and neutral components in terms of vierbein projections. The equations state that the analogs of the modified Dirac equation defined by charged components of the energy momentum tensor are satisfied separately.
   
   

5. In complex coordinate one expects that the two equations are complex conjugates of each other for Euclidian signature. For the Minkowskian signature an analogous condition should hold true. The dynamics enters the game in an essential manner: whether the equations can be satisfied depends on the coefficients a and b in the expression T= aG+bg implied by Einstein's equations in turn guaranteeing that the solution ansatz generalizing minimal surface solutions holds true (see this).
   
   
6. As a result one obtains three separate Dirac equations corresponding to the the neutral part D₀Ψ=0 and charged parts D_+/-Ψ=0 of the modified Dirac equation. By acting on the equations with these Dirac operators one obtains that also the commutators [D₊,D_-], [D₀,D_+/-] and also higher commutators obtained from these annihilate the induced spinor field model. Therefore entire - possibly- infinite-dimensional - algebra would annihilate the induced spinor fields. In string model the counterpart of Dirac equation when quantized gives rise to Super-Virasoro conditions. This analogy would suggest that modified Dirac equation gives rise to the analog of Super-Virasoro conditions in 4-D case. But what the higher conditions mean? Obviously these conditions resemble Virasoro conditions L_n|phys>=0 and their supersymmetric generalizations and might indeed correspond to a generalization of these conditions just as the field equations for preferred extremals could correspond to the Virasoro conditions if one takes seriously the analogy with the quantized string.
   

What could this additional symmetry mean from the point of view of the solutions of the modified Dirac equation? The field equations for the preferred extremals of Kähler action reduce to purely algebraic conditions in the same manner as the field equations for the minimal surfaces in string model. Could this happen also for the modified Dirac equation and could the condition on charged part of the Dirac operator help to achieve this?

1. CP₂ type vacuum extremals serve as a convenient test case. In this case the modified Dirac equation reduces to the ordinary Dirac equation in CP₂. One can construct the solutions of the ordinary Dirac equation from covariantly constant right-handed neutrino spinor playing the role of fermionic vacuum annihilated by the second half of complexified gamma matrices. Dirac equation reduces to Laplace equation for a scalar function and solution can be constructed from this "vacuum" by multiplying with the spherical harmonics of CP₂ and applying Dirac operator (see this). Similar construction works quite generally thanks to the existence of covariantly constant right handed neutrino spinor. Spherical harmonics of CP₂ are only replaced with those of space-time surface possessing either hermitian structure of Hamilton-Jacobi structure (corresponding to Euclidian and Minkowskian signatures of the induced metric, see this).
   
   
2. Could the properties of the preferred extremals make it possible to satisfy the additional conditions for the modified Dirac operator? The identical vanishing of the charged components of the energy momentum tensor would be an obvious manner to reduce the conditions as algebraic identifies. Note however that it could be an un-necessarily strong condition. In any case, the vanishing of charged components of the energy momentum tensor looks like a natural weakening of the conditions stating the vanishing of charged components of the induced gauge field.
   

There is also an interesting potential connection to the number theoretic vision about field equations.

1. The octonionic representation effectively replaces SO(7,1) as tangent space group with G₂ and means selection of preferred M²⊂ M⁴ having interpretation complex plane of octonionic space (see this). A more general condition is that the tangent space of space-time surface at each point contains preferred sub-space M²(x)⊂ M⁴ forming an integrable distribution. The same condition is involved with the definition of Hamilton-Jacobi structure. What puts bells ringing is that the modified Dirac equation for the octonionic representation of gamma matrices allows the conservation of electromagnetic charge in the proposed sense a observed for years ago. One can ask whether the conditions on the charged part of energy momentum tensor could relate to the reduction of SO(7,1) to G₂.
   
   
2. Octonionic gamma matrices appear also in the proposal stating that space-time surfaces are quaternionic in the sense that tangent space of the space-time surface is quaternionic in the sense that induced octonionic gamma matrices generate a quaternionic sub-space at a given point of space-time time. Besides this the already mentioned additional condition stating that the tangent space contains preferred sub-space M²⊂ M⁴ or integrable distribution of this kind of sub-spaces is required. It must be emphasized that induced rather than modified gamma matrices are natural in these conditions.
   

This raises two questions.

1. Could the quaternionicity of the space-time surface be equivalent with the reduction of the field equations to "stringy" field equations (minimal surface equations) stating that certain components of the induced metric in complex/Hamilton-Jacobi coordinates vanish in turn guaranteeing that field equations reduce to algebraic identifies following from the fact that energy momentum tensor and second fundamental form have no common components? This should be the case if one requires that the two solution ansätze are equivalent.
   
   
2. Could the above conditions for the modified Dirac equation imply that the space-time surface is quaternionic? Or could these conditions be seen as consistency conditions making the modified Dirac equation associated with the ordinary and octonionic gamma matrices equivalent at space-time level in the sense that the induced gamma matrices or modified gamma matrices are equivalent?