On the Origin of the Charge-Asymmetric Matter. II. Localized Dirac Waveforms

This paper continues the author's work \cite{PartI}, where a new framework of the matter-induced physical geometry was built and an intrinsic nonlinearity of the Dirac equation discovered. Here, the nonlinear Dirac equation is solved and the localized configurations are found analytically. Of the two possible types of the potentially stationary localized configurations of the Dirac field, only one is stable with respect to the action of an external field and it corresponds to a positive charge. A connection with the global charge asymmetry of matter in the Universe and with the recently observed excess of the cosmic positrons is discussed.


Introduction
This paper continues the author's study of the long-standing question of how the physical Dirac field of a real matter becomes a finite-sized particle, and it is approached here as a practical problem. The problem is posed and solved in a new framework of the matter-induced affine geometry [1], which deduces the geometric relations in the space-time continuum from the dynamic properties of the Dirac field. The intuitive argument of a possible auto-localization of the Dirac field followed from an observation [1] that the local time flows slower at higher invariant density, and then from the wave nature of the Dirac equation. Its further consequence must be the (well-known but not clearly understood) charge asymmetry of the observed localized matter. In the present work, these qualitative expectations are confirmed by explicit calculations.
The earlier developed [1] mathematical background for the present work is based on the following ideas and results. It is observed that if at a point in spacetime continuum (the principal differentiable manifold M) a physical Dirac field is defined, then the latter determines the tetrad of Dirac currents. These are linearly independent and Lorentz-orthogonal and can serve as local algebraic basis for any four-dimensional vector 2. The framework.
In the first part of this work we explored differential identities for the four Dirac currents, vector current j, axial current J, and two "charged currents", Θ and Φ. Using them, we found all components of the affine connection ω ABC , as well as connection Γ B of the Dirac field in principal manifold M, (2.1) The connection (2.1) determines the covariant derivative of the Dirac field and it enters the Dirac equation The nonzero elements of the ω ABC in the tetrad basis of the normalized Dirac currents e A are as follows, ω 030 = −ω 131 = −ω 232 = Q, ω 12D = 2eÃ [D] , (D = 0, 1, 2, 3), (2.3) where Q ≡ ∂ [3] ln R = −mP/R = −m sin Y is the derivative of the invariant density R in the direction of the axial current and it has an algebraic representation via the pseudoscalar density P. These formulae assume thatÃ [D] = +A D for the right-handed spatial triad e [1] , e [2] , e [3] with Θ = Re [1] , Φ = Re [2] and the naturally outward directed axial current J = Re 3 , i.e. [ e [1] × e [2] ] = e [3] [c.f Eqs. (A.4), (A. 5)]. When the latter is directed inward, but we still wish e [3] to point outward, then we have to take Θ = Re [2] , Φ = Re [1] and replace ω 12D → ω 21D = −ω 12D (orÃ [D] = −A D ) in Eqs.(2.3) 1 .
With the connection (2.4) the Dirac equation becomes a nonlinear equation and its explicit form reads as, [3] eÃ [3] [1] − eÃ [2] ψ + α [2] ∂ [2] − ieA [2] + eÃ [1] ψ + imρ [1] ψ = 0, (2.5) where anomalous term −3Q/2 singles out the direction of the axial current among others even when A µ = 0. This equation is valid in every connected domain where R 2 > 0 and the Dirac currents define a nondegenerate orthogonal tetrad e µ A (ψ). As anticipated, it is invariant in a most broad sense -it depends neither on choice of coordinates x µ in R 4 nor on a tetrad system h µ a (also in R 4 ) not even on a particular choice of the γ-matrices. The latter is always taken for granted since one can introduce a new Dirac field ψ ′ = Sψ leaving the gamma matrices unchanged. But this trick works only for re-parameterizations in R 4 , i.e. change of the Lorentz frame or transformations between orthogonal coordinates. It cannot be employed in the principal manifold M just because the Dirac field is a coordinate scalar.
Finally, Eq.(2.5) is nonlinear because both the connection ω ACB and the Dirac matrices α A = V A a (ψ)α a in it depend on the Dirac field ψ ∈ M. The dependence of ω ACB on the Dirac field is due to (2.3). The dependence of the Dirac matrices on ψ, α A = V A a (ψ)α a , is not so explicit but not less important and it cannot be avoided. Indeed, in the basis [A] each of the currents J A has only one nonzero component, e.g., The latter cannot be achieved without an explicit dependence α A (ψ). Indeed, with ψ ∈ M and numerical matrices α a the current j a will have all components. Obviously, this is a significant technical difficulty. However, only this dependence solves a conceptual problem of independence of the equation of motion for the physical Dirac field in M on a particular choice of the tetrad h a and of the matrices α a in tangent T p . Therefore, we begin with the establishing rules of transformation of the 16 Dirac matrices between M and R 4 .

Dirac matrices in principal manifold M
Historically, the Dirac equation for the free field ψ was formulated as iα a ∂ a ψ − mβψ = 0 with the aid of Hermitian Dirac matrices α a = (α a ) + and β = β + , which satisfy the commutation relations, Usually one assumes that α a = (1, α i ); a = 0, 1, 2, 3; i = 1, 2, 3 (so that α 0 = 1 is a unit matrix) but this is not required. An apparently symmetric form of commutation relations (3.1) emerges (along with the equation, iγ a ∂ a ψ − mψ = 0) in terms of the matrices γ a = (γ (0) , γ i ) = (β, βα i ), Neither of these matrices is uniquely defined. However, if there exist two sets of the matrices, γ a and γ [A] , that satisfy (3.2) then, according to the Pauli's fundamental theorem, there exists such a nonsingular S, that where κ = 0, 1, 2, 3, [κ] is a number standing for superscript A and (κ) is the same number for superscript a. There are sixteen linearly independent 4 × 4 matrices O p = (1, γ a , γ a γ b , ...), all of which are the products of 1,2,3 or 4 different gamma. Therefore, , ...), which is an indisputable technical advantage. By their definition, the matrices γ a are not Hermitian. However, since β and α i are Hermitian and anticommuting, the Hermit-conjugated γ-matrices are (γ a ) + = γ (0) γ a γ (0) . If, by the same token, , which yields, Multiplying this by S from the left and by γ [0] S + γ (0) from the right, we find, The matrix (Sγ [0] S + γ (0) ) commutes with all the matrices γ A and must be the unit matrix, viz., On the one hand, we can continue as where the transformation matrix V A a is real and has the properties, Then the commutation relations (3.1) are the same for γ a and γ A and S must be a solution of the matrix equation, Though V A a has a character of a Lorentz transformation, it has no infinitesimal prototype. Since S + = ρ 1 S −1 ρ 1 , we also have a habitual However, in the basis of matrices γ [A] , the Pauli-conjugated Dirac spinor must be defined as ψ = ψ + ρ 1 and not as ψ = ψ + γ [0] .
The set O p of 16 linearly independent elements of Clifford algebra comprised of various products of the γ a -(or the γ A -) matrices is in one-to-one correspondence with 16 Hermitian matrices, (1, Dirac matrices, ρ i and σ i , satisfy the same commutation relations as the Pauli matrices, σ i σ k = δ ik +iǫ ikl σ l , and ρ a ρ b = δ ab + iǫ abc ρ c . Finally, it is straightforward to check that the matrix ρ 3 = iγ 0 γ 1 γ 2 γ 3 (commonly known as −γ 5 ) is an invariant of transformations (3.3), Then the matrix ρ 2 = iρ 1 ρ 3 is transformed like ρ 1 , so that (as it should be for the spatial components of the axial current J a ) 2 .

The nonlinear Dirac equation, explicitly
So far, we have been studying the general geometric properties of the Dirac field in the scope of the affine geometry and carefully avoiding any assumptions about what a solution of the Dirac equation that has these properties can be. All the previously established [1] properties of the Dirac currents belong (along with the Dirac field itself) to the principal differentiable manifold M. Without resorting to any particular coordinate manifold R 4 we have established in [1] the following facts: (i) The congruence of lines of the vector field e µ [0] is normal. The family S (123) of hypersurfaces, τ (x) = const, of the constant world time τ is extrinsically flat; τ is a holonomic coordinate and it can be taken for x 0 in R 4 . (ii) The congruence of lines of the vector field e µ [3] is normal and geodesic. The hypersurfaces S (012) of the constant radius ρ have constant extrinsic curvature and the holonomic coordinate ρ can serve as The two-dimensional surfaces S (12) of constant τ and ρ are just spheres, i.e. umbilical (with two equal Gauss'curvatures) surfaces with constant mean (extrinsic) curvature H = mP/R = −m∂ [3] ln R. The latter is determined by the Dirac field within principal manifold M and depends only on the radius ρ. The intrin- [3] , is due to the external electromagnetic field. It coincides with projection of the magnetic field onto the direction of the axial current.
(iv) The two-dimensional surfaces S (03) are covered by a well-defined coordinate net formed by the streamlines of the vector and axial currents. This net can be identically mapped between the principal manifold M and the arithmetic R 4 .
These general observations can be summarized as follows. For any solution of the Dirac equation, which is not homogeneous in spatial directions, the spherical symmetry is the property of a solution, thus being a dynamic symmetry.
In order to find a solution of the Dirac equation, one has to specify a coordinate basis in R 4 and a basis of the Dirac matrices. Here, we shall employ the numerical matrices α a in spinor representation (A.2) and associate them with a tetrad h µ (a) . Then, α A = V A a α a , while the derivatives D [A] will stay in the basis e A , which is associated with coordinate surfaces determined in the principal manifold M. In this mixed representation, Dirac equation reads as The operators D A , which are copied from Eq.(2.5), are as follows, [2] , where ρ 3 differentiate between the right and left components and it stands for +1 for u R , d R and for −1 for u L , d L . The coordinate net formed by the integral lines of the tetrad vectors e [0] and e [3] that covers the two-dimensional surface S (03) in M is holonomic and the vectors h (0) , h (3) in R 4 can be chosen tangent to this surface. In order for the other two tetrad vectors, h (1) and h (2) , to be normal to this surface, it is necessary that the components V (1) [3] = 0. Just by inspection of Eqs.(A.4), we see that this is possible only when either d R = d L = 0 or u R = u L = 0. In both cases, as seen from Eqs.

Alexander Makhlin
Remembering about the sign due to ρ 3 , we obtain the following formulae for all the differential operators involved, [3] )], [3] )], In Eqs.(4.4) and (4.5), the operator D [0] + D [3] acts only on u R and d R while D [0] − D [3] only on u L and d L .

Solutions of the nonlinear equations
So far we were expanding the vector of spacetime displacement dx µ in terms of the basis e A of the tetrad determined by the Dirac currents dx µ = e µ A dS A . But the true physical variables are the world time τ and the distance ρ. They are holonomic coordinates, because dτ = RdS [0] and dρ = RdS [3] are the total differentials of the independent coordinates dx µ ∈ R 4 , [3] . (5.1) Here, the upper sign is for the L < 0 and the axial current is directed inward. The world time τ and the radial variable ρ, being defined as invariants in M, can immediately be used in arithmetic R 4 .

Reduction to the physical variables
At the points where j (3) = V [0] = 0 and J (0) = V (0) [3] = 0 (in general, a 2-d surface) the relation between spatial components, [ Θ × Φ]/R 2 = + J /R, is an algebraic identity. For the axial current directed outward, i.e. J 3 > 0, we take J µ = +Re µ [3] , Θ µ = Re µ [1] and Φ µ = Re µ [2] , so that e [3] = [ e [1] × e [2] ]. In this case, we change the variables in Eq.(4.4) as follows, [1] , eÃ [2] → e mÃ [2] (5.2) Adopting the physical variables (5.2) in Eqs.(4.4) we obtain the equations that eventually must be solved. In these equations, according to (4.6), there is an operator ( [2] u L e iφ u L = 0, where For the axial current directed inward, in order to preserve an intuitive physical understanding of a distance from an object, we want e [3] be directed outward. Then the triplet (e [1] , e [2] , e [3] ) will be left-handed. We have to take J µ = −Re µ [3] , Θ µ = Re µ [2] , and Φ µ = Re µ [1] in order for the vector product [ e [1] × e [2] ] = e [3] to represent the external normal and the triplet (e [1] , e [2] , e [3] ) to be right-handed. This results in the interchange of the tetrad indices 1 ↔ 2 in equations (2.3), or, equivalently, in the change of the sign of the tetrad components of the vector potential, eÃ B → −eÃ B . Thus, the string of the change of variables becomes RÃ τ , eÃ [3] → + e m RÃ ρ , eÃ [1] → − e mÃ [1] , eÃ [2] → − e mÃ [2] . (5.4) Note, that in the course of the change of variables outlined above, the sign of the eÃ [3] has been changed twice. Now, using the physical variables (5.4) in Eqs.(3.5) we arrive at a similar system, The difference between ψ u and ψ d is seen right in the equations of motion. The tetrad components of an external field along holonomic coordinates, A τ , A ρ ∈ S (03) , affect the ψ u -mode not the ψ dmode. Conversely, the associated with the non-holonomic coordinates angular components A [1] , A [2] ∈ S (12) are assembled as the ladder operators and affect only ψ d pushing it up to the state ψ u . This difference between the last two equations of systems (4.3) and (5.5) points to a generic instability of the ψ d -mode 3 . It is discussed in Sec.7 .

Reduction to the real-valued functions
As the last step before solving systems (5.3) and (5.5) we split real and imaginary parts of the first two equations of these systems and reduce equations to a form convenient for finding the solutions. For the mode ψ u the result reads as 3 Since e 1 and e 2 are the "angular"directions, it is instructive to recall that the operators L + = L 1 + iL 2 are ladder operators for the angular momentum that moves eigenstate of the Lz up. Both systems (5.3) and (5.5) contain only L + . While ψu cannot be pushed further up (and is stable), the ψ d is readily pushed up to the ψu. One can view these transitions as a manifestation of the ψ d -waveform's "motion". In fact, it is a flow of surrounding Dirac matter with R ≥ 1 that looks like a motion of the ψ d -dip (or void).
For the mode ψ d the result is somewhat different, The phases φ u R and φ u L are affected in ψ u by the right and left lightlike components of the vector potential, respectively, but with the coupling constant 2e. Conversely, the phases φ d L and φ d R of the ψ d are not affected at all.
Next, adding and subtracting equations (5.6.a ′ ) and (5.6.b ′ ) and recalling that φ u L − φ u R = Y u we find that where Z u = φ u L + φ u R and u L /u R = X u . Repeating the same for the mode ψ d we obtain, where Z d = φ d L + φ d R and d L /d R = X d . Eqs.(5.8.d) and (5.9.d) are easily obtained from Eqs.(5.3.c,d) and (5.5.c,d) because none of the amplitudes u R , u L and d R , d L and of the phase differences Y u , Y d depend on the angular variables S [1] and S [2] . We postpone discussion of the Eqs.(5.3.c,d) and (5.5.c,d), which are responsible for the stability or instability of the solutions, till Sec.7.
Before looking for the stationary modes of the nonlinear Dirac equation we are going to learn whether they can emerge as asymptotic configurations at τ → ∞ of a transient process that can begin from an arbitrary perturbation or are they ad hoc constructed isolated solutions. By adding and subtracting Eqs.(5.6.a,b), with the l.h.s. reduced to the logarithmic derivatives, and some simple algebra we obtain where ∂ ρ ln R = − sin Y u /R. Excluding from these two equations the ∂ ρ ln R, one finds a first-order wave equation, ∂ τ X + c(X )∂ ρ X = 0, with the wave velocity c(X ) = (1 − X 2 )/(1 + X 2 ). Because c(1) = 0, the "propagation"of X stops at X = 1. Since R depends only on ρ, both equations (5.10) are easily integrated, where the constants of integration C 1 (τ ) and C 2 (ρ) are arbitrary functions of only one argument. Since X u (∞, ρ) = 1 (and then C 1 (∞) = 0), we find that at the asymptotic world time τ the coefficients in front of cos Y u (ρ) in Eqs.(5.8.a) and (5.8.c) become 2 and 0, respectively. Assuming further that e = 0 (no external field), we find that ∂ ρ Z u = 0 and thus Z u = Z u (τ ). Now, ∂ τ Z u is the only potentially τ -dependent term in Eq.(5.8.a); then it cannot depend on τ . Therefore, the only option is ∂ τ Z u = −2E = const, Z u = −2Eτ , and it immediately follows that u 2 L = u 2 R = u 2 = R/2 (which is an evidence that the particle is at rest!). Eqs.(5.11) are compatible only in the limit of τ → ∞ since they imply ∂ τ R = 0; a transient process naturally requires that ∂ τ R = 0. Similar results are true for the mode ψ d .

Stationary solutions.
Being interested here only in stationary states we assume a trivial dependence of the phases of Dirac field components on τ , ψ ∝ e −iEτ , and replace, Taking further the coupling constant e = 0, which is, in fact, equivalent to a one-body approximation, we end up with an autonomous system of two ODEs for two unknown functions (the amplitude R(ρ) and the phase difference Y(ρ)) of the natural parameter ρ (and not the affine parameter s!) along the radial geodesic lines.

Localized solution for the ψ u -mode of the Dirac field.
In the stationary case, Eqs.(5.8) for the ψ u -mode with the axial current directed outward, read as where ǫ = E/m. The characteristic equation for this system, is easily solved in terms of w(R) = cos Y u . Then, Rw ′ R − 2w + 2ǫR = 0, and cos Y u = CR 2 + 2ǫR, (6.3) is the first integral of system (6.1) depending on one, yet undetermined, constant C. 1. General (periodic) solution. Solving Eq.(6.3) for R, and taking into account two possible signs of C, one can rewrite Eq.(6.1a) as Thus, the dependence ρ(Y) in the cases C > 0 and C < 0 is given by the quadratures [2], where b 2 = ǫ 2 /|C| > 0 and w = F (Φ|k 2 ) = sn −1 (sin Φ|k 2 ) is the incomplete elliptic integral of the first kind 4 , 4 These expressions have no practical value and will be used below for a sole purpose of proving that the modules of the elliptic integrals must equal +1 by the physics of the problem. Then, and only then is R(ρ) not oscillating in radial direction. This uniquely fixes the constant as |C| = ǫ 2 and guarantee that elliptic integrals become smooth elementary functions. The limits of integration in (6.5) are tentative.
Its inverse is a well-known Jacobi's amplitude function, Φ = am(w|k 2 ). Leaving aside for a while the case of C < 0, we readily find that sin Y 2 = sn(u|k 2 ), cos Y 2 = cn(u|k 2 ), sin Y = 2sn(u|k 2 )cn(u|k 2 ), cos Y = cn 2 (u|k 2 ) − sn 2 (u|k 2 ), (6.8) and, since sn(u|k 2 )cn(u|k 2 )du = −dn(u|k 2 )/k 2 [2], the latter equation is readily integrated, (6.10) In the second case of C < 0 we would have The Jacobi's elliptic functions sn(u|k 2 ), cn(u|k 2 ) and dn(u|k 2 ) are known to be double-periodic functions of their argument. While periodic behavior of the phase Y(ρ) cannot a priori be excluded, periodicity in radial direction is impossible for the invariant density R(ρ), simply because it would conflict with the physical localization.

Localized (aperiodic) solution.
There is, however, a special case when the module of the elliptic function k = 1 and the periodicity disappears (the period becomes infinite). For the Eq.(6.10), this means that b 2 = ǫ 2 /|C| = 1 so that dn(u|1) = 1/ cosh u (as well as cn(u|1) = 1/ cosh u and sn(u|1) = tanh u). For the Eq.(6.11) the same would mean b 2 = −1, which is impossible, since b 2 > 0, by definition. Hence, the case of C < 0 must be dropped from further consideration.

Dirac field in ψ d -mode.
We expect that in real world the mode ψ d with the axial current looking inward will be unstable and not similar, even qualitatively, to the mode ψ u . However, it is instructive to repeat the previous steps and consider only Eqs.(5.7) leaving aside Eqs. (5.5.c,d). Then most of the analysis remains the same and only Eqs.(6.1) and (6.12)-(6.16) are modified. Eqs.(6.1) now read as and the change of the sign of ǫ and of the slope does not affect the characteristic equation (6.3) except that we must replace ǫ → −ǫ, cos Y u → − cos Y d in it. Then the cases C > 0 and C < 0 must be swapped in Eqs.(6.4)-(6.11) with the conclusion that constant C must be determined as C = −ǫ 2 = −(E/m) 2 , and equation (6.3) of characteristics of system (6.18) reads as After using the constraint, 1 − ǫR = − √ 2 sin(Y d /2), the system (6.18) becomes, and its first equation is readily integrated as Acting as previously, we simplify the constrain and Eq.(6.20.b) to where the second equation is identical to (6.18.b) and is a consequence of the first one. When ρ → ∞, we have 1 − ǫR → 0, which is possible only when ǫ = E/m > 0. Here, the condition of a particle at rest requires that ǫ = E/m = +1. We also obtain the anticipated sin Y(∞) = 0 and cos Y d (∞) = 1, i.e. Y d (∞) = π. Thus the solution (in natural units) can be interpreted as an isolated particle at rest with the positive energy E = +m, which is 2m higher than that for the similar localized static ψ u -mode. Here, once again, R(∞) = 1. If the auto-localization is a real process it must favor localization not of ψ d that has a dip , but the bump of ψ u . This is also a hint that an ad hoc created ψ d can be unstable (as it is in Nature). We elaborate on it in the last section. Finally, for the mode with a dip of the invariant density in its interior, the invariant density reaches its theoretical minimum, R(ρ 0 ) = 0, at the inflection point ρ 0 = 1/m 5 . At this point we have sin Y d (ρ 0 ) = 1, i.e. Y d (ρ 0 ) = π/2. Inside this radius the density R, as formally defined by (6.23), becomes negative, which is impossible. This can be a yet another indication that an isolated localized negative charge is unstable (at least in the absence of external field or of stable third bodies nearby). In other words, even being localized, it most likely is "an agile shallow deepening on a hill". Indeed, in real world of a stable matter, all electrons are light and only weakly localized around atomic nuclei, so that normal matter is charge-neutral. The heavy inward-polarized particles (e.g., antiprotons) are found only rarely and they would not be detected without abundant normal matter nearby. These probably are "deep holes on a high hill". Verification of this hypothesis is not a one-body problem and is beyond the scope of this work.

Stability and an effective Lagrangian
The two exact solutions of the Dirac equation in one-body approximation, given by Eqs.(6.13) -(6.16) for the modes ψ u , and by Eqs.(6.21) -(6.23) for the mode ψ d , seem to be very similar to each other except that ψ u has a bump and ψ d has a dip of the invariant density near the center. According to the initial hypothesis, they correspond to positive and negative charges, respectively. The primary guess was [1,4] that the former must be localized better and (if being unstable) live longer than the later, solely because the proper time in their interior flows the slower, the higher the invariant density is. Beyond the one-body approximation, the difference between these solutions is encoded mainly in the last two equations of the system (5.3) for ψ u and (5.5) for ψ d . In the case of ψ u they do not depend on the external field A µ , while in the case of ψ d they do. Furthermore, the tetrad components A [1] + iA [2] in Eqs. (5.5.c,d) oscillate with time as e −2imτ and can cause a transition from ψ d to ψ u .
The field A µ in the Dirac equation is an external field. Remarkably, whatever this field is, the Dirac field determines world time across every auto-localized object. In a sense, all solutions of Eqs.(5.3) and (5.5) with the energy ǫ = E/m are the static solutions. But it is well-known that not all static solutions are stable. Solutions (6.16) and (6.23) obtained in absence of an external field are both truly static since there is nothing in Eqs.(6.1) and (6.18) that could have trigger instability. To investigate the effects of instability one must return to Eqs. (5.5.c,d) and also to Eqs.(5.8) and (5.9), which also account for the external field A D and dynamics of the sums of phases, Z = φ L + φ R . The problem has two different aspects, viz., formation of a perturbation and its decay.
Below, we try to specify both aspects and speculate regarding possible approaches/tools. The following terminology seems most appropriate for the discussion. Let us consider the components of ψ u and ψ d as the wave functions of the initial state and denote them as |u i , |d i . Next, let us contract Dirac equation with the Hermit conjugated wave function of a "final state", f u|, f d| and consider f ... i as "transition amplitudes".

Creation of perturbations in Dirac vacuum.
The problem of what may trigger the initial (and almost necessarily unstable) configuration is the most subtle one. Classically, one has to start with arbitrary initial field ψ and a plausible external field A τ ± A ρ (i.g., of the cosmic microwave background). In quasi-static regime, the interaction of reasonably well defined initial states |u i with the lightlike components A τ ± A ρ of the vector potential is not distracting, since Eqs. (5.3.a,b) can contribute only to diagonal (with respect to the spin) matrix elements, These are not the transitions between up-and down-states. Regardless how weak this interaction is, it takes place in enormous space and for astronomical times. It can collapse to a solitary excitation just because such excitations exist. This mechanism can be considered as a potential source of the cosmic positron excess (for an extensive review see Ref. [3]

Decay of an initial perturbation.
If an initial finite waveform is given, a reasonable theory must predict its decay into stable solitary configurations. Eqs.(5.5.c,d) (unlike (5.3.c,d)) prompt the interaction that affects stability of the inward-polarized state and causes its flip into a stable up-state. In these formulae, A (1) and A (2) are the components of vector potential with respect to a judiciously chosen basis (h 1 , h 2 ) on the surface S (12) ∈ M mapped onto R 4 . The transition from unstable mode to the stable one is due to the charged Dirac currents that naturally oscillate as e −2imτ , and this transition can be triggered by almost any external electromagnetic field. The latter can be random or regular and originate, e.g., from the cosmic background. Possibly, they can even stabilize the ψ d mode for a long time. This could explain the difference between an apparently stable particle in a storage ring and a visibly unstable particle in the natural world.

Similarity to magnetic resonance?
The matrix elements (7.2) are intimately connected with the dynamics of the spin 1/2 in magnetic field, where quantum and classical equations of motion coincide. Indeed, the sectional curvature 6 of the spherical surface S (12) (the curvature of the lines of the charged currents Θ and Φ), is totally due to the projection of the external magnetic field onto radial direction of the axial current. If such a projection is not zero, it will cause flip of the spin polarization into the outward direction of the stable ψ u -mode.

An effective Lagrangian.
More accurate approach that would allow one to go beyond the lowest order approximation can probably be based on the so-called effective Lagrangian, L = ψ + [iα A D A ψ − mρ 1 ]ψ, with the operator of Eq.(2.2) in brackets. The terms depending on A µ in it can be viewed as the interaction with the outside sources. Retaining the interaction term (e = 0), actually, leads beyond the one-body approximation. Below, solely for the purpose of stability analysis, we add the alien up-and/or down-components as a perturbation. The state is supposed to be stable if the alien components dissipate due to the interaction. It will be genuinely unstable if the interaction enforces dissipation of the native components. We continue to dub the configurations with u 2 L + u 2 R > d 2 L + d 2 R as ψ u (with native u and an admixture of alien d). Those with u 2 L + u 2 R < d 2 L + d 2 R are dubbed as ψ d (with native d and alien u). calized object 7 . Similar oscillations also show up in the theory of the Compton scattering as the t-channel transitions of electron into the negative energy states. These transitions are responsible for the classical part of the Compton cross-section (Thompson scattering) . (iii) The difference in degree and the time duration of the localization obviously makes the localized charges of opposite sign unequivocally different particles. The correlation between the signs of electric charge, shape and polarization explains the interdependence between the discrete C-and P -transformations as a natural property of the simplest localized waveforms. While C qualitatively stands for the charge conjugation, P is not an abstract reflection symmetry in a flat space; it stands for the interchange of inward and outward. In a sense, these two discrete transformations do not exist separately; in this sense, CP is a physical symmetry between the corresponding processes 8 .
3. The prospects. Our major perception of vacuum is absence of localized matter. This means that in the vacuum R is constant, e.g., R = 1. Since Dirac equation is a hyperbolic system, the Dirac field must experience refraction towards domains where R > 1, amplifying R even more, which resembles a well-known nonlinear effect of self-focusing. The opposite trend must be observed in domains where R < 1; the Dirac waves tend to escape them. This idea can be phrased more precisely as: Identification of the sign of logR with the sign of electric charge leads to a dynamic picture of an empirically known charge-asymmetric world in which stable positively charged elementary Dirac objects are highly localized (and presumably heavy), while negatively charged objects tend to be poorly localized (and presumably light). This mechanism of localization is generic and points to the picture that stunningly resembles the today's world. It must be worked out in greater details with the prospect that the issue of cosmological charge asymmetry, first addressed long ago by A.D.Sakharov [5], as well as the currently observed positron excess [3], could be better understood.
Meanwhile, to validate our approach in cosmological context, two major questions must be answered, (i) What (if anything) can trigger a spontaneous creation of a proton alone (without an antiproton)? This is the most formidable problem. (ii) Let a pp pair is created in an energetic process and the antiproton is thoroughly isolated from a normal matter (except for the cosmic background radiation). Will it live infinitely long? If not, then how will it decay? This question does not seem unbearable 9 and can be solved by methods developed in this one and previous author's papers (work in progress).

A. Notation and algebraic conventions.
All observables associated with the Dirac field are bilinear forms built with the aid of Hermitian Dirac matrices α i = (α i ) + and β = β + , which satisfy the commutation relations α a βα b + α b βα a = 2βη ab , α a β + βα a = 0, β 2 = 1, (A.1) Throughout this paper, the Dirac matrices associated with a tetrad h µ a ∈ R 4 are numeric and are chosen in the spinor representation, where τ i are the 2 × 2 Pauli matrices. If the Dirac spinor is written down in terms of modules and phases of its components, then, with the Dirac matrices (A.2), the scalars and the four Dirac currents have the following components, = Re iY , ,