In the previous papers of the author [1] [2] a novel framework of the matter- induced affine geometry (MIAG) was developed and the simplest (two-com- ponent) autolocalized solutions of the nonlinear Dirac equations were found in explicit form. The solitary autolocalized Dirac field waveforms in free space turned out to be spherically symmetric, and, most importantly, this symmetry is dynamical; it is a consequence of the equations of motion. Below, we continue our quest for the stationary/stable autolocalized solutions because only these are pertinent to the problem of cosmological charge asymmetry. The requirement of absolute stability seems to be imperative in interstellar or even intergalactic space, but it is not necessarily a prerequisite in laboratory experiments. These are conducted in “normal” charge-asymmetric world with primitive fragments of antimatter created artificially, and then thoroughly guarded in sophisticated traps.

The Problem. In this study, we aim at finding solitary static autolocalized solutions of the Dirac equation along with a proof that the generic nonlinear mechanism of autolocalization (the local time slowdown) favors matter over antimatter. From this perspective, the case of hydrogen or anti-hydrogen atoms is not a one-body problem, and it is not addressed here. Autolocalization of the Dirac field from fluctuations in a uniform background (which will be addressed in another paper) is most likely a very slow and rare transient process that ends up with a proton. Its timescale and relative weight of all the underlying processes and/or mechanisms are not yet clear, but the Universe definitely had enough time to conduct such an experiment. Unlike the pioneer work by A. Sakharov [3] , where the origin of currently observed charge asymmetry was attributed to the violation of CP-invariance and nonequilibrium processes in the early hot Universe, this work ascribes it to the fundamental mechanism of the Dirac field autolocalization.

The present work extends the previous results [2] to a more realistic case of the four-component Dirac field. Being otherwise unwieldy, calculations are significantly simplified by accounting, ab initio, for the earlier discovered [1] dynamical spherical symmetry; below, we call it the “spherical ansatz”. The explicit calculations confirm our previous conjecture that only one of the two major types of isolated localized solutions is genuinely stable (has well-defined energy and satisfies all the consistency conditions). It is identified as a particle with positive charge and negative energy. It is imperative to find, as the next step, the transient process that ends up with a stable autolocalized waveform.

The Method. The earlier developed [1] mathematical background for the present work is based on the following ideas and results. It has been observed long ago [4] that if a physical Dirac field is defined at a point in spacetime continuum (the principal differentiable manifold), then such field deter- mines the tetrad of Dirac currents. These are linearly independent and Lorentz- orthogonal. They can serve as local algebraic basis for any four-dimensional vector space, including the infinitesimal displacements in coordinate space. The singular case when the Dirac currents are lightlike, which is possible only on a two-dimensional surface in spacetime, is not considered here. With this excep- tion, the Dirac field provides the means to navigate through the spacetime so that geometry can be viewed a descendant of stable Dirac matter.

The Dirac currents (one timelike and three spacelike) are further employed as the Cartan’s moving frame in spacetime, which, in its turn, resulted in the technique of covariant derivatives for the vector and spinor fields. The physics was naturally brought into this mathematical picture by the equations of motion of the Dirac field. The 28 primary differential identities that were derived in [1] from equations of motion fully determined all the components of the matter- induced affine connection (the Ricci coefficients of rotation of the tetrad) in and without resorting to a particular coordinate system. Connections determi- ned this way completely defined an affine geometry (endowed with the connec- tion but with no metric), which was dubbed [1] a matter-induced affine geometry. With known connections, it became possible to find the coordinate lines and surfaces of the MIAG, all of which have a clear physical meaning and quite high degree of symmetry. Notably, the MIAG uniquely determines the hypersurfaces of constant world time of a waveform, though inside a stable solitary waveform there can be neither events nor propagating signals (not to speak of rods and synchronized clocks)¹.

The general properties of coordinate surfaces in are discovered in [1] without any assumptions regarding the nature of an ambient space or the Dirac field. It appears that the main qualitative characteristic of the stationary Dirac object is the direction of axial current, which can point only outward or inward. It was realized that the locally defined notions of outward and inward are prerequisites for any reasonable discussion of the localization phenomenon. The framework of matter-induced affine geometry not only ideally fits this goal but also explains the autolocalization, as it is seen in the real world, as an intrinsic property of the Dirac field. For a waveform with the mass parameter m, the nonlinearity of Dirac equation is effective at distances comparable with the Compton wavelength,.

The Outline and Results. The paper is organized as follows. Section 2 provides a concise overview of the results of the author’s papers [1] and [2] , where the concept of MIAG was introduced. In Section 3.1 we explicitly translate the previously established spherical symmetry into the requirement that the axial current only have one nonzero component. The spherical ansatz carries no ambiguity, since the congruence of lines of the axial current is both normal and geodetic. Section 3.2 accumulates observations from a frontal attack on the nonlinear system of equations for the four-component Dirac spinors, which point to the most rational choice of variables. At the end of this section we describe the complete protocol of mapping the ambient space onto physical manifold, where all “geometric quantities” are defined by the Dirac field. The radial coordinate r, which is compatible with the matter-defined affine curvature and can serve as a usual coordinate in the ambient, is introduced in Section 3.3. The 16 additional differential identities (this time, for the convection currents), that serve as a test for the stability of the solitary waveforms are derived in Section 4. Using the “optimal variables” found in Section 3.2, the original system (written down explicitly in Appendix B) is reduced to the real-valued equations in Section 6, which are thoroughly analyzed in Section 0 for and in Section 7 for. The analytic solutions in natural variables of the physical manifold are found in Section 6.3. Here, we find more evidence that the mode is unstable. This issue cannot be resolved within a one-body problem. Essentially, for both modes the vacuum level follows from equations of motion and it cannot be altered by a fiat.

The analytic solutions in variables of the ambient space are found in Section 7.2. When the energy of a waveform is defined with respect to the world time of the ambient Minkowski space, then there is no way to resolve the proper time slowdown. There is no difference between the shapes of and; the vacuum level of invariant density for both modes appears to be. Furthermore, their shape is not fixed any by the nonlinear dynamics (as it should be for all autolocalized waveforms, e.g. solitons). This is an ultimate proof of the previous conjecture that proper time slowdown is the major mechanism behind autolocalization in physical manifold.

The last section points to several intriguing results which did no receive the discussion they deserve. All of them are challenges that could not be addressed in this paper, because they probably cannot be met within the scope of one-body problem. We are just making physically motivated conjectures. For example, we propose to look for a correlation between the excess of positrons in cosmic rays and strength of magnetic fields in their sources.

Looking for a solution of the practical problem of autolocalization of the Dirac field into solitary waveforms I proposed a novel concept of matter-induced affine geometry (MIAG) [1] . It stems from the observation that Dirac field, being a coordinate scalar, naturally generates at a point an affine centered vector space (spanned by the Dirac currents), which is similar to the tangent space of the four-dimensional manifold (spanned by the vectors or). It determines, at any point of spacetime, a locally Minkowskian basis comprised of four Dirac currents defined on the principal physical manifold. These are the vector current , the axial current, and the two “charged currents”, which are the real and imaginary parts of the. Here, is the invariant density of the Dirac field. The latter is defined as , which is small subset of many Fierz identities [5] , and, by definition,. The superscripts numerate the Dirac matrices and also the components of the Dirac currents with respect to tetrad basis , which belongs to an “intermediate” manifold. By means of yet another Fierz identity, viz.

where is a usual pseudo-Euclidean Minkowski metric, the quadruples, , form a complete set of orthogonal unit vectors (even though no notion of length has been introduced). The quadruple is the solution of the linear system,. Therefore, all indices are moved up and down by the Minkowski or, which is nothing but a consequence of the Fierz identities². The quantities and are the scalar and pseudoscalar densities, respectively, and. Expressions for the Dirac currents and scalars in terms of ampli- tudes and phases of the Dirac field’s components are presented in Appendix A.

The covariant derivative of the Dirac field is of a standard form, . It is determined without leaving the principal manifold. The connections of the Dirac field are

in tetrad basis of principal manifold and in auxiliary basis, respectively. We adopted, without discussion, the usual Dirac equations of motion in these bases,

with an arbitrary mass parameter m. Using the Dirac currents as a moving frame, we explored differential identities for the curls and divergences of all four Dirac currents and found all components of the affine connection (the coefficients of rotation of the tetrad). The nonzero elements of the in the tetrad basis of the normalized Dirac currents are as follows:

where is the derivative of invariant density in direction of the axial current, and it has an algebraic representation via the pseudoscalar density. Hence, the equation of motion (2.3) acquires a term, which is linear in Dirac field and in pseudoscalar, which, in its turn, is bilinear in. Therefore, Dirac equation becomes a nonlinear system,

where the anomalous term, , singles out the direction of axial current among others even when an external field. The notation discerns between the cases of outward and inward directions of the axial current of the localized Dirac waveform, which must be considered separately. In the first case, and in the second case, where is the “natural” tetrad component of the vector potential in the connection (2.2). This difference came from the requirement that the radial coordinate, as measured along the congruence of axial current, increases in the outward direction. The Dirac matrix (a.k.a.) differentiates between the right and left components, and it stands for +1 for and for −1 for.

The dynamics of the Dirac currents determines, in a unique way, the hypersurfaces of constant world time and of constant radius in; and are the natural parameters along normal congruences of vector and axial currents, respectively. Because the differentials and are complete (integrable), both and are the matter-defined holonomic coordinates over.

Coefficients of rotation of the tetrad basis are determined within the principal manifold, which guarantees that nothing in depends on the choice of coordinates in arithmetic. The analytic two-component solutions of the nonlinear Dirac equation (2.5), which were found in Ref. [2] , in absence of external electromagnetic field, are of two types. One of them, , has the directed outward (or up) axial current and a magnified invariant density, , in its interior. There, the proper time flows slower than the world time (which is the same across the entire waveform). These waveforms are supposed to be small, heavy, and positively charged particles. Waveforms of the second type, , with the axial current directed inward (or down), have a reduced invariant density, , in their interior, so that the proper time flows faster than the world time. They must be light, negatively charged, and cannot be truly localized or even stationary unless there exists an external “attractive field”; otherwise, waves of the Dirac field will tend to escape into ambient space.

Since and, these two directional derivatives yield two additional nonlinear terms to the Dirac equation (2.5). The first of them is solely responsible for the local time slowdown, which gives a major contribution to autolocalization. In fact, this is the main physical mechanism behind autoloca- lization of any wave field, which always tend to concentrate in domains with the minimal phase velocity.

In this paper, we consider the general case of four-component Dirac spinors and address the issue of stability more thoroughly. Namely, we examine whether the autolocalized Dirac waveforms, which were discovered within the MIAG framework, satisfy an extended set of differential identities (unconditionally or under specific conditions). For some of them, the answer will be affirmative. These are the true physical solutions. All others must be rejected. In a sense, the additional identities play the same role as boundary conditions that can validate or invalidate certain solutions as being pertinent to a physical problem. To derive them, we will also need the Dirac tensors,

where stands for a skew-symmetric product. These tensors satisfy the following Fierz identities, which allow one to express them via vectors and scalars [5] ,

where. Mathematically, because the covariant derivatives of the tetrad vectors are just the coefficients of rotation, , it will be easy to calculate the derivatives of these tensors.

Below, we explore the properties of the four-component solutions. In this paper, advancement became possible due to the following earlier observation. Three of four Dirac currents (the vector current, , and two charged currents,) constitute a canonical system with respect to the congruence of axial current [1] ³. Therefore the entire tetrad is Fermi-transported along the lines of vector field. These lines point into radial direction, and their congruence appears to be both normal and geodesic over the principal manifold.

Equations (2.5) with the coefficients of rotation (2.4) are descendants of the nonlinear Dirac system (2.3). These equations incorporate only the same 28 differential identities that were derived in Ref. [1] and completely determine the geometry of the Dirac field of the solitary waveforms. Since these identities are derived from the equations of motion, the properties of the waveforms found so far (like being stationary and spherically symmetric) are the dynamic symme- tries. However, not all solutions of these equations are physically acceptable. Of the two two-component analytic solutions found in Ref. [2] , only one is unques- tionably stable, which indicates that, possibly, not all relevant constrains were found and/or employed. In this section, we derive more identities that fill in these blanks.

The well-known Gordon’s decomposition of the vector current is one more differential identity that follows from the Dirac equation of motion. It aims at a qualitative dissecting of the vector current into flux of electric charge (bulk motion of localized charged particles, usually dubbed as convection/conduction current) and local electromagnetic polarization (e.g., proper or induced electric and magnetic moments) frozen into this flux. After the bulk transport is separated, the internal dynamics of a localized waveform is encoded in polariza- tion tensors (2.4). In classical electrodynamics, charged particles are considered pointlike and governed by ordinary differential equations of their trajectories. A gap with continuous nature of field described by PDE has never been consis- tently bridged, mostly because an intrinsic mechanism for localization of a realistic field of a matter has not been found until recently [1] [2] .

Originally, the Gordon’s decomposition was introduced as an identity for the vector current, in which transport and polarization parts are explicitly split. Here, we are dealing with four Dirac currents, and each of them allows for such decomposition. The procedure and result appear to be very similar for the Dirac currents, , and. For the axial current, the result is even qua- litatively different, but it prompts more useful identities involving fluxes. Flux of the pseudoscalar density, is of special interest for the unstable mode, since its decay must result in additional “propagating waveforms”. Overall, the Gordon decompositions of the Dirac currents provide 16 differential identities that must be satisfied for stable solitary waveforms. While we have no compre- hensive approach, a picture of fluxes in ambient space seems to be the only way to learn what the products of decay can be.

1) The vector current. The Gordon’s decomposition of the vector current is readily obtained by replacing and then in the definition with the r.h.s. of the Dirac equation (2.2) and its conjugate, viz.,

After taking the half-sum and splitting the products of Dirac matrices as the result reads as follows:

and, as long as satisfies equations of motion (is an on-mass-shell solution), this is just yet another identity. By virtue of (2.1), the expression in brackets in second term becomes, so that (c.f. Eq. (2.6))

In holonomic coordinate basis, this would be a well-known result. To com- pute, we resort to the Fierz identity (2.7.a). Using the previously found coefficients of rotation, Eqs. (2.3), we find that the second term in (2.7.a) does not contribute to and

Collecting all terms proportional to in the l.h.s., we obtain the final result for the convection flux of the scalar density,

which differs from the commonly known in three respects. First, only the current of unstable mode, where, is affected by the electro- magnetic field. Secondly, the directions of convection and total currents do not necessarily coincide. Third, for stable mode, where, the convec- tion part in the r.h.s. of the last equation differs from total vector current only by a scalar factor, so that convection and total currents are parallel (which hints an intrinsic stability). Finally, one can determine the fraction of polarization current within the total one in both modes.

2) The axial current. The Gordon’s decomposition of the axial current is surprisingly different. Since and we can rewrite the axial current in two ways,

Taking the half-sum of these two expressions and separating symmetric and skew-symmetric products of Dirac matrices, we obtain

Unlike the previous case, we find in the axial current neither convection flux, nor a recognizable polarization component. Instead, the last equations allows one to discover that the pseudoscalar density of the Dirac field has the intrinsic property of propagation and can be viewed as a relativistic field in its own right [10] .

The expected pattern of the convection current of the pseudoscalar density emerges if, instead of the sum, we take difference of the substitutions (4.5). This results in an identity,

which looks similar to Eq. (4.10), except that there is no full axial current in its l.h.s. Proceeding as previously, we find that the second term in (2.7.b) does not contribute to and

Therefore, the convection current of the pseudoscalar density does exist and is as follows:

Quite understandably, the pseudoscalar density is carried not by a spacelike axial current, , but by a timelike vector current,.

3) The “charged currents”. The Gordon’s decomposition of the charged currents employs two representations,

This case is very similar to the first one, and the result

is similar to (4.2). There is no counterpart to the second term of Eq. (4.2) here simply because, which is one of the Fierz identities [5] . The r.h.s. of Eq. (4.10) can be computed using the Fierz identity (2.7.c). Since the Dirac current is complex-valued and thus gauge variant (unlike the real currents and), its gauge-invariant covariant derivative consists of two parts,. Accordingly,. Once again, using Eqs. (2.3) and exercising some algebra we obtain,

and rearrange Eq. (4.10) as

with the same observations as for Eq. (4.4). Putting here for its explicit representation (2.7.c) we obtain

where and the tetrad index of the basis can be replaced by any other tetrad index, including the coordinate index. In the last equation, the component of the vector potential can be eliminated by a gauge transformation (c.f. footnote¹¹). For the convection part of the charged current (A.8) in the l.h.s. of (4.13) to be nonzero, the Dirac field of a waveform must have both and components, which is not required in the r.h.s.

It should be noted that the density, corresponding to the convection current (4.13), is zero, which is just one of many the Fierz identities,. The physical meaning of this current is unclear and we will refrain from using this equation as an additional constraint.

In this section,which is mostly technical, we carry out the program outlined in Section 3.2. The cases and are considered separately. They belong to the different spacetime domains separated by a singular two-dimen- sional surface, where all four Dirac currents become lightlike. The question that remains open is whether these domains can be parts of a single solitary waveform. We begin with reduction of the system (3.5), which is written down in expanded form in Appendix B, to real-valued equations. Subsequent analysis drastically simplifies the coordinate dependencies, so that we end up with the system of ODE.

The differences of first equations in couples (B.4.a)-(B.5.a) and in (B.11.a)- (B.12.a) yield the expected general result, , well known as a non- conservation of the axial current. Namely,

where, we remind, , and (cf. Eq. (3.1)). In fact, these phase differences determine the shape of waveforms, which is rather a rule than exception for autolocalized solitary solutions of all wave fields.

The sums of Eqs. (B.4.a)-(B.5.a) and (B.11.a)-(B.12.a), after using Eqs. (5.1) to exclude and exercising simple algebra, result in

where the box emphasizes results, which will be used later without a notice (especially in lengthy equations). Hence, away from a singular surface, the rapidity in tetrad (3.17) does not change in radial direction, making invariant density and phase differences the only r-dependent functions.

In Eqs. (B.4.a)-(B.5.a) and in (B.11.a)-(B.12.a), like, one can safely put. Therefore, the sums of phases, and, do not depend on radial variable either. (As a matter of fact, neither, nor show up in any of the equations below¹¹.) But according to (3.1) and (3.3), we have. Therefore, is also r-independent. Finally, the phase differences and obviously do not depend on. Recalling Eqs. (3.4), we conclude that the only r-dependent phase differences are and. It is also clear that the ratio (when) (or, (when)) also does not depend on. These observations lay firm ground for the future separation of radial variable in equations that determine localization.

Analysis of the Eqs. (B.6), (B.7) and (B.13), (B.14) is more cumbersome, and the cases of and will be analyzed one-by-one.

1) The -mode,.

Let us multiply Eqs. (B.6) and (B.7) by and, respectively, add them up, and divide the result by. Next, we make the following substitutions:

remembering that while does not depend on. After separating the real and imaginary parts, simple but lengthy algebra yields two Eqs. (5.4.a,c) below. If we multiply Eqs. (B.6) and (B.7) by and, respectively, and add them up, then the same procedure yields two Eqs. (5.4.b,d) below. Next, consider the half-sum and half-difference of the last two equations and split their real and imaginary parts. These algebraic calculations are bulky but straightforward. In terms of the variables, which were introduced in previous section (Section 3), we have

2) The -mode,.

In this case, we multiply Eqs. (B.13) and (B.14) by and, respec- tively, add them up, and divide the result by. Next, we make the following substitutions:

After separating real and imaginary parts, a simple but lengthy algebra yields two Eqs. (5.6.a,c) below. If we multiply Eqs. (B.13) and (B.14) by and, respectively and add them up, then the same procedure yields two Eqs. (5.6.b,d) below. Here, and.

and in Eqs. (5.4) and (5.6), are the components of the vector potential with respect to the tetrad (cf. Eqs. (3.21)). For the sake of completeness, both systems include Eq. (5.1) as (e) and Eq. (4.4) as (f). In the latter, we have replaced the ratio by the and simplified it. Notably, the roles of electric potential and magnetic in Eqs. (5.4) and (5.6), are interchanged, which prompts the differences in physical mechanisms of autolocalization for the - and -modes.

In what follows, we are interested only in the stationary solitary waveforms. Therefore, we continue with an ad hoc assumption that the components are static with respect to the world time of a stable waveform,

According to Eqs. (3.14), the tetrad vectors and are not orthogonal. Nevertheless, since localization occurs in locally defined world time but can be observed directly only in, the fully adequate variables are and. Furthermore, it is natural to define the external field in also. However, the intermediate calculations are more transparent in terms of the couple. This is the simplest way to detect and eliminate the redundant dependencies.

Then we can immediately rely on the following previously established general properties of stationary solitary waveforms:

1) The quantities, and depend only on radial variable and not on any other variables.

2) The sum of two phases and the differences and, as well as “rapidity” do not depend on; a priori, they can depend on any of the three other variables.

Under assumption (5.7), Eqs. (5.4) and (5.6) will yield even more similar relations that allow one to drastically simplify both systems:

3) The Lorentz parameter does not depend on,; then by virtue of Eq. (3.14),.

4) The second derivatives and. The quantities and depend only on and this dependence is linear, and.

5) The component can depend only on, and for both modes.

Other relations of this kind are mode-specific, partially because the choice of the meaningful physical variables critically depends on whether the mode is stable. This is not known in advance .

It appears that there are only two viable options that we will employ here and motivate later on in Section 0 after we derive in Section 0 additional differential identities that involve convection currents.