On the Origin of the Charge-Asymmetric Matter. II. Localized Dirac Waveforms
aa r X i v : . [ phy s i c s . g e n - ph ] M a y Open Journal of Modern Physics, 2016, 7, 662-679
On the Origin of Charge-Asymmetric Matter.II. Localized Dirac Waveforms
Alexander Makhlin Rapid Research Inc, Southfield, MI, USAEmail: [email protected] 25 February 2016; Accepted 25 April 2016; Published 28 April 2016Copyright c (cid:13)
Abstract
This paper continues the author’s work [1], where a new framework of the matter-inducedphysical 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 ana-lytically. Of the two possible types of the potentially stationary localized configurations of theDirac field, only one is stable with respect to the action of an external field and it correspondsto a positive charge. A connection with the global charge asymmetry of matter in the Universeand with the recently observed excess of the cosmic positrons is discussed.
Keywords
Nonlinear Dirac field; Localization; Cosmological charge asymmetry
1. Introduction
This paper continues the author’s study of the long-standing question of how the physical Dirac field of areal matter becomes a finite-sized particle, and it is approached here as a practical problem. The problemis posed and solved in a new framework of the matter-induced affine geometry [1], which deduces thegeometric relations in the space-time continuum from the dynamic properties of the Dirac field. The intuitiveargument of a possible auto-localization of the Dirac field followed from an observation [1] that the localtime flows slower at higher invariant density, and then from the wave nature of the Dirac equation. Itsfurther consequence must be the (well-known but not clearly understood) charge asymmetry of the observedlocalized 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 ideasand 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 linearlyindependent and Lorentz-orthogonal and can serve as local algebraic basis for any four-dimensional vector
How to cite this paper:
Makhlin, A. (2016) On the Origin of Charge-Asymmetric Matter. II. LocalizedDirac Waveforms. Journal of Modern Physics, , 662-679 http://dx.doi.org/10.4236/jmp.2016.77066662lexander Makhlinspace, including the infinitesimal displacements in coordinate space.The Dirac currents are employed as the Cartan’s moving frame in spacetime which, in its turn, results inthe technique of covariant derivatives for the vector and spinor fields. The physics is naturally brought intothis mathematical picture by the equations of motion of the Dirac field, which made unnecessary an artificialtangent (pseudo)Euclidean space. Differential identities derived from equations of motion fully determine allthe components of the matter-induced affine connection (the Ricci coefficients of rotation of the tetrad) in M and without resorting to a particular coordinate system. Thus determined connections completely definean affine geometry (endowed with the connection but with no metric). Thus defined connection depends onthe Dirac field which makes the Dirac equation nonlinear.With known connections, it became possible to find the coordinate lines and coordinate surfaces of thematter-induced affine geometry, which have a clear physical meaning and quite high degree of symmetry. Thecongruence of lines of the timelike vector current appeared to be normal, thus determining the family of thehypersurfaces of the constant world time τ . The lines of the spacelike axial current appeared to be straightand their congruence normal. They define the surfaces of the constant distance ρ . The two-dimensionalsurfaces of constant ρ at a given time τ were proved to be just spherical surfaces.Below, the inevitable localization of the Dirac field into particles observed in real world, but not explainedby any theory so far, is confirmed by the analytic solutions of the nonlinear Dirac equation in one-bodyapproximation. One of the solutions has maximum near its center and is clearly associated with a stablelocalized positive charge. Another one has minimum and is sought to be an intrinsically unstable negativecharge, which can be only weakly localized by an external field.The content of the paper is organized as follows. In Sec.2 we use the previously developed [1] tools of thematter-induced affine geometry to write down the Dirac equation in its most general coordinate-independentform. Then, in Sec.3 we derive the formulae that connect the Dirac matrices in the principal manifold M andin arithmetic R . In Sec.4, the Dirac equation in written down in a mixed representation, with derivativesin M , and coordinates and Dirac matrices in R . This representation is well suited for finding the analyticsolution. These solutions are found in Sec.5 and their stability is discussed in Sec.7. The conceptual questionsof the charge-asymmetric real world are briefly discussed in the Summary.
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 affineconnection ω ABC , as well as connection Γ B of the Dirac field in principal manifold M ,Γ B = ieA B + (1 / ω ACB ρ α A ρ α C . (2.1)The connection (2.1) determines the covariant derivative of the Dirac field and it enters the Dirac equationas α B Γ B , α B [ ∂ B ψ − Γ B ψ ] = − imρ ψ. (2.2)The nonzero elements of the ω ABC in the tetrad basis of the normalized Dirac currents e A are as follows, ω = − ω = − ω = Q, ω D = 2 e ˜ A [ D ] , ( D = 0 , , , , (2.3)where Q ≡ ∂ [3] ln R = − m P / R = − m sin Y is the derivative of the invariant density R in the directionof the axial current and it has an algebraic representation via the pseudoscalar density P . These formulaeassume that ˜ A [ D ] = + A D for the right-handed spatial triad e [1] , e [2] , e [3] with Θ = R e [1] , Φ = R e [2] and thenaturally outward directed axial current J = R e , i.e. [ ~e [1] × ~e [2] ] = ~e [3] [c.f Eqs. (A.4), (A.5)]. When thelatter is directed inward, but we still wish e [3] to point outward, then we have to take Θ = R e [2] , Φ = R e [1] and replace ω D → ω D = − ω D (or ˜ A [ D ] = − A D ) in Eqs.(2.3) . Throughout this paper, when uppercase index A of the basis e A ≡ e [ A ] , ( A = 0 , , ,
3) takes a particular numeric value weput it in brackets, [0] , [1] , ... . The lowercase indices a that are related to the tetrad h a ≡ h ( a ) are put in parentheses, (0) , (1) , ... . D A = ∂ A − Γ A carries out the parallel transport of the Diracspinor ψ in different directions. Substituting the results (2.3) into connection (2.1), it is straightforward toobtain, α [0] Γ = 12 Qα [3] + 2 ieA [0] α [0] Π = 12 Qα [3] + ie [ A [0] α [0] − ˜ A [0] ρ [3] α [3] ] ,α [3] Γ = +2 ieA [3] α [3] Π = ie [ A [3] α [3] − ˜ A [3] ρ [3] α [0] ] ,α [1] Γ = 12 Qα [3] + 2 ieA [1] α [1] Π = 12 Qα [3] + ie [ A [1] α [1] + i ˜ A [1] α [2] ] ,α [2] Γ = 12 Qα [3] + 2 ieA [2] α [2] Π = 12 Qα [3] + ie [ A [2] α [2] − i ˜ A [2] α [1] ] , (2.4)where Π = (1 ∓ iγ [1] γ [2] ) / S − (1 ± σ (3) ) S/
2. The upper and lower signs in the projector Π (accordingly,the sign in ˜ A [ D ] = ± A [ D ] ) correspond to the outward and inward directions of the axial current, respectively,which then determines the right- and left- oriented spatial triplets e [1] , e [2] , e [3] . It will be shown below,that, from the perspective of the localized solutions, this orientation is translated into the bump of thepositive charge and to the dip of the negative one, respectively, i.e. ± = − sign( ∂ [3] R ). Therefore, dependingon this sign, only the locally inward or locally outward components, ( d L , d R ) or ( u L , u R ), interact with theelectromagnetic potential but with the doubled coupling constant 2 e . In a sense, the charge conjugationgoes together with spatial reflection. The matrix ρ differentiate between the right and left components.With the connection (2.4) the Dirac equation becomes a nonlinear equation and its explicit form readsas, α [0] (cid:20) ∂ [0] − ieA [0] + iρ [3] e ˜ A [3] (cid:21) ψ + α [3] (cid:20) ∂ [3] − ieA [3] + iρ [3] e ˜ A [0] − (3 / Q (cid:21) ψ + α [1] (cid:20) ∂ [1] − ieA [1] − e ˜ A [2] (cid:21) ψ + α [2] (cid:20) ∂ [2] − ieA [2] + e ˜ A [1] (cid:21) ψ + imρ [1] ψ = 0 , (2.5)where anomalous term − Q/ A µ = 0.This equation is valid in every connected domain where R > e µA ( ψ ). As anticipated, it is invariant in a most broad sense – it depends neitheron choice of coordinates x µ in R nor on a tetrad system h µa (also in R ) 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 leavingthe gamma matrices unchanged. But this trick works only for re-parameterizations in R , i.e. change of theLorentz frame or transformations between orthogonal coordinates. It cannot be employed in the principalmanifold 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 Aa ( 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). Thedependence of the Dirac matrices on ψ , α A = V Aa ( ψ ) α a , is not so explicit but not less important and itcannot be avoided. Indeed, in the basis [ A ] each of the currents J A has only one nonzero component, e.g., j A = ψ + α A ψ = V Aa j a = R V Aa V a = R δ A [0] . The latter cannot be achieved without an explicit dependence α A ( ψ ). Indeed, with ψ ∈ M and numericalmatrices 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 forthe 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 . 664lexander Makhlin
3. 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, α a βα b + α b βα a = 2 βη ab , α a β + βα a = 0 , β = 1 . (3.1)Usually one assumes that α a = (1 , α i ); a = 0 , , , i = 1 , , α = 1 is a unit matrix) but thisis not required. An apparently symmetric form of commutation relations (3.1) emerges (along with theequation, iγ a ∂ a ψ − mψ = 0) in terms of the matrices γ a = ( γ (0) , γ i ) = ( β, βα i ), γ a γ b + γ b γ a = 2 η ab . (3.2)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 γ [ κ ] = S − γ ( κ ) S, (3.3)where κ = 0 , , ,
3, [ κ ] is a number standing for superscript A and ( κ ) is the same number for superscript a . There are sixteen linearly independent 4 × O p = (1 , γ a , γ a γ b , ... ), all of which are the productsof 1,2,3 or 4 different gamma. Therefore, O [ p ] = S − O p S = (1 , γ [ a ] , γ [ a ] γ [ b ] , ... ), which is an indisputabletechnical advantage.By their definition, the matrices γ a are not Hermitian. However, since β and α i are Hermitian and anti-commuting, the Hermit-conjugated γ -matrices are ( γ a ) + = γ (0) γ a γ (0) . If, by the same token, γ [ A ] = γ [0] α [ A ] (with Hermitian γ [0] and α [ A ] ), then ( γ [ A ] ) + = γ [0] γ [ A ] γ [0] , which yields, S − γ ( A ) S = γ [ A ] = γ [0] ( γ [ A ] ) + γ [0] = γ [0] ( S − γ A S ) + γ [0] = ( γ [0] S + γ (0) ) γ A ( γ [0] S + γ (0) ) − . Multiplying this by S from the left and by γ [0] S + γ (0) from the right, we find, γ A ( Sγ [0] S + γ (0) ) = ( Sγ [0] S + γ (0) ) γ A . (3.4)The matrix ( Sγ [0] S + γ (0) ) commutes with all the matrices γ A and must be the unit matrix, viz. , γ [0] S − = S + γ (0) . (3.5)On the one hand, we can continue as α [ A ] = γ [0] γ [ A ] = γ [0] S − γ A S = S + γ (0) γ A S = S + α A S. (3.6)On the other hand, condition (3.5) means that γ [0] = S + γ (0) S = S − γ (0) S , which conflicts with Eq.(3.3),because matrix S is not unitary. This conflict can be avoided by adopting a slightly different agreement(that does not affect any of the common usages of the gamma-matrices). Namely, let us denote β = ρ anddefine γ -matrices as γ a = ρ α a and γ [ A ] = ρ α [ A ] . Now we must replace both γ [0] and γ (0) in Eq.(3.4) by ρ , so that S + ρ = ρ S − and γ [ a ] = ρ α [ a ] = ρ S + α ( a ) S = S − ρ α ( a ) S = S − γ ( a ) S, a = 0 , , , , in compliance with (3.3). Choosing α (0) = 1, we have γ (0) = ρ , α [0] = S + S , γ [0] = ρ S + S = S + Sρ =( γ [0] ) + .Throughout this paper, we are only interested in a special case of the transformations (3.3) and (3.6), γ [ A ] = V Aa γ a , α [ A ] = V Aa α a , (3.7)where the transformation matrix V Aa is real and has the properties, V aA V Ba = δ BA , V aA V Ab = δ ab . (3.8)665lexander MakhlinThen the commutation relations (3.1) are the same for γ a and γ A and S must be a solution of the matrixequation, α [ A ] = S + α A S = V Aa ( ψ ) α a . (3.9)Though V Aa has a character of a Lorentz transformation, it has no infinitesimal prototype. Since S + = ρ S − ρ , we also have a habitual γ [ A ] = ρ α [ A ] = S − γ A S = V Aa ( ψ ) γ a . However, in the basis of matrices γ [ A ] , the Pauli-conjugated Dirac spinor must be defined as ψ = ψ + ρ 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 , ρ i , σ i , ρ i σ k = σ k ρ i ), i, k = 1 , ,
3, where ρ = γ , ρ = γ γ γ , ρ = iγ γ γ γ = iρ ρ and σ i = iρ γ i = iγ γ γ γ i = ρ α i . TheDirac 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 = iγ γ γ γ (commonlyknown as − γ ) is an invariant of transformations (3.3), ρ [3] = i ǫ ABCD γ A γ B γ C γ D = i ǫ ABCD V Aa V Bb V Cc V Dd γ a γ b γ c γ d = i ǫ abcd γ a γ b γ c γ d = ρ (3.10)Then the matrix ρ = iρ ρ is transformed like ρ , so that ρ [3] = S − ρ S = ρ , ρ [ i ] = S + ρ i S = ρ i , i = 1 , . (3.11)As long as S + ρ S = S + Sρ = ρ S + S = ρ α [0] , the matrices σ on the M , being defined as σ [ I ] = ρ α [ I ] , aretransformed as σ [ I ] = S + σ I S = ρ ( V I (0) α (0) + V Ij α j ) = V I (0) ρ + V Ij σ j (3.12)(as it should be for the spatial components of the axial current J a ) .
4. The nonlinear Dirac equation, explicitly
So far, we have been studying the general geometric properties of the Dirac field in the scope of the affinegeometry and carefully avoiding any assumptions about what a solution of the Dirac equation that has theseproperties can be. All the previously established [1] properties of the Dirac currents belong (along with theDirac field itself) to the principal differentiable manifold M . Without resorting to any particular coordinatemanifold R 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 in R . (ii) The congruence of lines of the vector field e µ [3] is normal and geodesic. The hypersurfaces S (012) of theconstant radius ρ have constant extrinsic curvature and the holonomic coordinate ρ can serve as x in R . (iii) The two-dimensional surfaces S (12) of constant τ and ρ are just spheres, i.e. umbilical (with two equalGauss’curvatures) surfaces with constant mean (extrinsic) curvature H = m P / R = − m∂ [3] ln R . The latteris determined by the Dirac field within principal manifold M and depends only on the radius ρ . The intrin-sic (sectional) curvature, R t = 2 e ( ∂ [1] A [2] − ∂ [2] A [1] ) − e ( A + A ) = 2 eF = 2 eB [3] , is due to the external electromagnetic field. It coincides with projection of the magnetic field onto the direction of theaxial current. (iv) The two-dimensional surfaces S (03) are covered by a well-defined coordinate net formed by the stream-lines of the vector and axial currents. This net can be identically mapped between the principal manifold M and the arithmetic R . Then the charge-conjugated spinor ψ c = C ψ ∗ = ρ σ ψ ∗ becomes ψ c = ρ σ [2] ψ ∗ . In particular, Λ a ( − ) = ψ + α a ψ c → Λ [ a ]( − ) = ψ + α [ a ] S − ρ σ Sψ ∗ = ψ + S + α a ρ σ Sψ ∗ . At the same time, γ [0] γ [ I ] = S − α I S and iγ [1] γ [2] = S − σ S , ... the spherical symmetry is the property of a solution, thus being adynamic symmetry .In order to find a solution of the Dirac equation, one has to specify a coordinate basis in R and a basisof the Dirac matrices. Here, we shall employ the numerical matrices α a in spinor representation (A.2) andassociate them with a tetrad h µ ( a ) . Then, α A = V Aa 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 mixedrepresentation, Dirac equation reads as V A (0) + V A (3) V A (1) − iV A (2) V A (1) + iV A (2) V A (0) − V A (3) V A (0) − V A (3) − V A (1) + iV A (2) − V A (1) − iV A (2) V A (0) + V A (3) D A ( u R e iφ uR ) D A ( d R e iφ dR ) D A ( u L e iφ uL ) D A ( d L e iφ dL ) = − im u L e iφ uL d L e iφ dL u R e iφ uR d R e iφ dR . (4.1)The operators D A , which are copied from Eq.(2.5), are as follows, D [0] = ∂ [0] − ieA [0] + iρ e ˜ A [3] , D [1] = ∂ [1] − ieA [1] − e ˜ A [2] , D [3] = ∂ [3] − ieA [3] + iρ e ˜ A [0] − Q/ , D [2] = ∂ [2] − ieA [2] + e ˜ A [1] , (4.2)where ρ differentiate between the right and left components and it stands for +1 for u R , d R and for − u L , d L . The coordinate net formed by the integral lines of the tetrad vectors e [0] and e [3] that coversthe two-dimensional surface S (03) in M is holonomic and the vectors h (0) , h (3) in R can be chosen tangentto this surface. In order for the other two tetrad vectors, h (1) and h (2) , to be normal to this surface, it isnecessary that the components V (1)[0] = V (2)[0] = V (1)[3] = V (2)[3] = 0. Just by inspection of Eqs.(A.4), we see thatthis is possible only when either d R = d L = 0 or u R = u L = 0. In both cases, as seen from Eqs.(A.5), wehave V (0)[1] = V (3)[1] = V (0)[2] = V (3)[2] = 0. In other words, the spacetime with the matter-induced anholonomicbasis can be viewed as a direct product of the two-dimensional subspaces, S (03) N S (12) . This is sufficient totreat the up- and down-polarizations separately, ψ u = u R exp( iφ uR )0 u L exp( iφ uL )0 , ψ d = d R exp( iφ dR )0 d L exp( iφ dL ) . (4.3)Having only u R , u L or d R , d L components, the states ψ u and ψ d cannot bear quantum numbers of anangular momentum. For the up-polarized ψ u , we have J (3) = + |J (3) | , Q ≡ ∂ [3] ln R = − m sin Y < R = R u = 2 u R u L and the matrix α ( a ) V [ A ]( a ) in the l.h.s. of Eq.(4.1) simplifies to V [0](0) + V [0](3) = V [3](0) + V [3](3) = u R /u L , V [0](0) − V [0](3) = V [3](3) − V [3](0) = u L /u R , V [1](1) ± iV [1](2) = ∓ i ( V [2](1) ± iV [2](2) ) = e ∓ i ( φ uL + φ uR ) . Accordingly, system (4.1) for ψ u becomes u R [ D [0] + D [3] ] u R e iφ uR = − imu L e iφ uL , e − i ( φ uR + φ uL ) [ D [1] + i D [2] ] u R e iφ uR = 0 ,u L [ D [0] − D [3] ] u L e iφ uL = − imu R e iφ uR , e − i ( φ uR + φ uL ) [ D [1] + i D [2] ] u L e iφ uL = 0 . (4.4)For the down-polarized ψ d , we have J (3) = −|J (3) | , Q = ∂ [3] ln R = + m sin Y >
0. Here, R = R d =2 d R d L and the elements of the matrix in the l.h.s. of Eq.(4.1) become, V [0](0) − V [0](3) = V [3](0) − V [3](3) ) = d R /d L , V [0](0) + V [0](3) = − ( V [3](0) + V [3](3) ) = d L /d R , − ( V [1](1) ± iV [1](2) ) = ∓ i ( V [2](1) ± iV [2](2) ) = e ∓ i ( φ dL + φ dR ) . Now, the system (4.1) reads as d R [ D [0] + D [3] ] d R e iφ dR = − imd L e iφ dL , e − i ( φ dR + φ dL ) [ D [1] + i D [2] ] d R e iφ dR = 0 ,d L [ D [0] − D [3] ] d L e iφ dL = − imd R e iφ dR , e − i ( φ dR + φ dL ) [ D [1] + i D [2] ] d L e iφ dL = 0 . (4.5)667lexander MakhlinRemembering about the sign due to ρ , we obtain the following formulae for all the differential operatorsinvolved, D [0] + D [3] = ∂ [0] + ( ∂ [3] − (3 / Q ) − ie [( A [0] − ˜ A [0] ) + ( A [3] − ˜ A [3] )] , D [0] − D [3] = ∂ [0] − ( ∂ [3] − Q ) − ie [( A [0] − ˜ A [0] ) − ( A [3] − ˜ A [3] )] , D [1] + i D [2] = ∂ [1] + i∂ [2] − ie [( A [1] − ˜ A [1] ) + i ( A [2] − ˜ A [2] )] , D [1] − i D [2] = ∂ [1] − i∂ [2] − ie [( A [1] + ˜ A [1] ) − i ( A [2] + ˜ A [2] )] . (4.6)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 .
5. 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 tetraddetermined by the Dirac currents dx µ = e µA dS A . But the true physical variables are the world time τ and thedistance ρ . They are holonomic coordinates, because dτ = R dS [0] and dρ = R dS [3] are the total differentialsof the independent coordinates dx µ ∈ R , τ − τ = Z x ( τ ) x ( τ ) j µ ( x ) dx µ = Z R dS [0] , ρ − ρ = Z x ( ρ ) x ( ρ ) J µ ( x ) dx µ = Z ±R dS [3] . (5.1)Here, the upper sign is for the ψ u , where J = u R + u L >
0. The lower sign is for ψ d , where J = − d R − d L < τ and the radial variable ρ , being defined asinvariants in M , can immediately be used in arithmetic R . At the points where j (3) = V (3)[0] = 0 and J (0) = V (0)[3] = 0 (in general, a 2-d surface) the relation betweenspatial components, [ ~ Θ × ~ Φ] / R = + ~ J / R , is an algebraic identity. For the axial current directed outward,i.e. J >
0, we take J µ = + R e µ [3] , Θ µ = R e µ [1] and Φ µ = R e µ [2] , so that ~e [3] = [ ~e [1] × ~e [2] ]. In this case, wechange the variables in Eq.(4.4) as follows, ∂ [0] → m R ∂ τ , ∂ [3] → m R ∂ ρ , eA [0] → em R A τ , eA [3] → em R A ρ ,ω e ˜ A [0] → em R ˜ A τ , ω e ˜ A [3] → em R ˜ A ρ , e ˜ A [1] → em ˜ A [1] , e ˜ A [2] → em ˜ A [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 ( ∂ [3] − ∂ [3] ln R ) f = R / ∂ [3] ( R − / f ) = R ·R / ∂ ρ ( R − / f ). Since ∂ A R = ∂ A S = ∂ A P = ∂ A Y = 0 for A = 0 , ,
2, a simple calculation with ∂ τ R = ∂ τ Y = 0 yields the system, R ∂ ρ + ∂ τ ) (cid:0) u R R (cid:1) + i R (cid:0) u R R (cid:1) ( ∂ ρ + ∂ τ ) φ uR = − i (cid:0) u L R (cid:1) e + i Y u , (a) − R ∂ ρ − ∂ τ ) (cid:0) u L R (cid:1) − i R (cid:0) u L R (cid:1) ( ∂ ρ − ∂ τ ) φ uL = − i (cid:0) u R R (cid:1) e − i Y u , (b) (5.3) e − i ( φ uL + φ uR ) (cid:20) ∂ [1] + i∂ [2] (cid:21) u R e iφ uR = 0 , (c) e − i ( φ uL + φ uR ) (cid:20) ∂ [1] + i∂ [2] (cid:21) u L e iφ uL = 0 , (d)where Y u = φ uL − φ uR . For the axial current directed inward, in order to preserve an intuitive physicalunderstanding of a distance from an object, we want e [3] be directed outward. Then the triplet ( e [1] , e [2] , e [3] )668lexander Makhlinwill be left-handed. We have to take J µ = −R e µ [3] , Θ µ = R e µ [2] , and Φ µ = R e µ [1] in order for the vectorproduct [ ~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 ↔ e ˜ A B → − e ˜ A B . Thus, the string of the changeof variables becomes ∂ [0] → m R ∂ τ , ∂ [3] → − m R ∂ ρ , eA [0] → em R A τ , eA [3] → − em R A ρ ,e ˜ A [0] → − em R ˜ A τ , e ˜ A [3] → + em R ˜ A ρ , e ˜ A [1] → − em ˜ A [1] , e ˜ A [2] → − em ˜ A [2] . (5.4)Note, that in the course of the change of variables outlined above, the sign of the e ˜ A [3] has been changedtwice. Now, using the physical variables (5.4) in Eqs.(3.5) we arrive at a similar system, − R ∂ ρ − ∂ τ ) (cid:0) d R R (cid:1) + i R (cid:0) d R R (cid:1)(cid:20) ( ∂ τ − ∂ ρ ) φ dR − em ( A τ − A ρ ) (cid:21) = − i (cid:0) d L R (cid:1) e i Y d , (a) R ∂ ρ + ∂ τ ) (cid:0) d L R (cid:1) + i R (cid:0) d L R (cid:1)(cid:20) ( ∂ τ + ∂ ρ ) φ dL − em ( A τ + A ρ ) (cid:21) = − i (cid:0) d R R (cid:1) e − i Y d , (b) e − i ( φ dL + φ dR ) (cid:20) ∂ [1] + i∂ [2] − iem ( A [1] + iA [2] ) (cid:21) d R e iφ dR = 0 , (c) (5.5) e − i ( φ dL + φ dR ) (cid:20) ∂ [1] + i∂ [2] − iem ( A [1] + iA [2] ) (cid:21) d L e iφ dL = 0 , (d)where Y d = φ dL − φ dR . The difference between ψ u and ψ d is seen right in the equations of motion. The tetradcomponents of an external field along holonomic coordinates, A τ , A ρ ∈ S (03) , affect the ψ u -mode not the ψ d -mode. 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 differencebetween the last two equations of systems (4.3) and (5.5) points to a generic instability of the ψ d - mode .It is discussed in Sec.7 . As the last step before solving systems (5.3) and (5.5) we split real and imaginary parts of the first twoequations of these systems and reduce equations to a form convenient for finding the solutions. For the mode ψ u the result reads as R (cid:18) ∂∂ρ + ∂∂τ (cid:19)(cid:0) u R R (cid:1) = (cid:0) u L R (cid:1) sin Y u , (a) R (cid:18) ∂∂ρ + ∂∂τ (cid:19) φ uR = − u L u R cos Y u , (a ′ ) R (cid:18) ∂∂ρ − ∂∂τ (cid:19)(cid:0) u L R (cid:1) = (cid:0) u R R (cid:1) sin Y u , (b) R (cid:18) ∂∂ρ − ∂∂τ (cid:19) φ uL = u R u L cos Y u . (b ′ ) (5.6) Since e and e are the “angular”directions, it is instructive to recall that the operators L + = L + iL are ladder operatorsfor the angular momentum that moves eigenstate of the L z up. Both systems (5.3) and (5.5) contain only L + . While ψ u cannotbe pushed further up (and is stable), the ψ d is readily pushed up to the ψ u . One can view these transitions as a manifestationof the ψ d -waveform’s “motion”. In fact, it is a flow of surrounding Dirac matter with R ≥ ψ d -dip (or void). ψ d the result is somewhat different, R (cid:18) ∂∂ρ − ∂∂τ (cid:19)(cid:0) d R R (cid:1) = − (cid:0) d L R (cid:1) sin Y d , (a) R (cid:18) ∂∂ρ − ∂∂τ (cid:19) φ dR = d L d R cos Y d + 2 em R ( A ρ − A τ ) , (a ′ ) R (cid:18) ∂∂ρ + ∂∂τ (cid:19)(cid:0) d L R (cid:1) = − (cid:0) d R R (cid:1) sin Y d , (b) R (cid:18) ∂∂ρ + ∂∂τ (cid:19) φ dL = − d R d L cos Y d + 2 em R ( A ρ + A τ ) . (b ′ ) (5.7)The phases φ uR and φ uL are affected in ψ u by the right and left lightlike components of the vector potential,respectively, but with the coupling constant 2 e . Conversely, the phases φ dL and φ dR of the ψ d are not affectedat all.Next, adding and subtracting equations (5.6. a ′ ) and (5.6. b ′ ) and recalling that φ uL − φ uR = Y u we findthat R ∂ Y u ∂ρ = R ∂ Z u ∂τ + (cid:18) X u + 1 X u (cid:19) cos Y u , (a) d R dρ = − sin Y u , (b) R ∂ Z u ∂ρ = − (cid:18) X u − X u (cid:19) cos Y u . (c) (5.8) (cid:2) ∂ [1] + i∂ [2] (cid:3) Z u = 0 , (d)where Z u = φ uL + φ uR and u L /u R = X u . Repeating the same for the mode ψ d we obtain, − R ∂ Y d ∂ρ = R ∂ Z d ∂τ + (cid:18) X d + 1 X d (cid:19) cos Y d − em R A τ , (a) d R dρ = + sin Y d , (b) R ∂ Z d ∂ρ = − (cid:18) X d − X d (cid:19) cos Y d + 4 em R A ρ . (c) (5.9) (cid:2) ∂ [1] + i∂ [2] (cid:3) Z d = 4 em ( A [1] + iA [2] ) , (d)where Z d = φ dL + φ dR 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 onthe angular variables S [1] and S [2] . We postpone discussion of the Eqs.(5.3.c,d) and (5.5.c,d), which areresponsible 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 theycan emerge as asymptotic configurations at τ → ∞ of a transient process that can begin from an arbitraryperturbation 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 ∂ X u ∂τ = X u (cid:18) X u − X u (cid:19) ∂ ln R ∂ρ , ∂ X u ∂ρ = X u (cid:18) X u − X u (cid:19) ∂ ln R ∂ρ , (5.10)where ∂ ρ ln R = − sin Y u / R . Excluding from these two equations the ∂ ρ ln R , one finds a first-order waveequation, ∂ τ X + c ( X ) ∂ ρ X = 0, with the wave velocity c ( X ) = (1 − X ) / (1 + X ). 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, X u ( τ, ρ ) = 1 − ∂ ρ ln R · τ + C ( ρ ) , X u ( τ, ρ ) = R − C ( τ ) R + C ( τ ) , (5.11)670lexander Makhlinwhere the constants of integration C ( τ ) and C ( ρ ) are arbitrary functions of only one argument. Since X u ( ∞ , ρ ) = 1 (and then C ( ∞ ) = 0), we find that at the asymptotic world time τ the coefficients in front ofcos Y u ( ρ ) in Eqs.(5.8.a) and (5.8.c) become 2 and 0, respectively. Assuming further that e = 0 (no externalfield), 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 inEq.(5.8.a); then it cannot depend on τ . Therefore, the only option is ∂ τ Z u = − E = const , Z u = − Eτ ,and it immediately follows that u L = u R = u = R / τ → ∞ since they imply ∂ τ R = 0; a transient process naturallyrequires that ∂ τ R 6 = 0. Similar results are true for the mode ψ d .
6. Stationary solutions.
Being interested here only in stationary states we assume a trivial dependence of the phases of Dirac fieldcomponents on τ , ψ ∝ e − iEτ , and replace, φ R → φ R ( ρ ) − Eτ , φ L → φ L ( ρ ) − Eτ , throughout this section.Then, u L = u R = u = R / d L = d R = d = R /
2. Taking further the coupling constant e = 0, whichis, in fact, equivalent to a one-body approximation, we end up with an autonomous system of two ODEs fortwo unknown functions (the amplitude R ( ρ ) and the phase difference Y ( ρ )) of the natural parameter ρ (andnot the affine parameter s !) along the radial geodesic lines. 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 R ( ρ ) d Y u ( ρ ) dρ = − ǫ R ( ρ ) + 2 cos Y u ( ρ ) , (a) R ( ρ ) d R ( ρ ) dρ = −R ( ρ ) sin Y u ( ρ ) (b) , (6.1)where ǫ = E/m . The characteristic equation for this system, d Y u − ǫ R + 2 cos Y u = − d RR sin Y u , (6.2)is easily solved in terms of w ( R ) = cos Y u . Then, R w ′R − w + 2 ǫ R = 0, andcos Y u = C R + 2 ǫ R , (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 signsof C , one can rewrite Eq.(6.1a) as d Y dρ = ∓ √ C · r ǫ C + cos Y , C > d Y dρ = ∓ p | C | · s ǫ | C | − cos Y . C < . (6.4)Thus, the dependence ρ ( Y ) in the cases C >
C < ρ ( Y ) = ∓ p C (1 + b ) Z Y / dφ q − b sin φ = ∓ p C (1 + b ) F (cid:0) Y (cid:12)(cid:12)
21 + b (cid:1) ] , C > , (6.5) ρ ( Y ) = ∓ p | C | ( b − Z Y / dφ q − − b sin φ = ∓ p C ( b − F (cid:0) Y (cid:12)(cid:12) − b (cid:1) , C < , (6.6)where b = ǫ / | C | > w = F (Φ | k ) = sn − (sin Φ | k ) is the incomplete elliptic integral of the first kind , F (Φ | k ) = Z Φ0 (1 − k sin φ ) − / dφ = Z X [(1 − x )(1 − k x )] − / dx, X = sin Φ . (6.7) These expressions have no practical value and will be used below for a sole purpose of proving that the modules of theelliptic 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 and guarantee that elliptic integrals become smooth elementary functions. Thelimits of integration in (6.5) are tentative. w | k ). Leaving aside for a while the case of C <
0, we readily find thatsin Y u | k ) , cos Y u | k ) , sin Y = 2sn( u | k )cn( u | k ) , cos Y = cn ( u | k ) − sn ( u | k ) , (6.8)where u = √ ǫ + C ρ = F ( Y / | / (1 + b )), k = 2 / (1 + b ). Now Eq.(6.1b) becomes, d R ( ρ ) dρ = − sin Y ( ρ ) = − u | k )cn( u | k ) , (6.9)and, since R sn( u | k )cn( u | k ) du = − dn( u | k ) /k [2], the latter equation is readily integrated, R ( ρ ) = √ ǫ + CC dn (cid:0)p ǫ + C ρ (cid:12)(cid:12)
21 + b (cid:1) , C > . (6.10)In the second case of C < R ( ρ ) = p ǫ − | C || C | dn (cid:0)p ǫ − | C | ρ (cid:12)(cid:12) − b (cid:1) , C < . (6.11)The Jacobi’s elliptic functions sn( u | k ), cn( u | k ) and dn( u | k ) are known to be double-periodic functions oftheir argument. While periodic behavior of the phase Y ( ρ ) cannot a priori be excluded, periodicity in radialdirection is impossible for the invariant density R ( ρ ), simply because it would conflict with the physicallocalization.
2. Localized (aperiodic) solution.
There is, however, a special case when the module of the ellipticfunction k = 1 and the periodicity disappears (the period becomes infinite). For the Eq.(6.10), this meansthat b = ǫ / | 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 ). Forthe Eq.(6.11) the same would mean b = −
1, which is impossible, since b >
0, by definition. Hence, thecase of
C < C of integration in the Eq.(6.3) is now uniquely determined as C = ǫ = ( E/m ) , and theequation of characteristics of system (6.1) becomescos Y + 1 = 2 cos ( Y /
2) = ( ǫ R + 1) . (6.12)Since the Jacobi’s elliptic functions with module k = 1 are elementary functions, it is much easier to return tothe original system (6.1) and the characteristic equation (6.12) with C = ǫ , using the latter as a constraint.After using the constraint (with the signs to be determined later), ǫ R + 1 = ±√ Y u / d Y u dρ = − / ǫ cos Y u , (a) d R dρ = − sin Y u = − Y u Y u , (b) (6.13)and its first equation is readily integrated to ρ ( Y ) first, and then yields Y ( ρ ) √ ǫρ = tanh − (sin Y u , sin Y u − tanh( √ ǫρ ) , cos Y u √ ǫρ ) . (6.14)When ρ → ∞ , we have ǫ R +1 →
0, which is possible only when ǫ = E/m <
0. We also obtain the anticipatedsin Y ( ∞ ) = 0 and cos Y u ( ∞ ) = −
1, i.e. Y u ( ∞ ) = π . Returning the result of integration into Eqs.(6.12) and(6.13b), we simplify the latter to ǫ R + 1 = −√ √ ǫρ ) , d R dρ = − sin Y u ( ρ ) = − √ | ǫ | ρ )cosh ( √ ǫρ ) . (6.15)672lexander MakhlinIn order for this solution to be interpreted as an isolated particle at rest, we must require that E = − m .Thus the solution R ( ρ ) = √ √ ρ ) + 1 , (6.16)is the mode with the negative energy with respect to the vacuum level zero attributed to R = 1. Finally, innatural units, sin Y u √ mρ ) , R ( ρ ) = √ √ mρ ) + 1 . (6.17)This result also follows from Eq.(6.9), since dn( u |
1) = 1 / cosh u . We can take the radius ρ of the spher-ical surface, where d R /dρ reaches its maximum (the inflection point) for the size of the particle. Here,sin Y u ( ρ ) = 1, and, consequently, sinh( √ mρ ) = 1, cosh( √ mρ ) = √
2. Therefore (in natural units), ρ = sinh − (1) √ m = 0 . m and s = ρ R ( ρ ) = 1 m , as it was previously contemplated. At the radius ρ , also as expected, the phase is Y u ( ρ ) = π/
2. Indeed,cos( Y u ( ρ ) /
2) = 1 / √ π/
4) and sin Y u ( ρ ) = 1, R u ( ρ ) = 2. The peak amplitude R u (0) = 1 + √ ψ d -mode. We expect that in real world the mode ψ d with the axial current looking inward will be unstable andnot similar, even qualitatively, to the mode ψ u . However, it is instructive to repeat the previous steps andconsider only Eqs.(5.7) leaving aside Eqs.(5.5.c,d). Then most of the analysis remains the same and onlyEqs.(6.1) and (6.12)-(6.16) are modified. Eqs.(6.1) now read as R ( ρ ) d Y d ( ρ ) dρ = +2 ǫ R ( ρ ) − Y d ( ρ ) , (a) R ( ρ ) d R ( ρ ) dρ = + R ( ρ ) sin Y ( ρ ) , (b) (6.18)and the change of the sign of ǫ and of the slope does not affect the characteristic equation (6.3) except thatwe must replace ǫ → − ǫ, cos Y u → − cos Y d in it. Then the cases C >
C < C must be determined as C = − ǫ = − ( E/m ) , andequation (6.3) of characteristics of system (6.18) reads as1 − cos Y = 2 sin ( Y /
2) = (1 − ǫ R ) . (6.19)After using the constraint, 1 − ǫ R = −√ Y d / d Y d dρ = − / | ǫ | sin Y d , (a) d R dρ = sin Y d = 2 sin Y d Y d , (b) (6.20)and its first equation is readily integrated as √ mρ = tanh − (cid:0) cos Y d (cid:1) , cos Y d √ ǫρ ) , sin Y d √ ǫρ ) . (6.21)Acting as previously, we simplify the constrain and Eq.(6.20.b) to1 − ǫ R = √ √ ǫρ ) , sin Y d ( ρ ) = +2 sinh( √ ǫρ )cosh ( √ ǫρ ) = d R dρ , (6.22)where the second equation is identical to (6.18.b) and is a consequence of the first one. When ρ → ∞ , wehave 1 − ǫ R →
0, which is possible only when ǫ = E/m >
0. Here, the condition of a particle at rest requires673lexander Makhlinthat 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) R ( ρ ) = 1 − √ √ mρ ) , (6.23)can be interpreted as an isolated particle at rest with the positive energy E = + m , which is 2 m higher thanthat for the similar localized static ψ u -mode. Here, once again, R ( ∞ ) = 1. If the auto-localization is a realprocess 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 itstheoretical minimum, R ( ρ ) = 0, at the inflection point ρ = 1 /m . At this point we have sin Y d ( ρ ) = 1,i.e. Y d ( ρ ) = π/
2. Inside this radius the density R , as formally defined by (6.23), becomes negative, whichis impossible. This can be a yet another indication that an isolated localized negative charge is unstable (atleast 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 electronsare light and only weakly localized around atomic nuclei, so that normal matter is charge-neutral. Theheavy inward-polarized particles (e.g., antiprotons) are found only rarely and they would not be detectedwithout abundant normal matter nearby. These probably are “deep holes on a high hill”. Verification ofthis hypothesis is not a one-body problem and is beyond the scope of this work.
7. Stability and an effective Lagrangian
The two exact solutions of the Dirac equation in one-body approximation, given by Eqs.(6.13) -(6.16) forthe 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 formermust be localized better and (if being unstable) live longer than the later, solely because the proper timein 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 theydo. Furthermore, the tetrad components A [1] + iA [2] in Eqs.(5.5.c,d) oscillate with time as e − imτ and cancause 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 Diracfield 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 arestable. Solutions (6.16) and (6.23) obtained in absence of an external field are both truly static since thereis nothing in Eqs.(6.1) and (6.18) that could have trigger instability. To investigate the effects of instabilityone 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. , formationof a perturbation and its decay.Below, we try to specify both aspects and speculate regarding possible approaches/tools. The followingterminology seems most appropriate for the discussion. Let us consider the components of ψ u and ψ d as thewave functions of the initial state and denote them as | u i i , | d i i . Next, let us contract Dirac equation with theHermit conjugated wave function of a “final state”, f h u | , f h d | and consider f h ... i i as “transition amplitudes”. The problem of what may trigger the initial (and almost necessarily unstable) configuration is the mostsubtle one. Classically, one has to start with arbitrary initial field ψ and a plausible external field A τ ± A ρ In general, none of the Dirac currents vanishes at R = 0; they all become proportional to one lightlike vector that musthave both up- and down-components. Then nothing can identify the surface S (12) of constant τ and ρ as a two-dimensionalsphere. | u i i with the lightlike components A τ ± A ρ of the vector potential is not distracting, sinceEqs.(5.3.a,b) can contribute only to diagonal (with respect to the spin) matrix elements, − ie f h d R | ( A τ + A ρ ) | d R i i , − ie f h d L | ( A τ − A ρ ) | d L i i . (7.1)These are not the transitions between up- and down-states. Regardless how weak this interaction is, it takesplace in enormous space and for astronomical times. It can collapse to a solitary excitation just because suchexcitations exist. This mechanism can be considered as a potential source of the cosmic positron excess (foran extensive review see Ref. [3]). Furthermore, in Eqs.(5.3.c,d) that could have trigger transition from up-to down- states, there is no interaction terms at all. Thus, solution (6.17) of Eqs.(5.3), which is associatedwith a positive charge, is expected to be stable. If an initial finite waveform is given, a reasonable theory must predict its decay into stable solitaryconfigurations. Eqs.(5.5.c,d) (unlike (5.3.c,d)) prompt the interaction − ie f h u R | e i ( φ dL + φ dR ) ( A (1) + iA (2) ) | d R i i and − ie f h u L | e i ( φ dL + φ dR ) ( A (1) + iA (2) ) | d L i i , (7.2)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 , h ) onthe surface S (12) ∈ M mapped onto R . The transition from unstable mode to the stable one is due to thecharged Dirac currents that naturally oscillate as e − imτ , and this transition can be triggered by almostany external electromagnetic field. The latter can be random or regular and originate, e.g., from the cosmicbackground. Possibly, they can even stabilize the ψ d mode for a long time. This could explain the differencebetween an apparently stable particle in a storage ring and a visibly unstable particle in the natural world. The matrix elements (7.2) are intimately connected with the dynamics of the spin 1 / of the sphericalsurface S (12) (the curvature of the lines of the charged currents Θ and Φ ), R t = 2 e ( ∂ [1] A [2] − ∂ [2] A [1] ) − e ( A + A ) = 2 eF = 2 eB [3] , (7.3)is totally due to the projection of the external magnetic field onto radial direction of the axial current. Ifsuch a projection is not zero, it will cause flip of the spin polarization into the outward direction of the stable ψ u -mode. More accurate approach that would allow one to go beyond the lowest order approximation can probablybe based on the so-called effective Lagrangian, L = ψ + [ iα A D A ψ − mρ ] 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 forthe purpose of stability analysis, we add the alien up- and/or down-components as a perturbation. The stateis supposed to be stable if the alien components dissipate due to the interaction. It will be genuinely unstableif the interaction enforces dissipation of the native components. We continue to dub the configurations with u L + u R > d L + d R as ψ u (with native u and an admixture of alien d ). Those with u L + u R < d L + d R aredubbed as ψ d (with native d and alien u ). The sectional curvature of a surface spanned by a net of the lines of the vectors e and e equals to the angle by which thebasis ( e , e ) is rotated after moving along an infinitesimal loop within this surface. a and Φ a and consider the matrix element, T ab = < b | T | a > = ψ + b (cid:20) α [1] (cid:0) − ieA [1] − e ˜ A [2] (cid:1) ψ + α [2] (cid:0) − ieA [2] + e ˜ A [1] (cid:1)(cid:21) ψ a , (7.4)between the configurations ψ a and ψ b . Here, ˜ A D stands for A D when the triplet ( e [1] , e [2] , e [3] ) forms theright-handed system, and for − A D when this triplet is left-handed. As an illustration, consider a particularterm assuming native u L , u R and alien d L , d R ; then T ab is ψ + b T + ψ a = ψ + b [ eA [1] ( − iα [1] + α [2] ) + eA [2] ( − iα [2] − α [1] )] ψ a (7.5)= − ie ( A [1] − iA [2] ) · ψ + b ρ σ + ψ a = − eA µ ( e µ [1] − ie µ [2] ) ψ + b O u ρ σ + O d ψ a . Here, σ + = ( σ [1] + iσ [2] ) / e [3] of the right-hand oriented triplet. Let us recall that O u/d = (1 ± σ ) / T + isas follows. The ladder operator ρ σ + eliminates the native components u R and u L (acting on ψ a as O d )and replaces them with the alien d R and d L , producing ψ ′ = ( d R , , − d L , σ + O d = σ + , this canbe viewed as a two-step action. Namely, the O d (inherited from connection (2.4)) filters out the d R and d L in their alien position, and then σ + moves them “up”, thus filtering out the positive helicity of the native“up”-final state ψ + b O u . In other words, ψ + b T + ψ a ∝ ( u ∗ bR d aR − u ∗ bL d aL ) e − imτ . If the state ψ a was a pureup-state ψ u and had no components ( d R , d L ) at all, then ψ + b T + ψ u = 0; this is the case of Eqs.(4.3)– the ψ u does not interact with the external A [1] , A [2] . Conversely, the state ψ d that has only ( d R , d L ) is unstableunder this interaction and the charged currents will convert it to the up-state. This reproduces the primitiveanalysis of Eqs.(7.1) and (7.2).Since the effective Lagrangian is nonlinear, there are many open questions, which cannot be addressedcomprehensively within the scope of the present work. For example, it is not clear a priori , which of states,initial or final, should determine the nonlinear terms. These issues will be discussed separately. Of highestpriority are the questions about time scales of the processes that contribute to the transition amplitudes(7.2) as well as about stability of the uniform distribution of the invariant density.
8. Summary
1. The method.
The most intriguing discovery of this work is that Dirac field endows spacetime witha matter-induced affine geometry (MIAG), which is fully determined by a real matter . This is possible solelybecause the Dirac field satisfies equations of motion. Then, and only then, the geometry is independent of aparticular coordinate background. Possibly, this result can look strange for mathematicians. But it shouldnot surprise physicists, who know very well that nothing in spacetime can be measured without localizedmaterial objects. So far, the method of MIAG determined the shape of a solitary localized object as sphericaldynamically and with no conjectures. The problem of signals still has to be worked out.
2. The results.
The author’s conjecture [4] that there exists a generic mechanism of the Dirac fieldauto-localization into finite-sized positively charged Dirac particles is rigorously confirmed. The explicitsolution representing such a particle is found. It possesses the following properties,(i) A solitary Dirac field waveform in free space can be stable with respect to the interaction with anexternal electromagnetic field A µ only if this waveform is formed solely by outward polarized components.The solution that represents such a waveform has negative energy E = − m .(ii) An apparently complementary inward-polarized solution with negative charge has positive energy E = + m . It cannot be stable as a strongly localized object; its instability is due to the indispensable“charged currents”Θ and Φ. They oscillate twice faster than stationary Dirac field, Θ ± i Φ ∝ e ± iEτ . Thecorresponding tetrad components A [1] , A [2] of the vector potential affect only the inward polarized waveform,thus making it unstable. This “motion”is confined to within the spheres of a constant radius within a lo-676lexander Makhlincalized object . Similar oscillations also show up in the theory of the Compton scattering as the t -channeltransitions of electron into the negative energy states. These transitions are responsible for the classical partof the Compton cross-section (Thompson scattering) .(iii) The difference in degree and the time duration of the localization obviously makes the localized chargesof opposite sign unequivocally different particles. The correlation between the signs of electric charge, shapeand polarization explains the interdependence between the discrete C - and P -transformations as a naturalproperty of the simplest localized waveforms. While C qualitatively stands for the charge conjugation, P isnot an abstract reflection symmetry in a flat space; it stands for the interchange of inward and outward . Ina sense, these two discrete transformations do not exist separately; in this sense, CP is a physical symmetrybetween the corresponding processes .
3. The prospects.
Our major perception of vacuum is absence of localized matter. This means thatin the vacuum R is constant, e.g., R = 1. Since Dirac equation is a hyperbolic system, the Dirac field mustexperience refraction towards domains where R >
1, amplifying R even more, which resembles a well-knownnonlinear effect of self-focusing. The opposite trend must be observed in domains where R <
1; the Diracwaves tend to escape them. This idea can be phrased more precisely as:
Identification of the sign of log R with the sign of electric charge leads to a dynamic picture of an empirically known charge-asymmetric worldin which stable positively charged elementary Dirac objects are highly localized (and presumably heavy), whilenegatively charged objects tend to be poorly localized (and presumably light). This mechanism of localizationis generic and points to the picture that stunningly resembles the today’s world. It must be worked out ingreater details with the prospect that the issue of cosmological charge asymmetry, first addressed long agoby 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)? Thisis the most formidable problem.(ii) Let a p ¯ p pair is created in an energetic process and the antiproton is thoroughly isolated from a normalmatter (except for the cosmic background radiation). Will it live infinitely long? If not, then how will itdecay? This question does not seem unbearable and can be solved by methods developed in this one andprevious author’s papers (work in progress). Acknowledgments
I am indebted M.E. Osinovsky for his advice on subtle issues of spinor analysis and differential geometryand for critically reading the manuscript. This work is supported by the Rapid Research, Inc.
References [1] Makhlin, A. (2016)
Journal of Modern Physics , , 587-610 http://dx.doi.org/10.4236/jmp.2016.77061[2] Byrd, P.F.and Friedman, M.D. (1971) Handbook of elliptic integrals for engineers and scientists, 2dedition, Springer-Verlag,Berlin. http://dx.doi.org/10.1007/978-3-642-65138-0[3] Serpico, P.D. (2012) Astroparticle Physics , , 2-11. http://dx.doi.org/10.1016/j.astropartphys.2011.08.007[4] A.Makhlin,Localization,CP-symmetry and neutrinosignalsof theDiracmatter,arxiv:1005.2693[math-ph][5] Sakharov, A.D. (1967) JETP Letters , , 24-27. This motion cannot be interpreted as an oscillation of a mean coordinate – the famous Schr¨ o dinger Zitterbewegung . This is in contrast with the view of Dirac field as the representation of the Lorentz group. In that framework, the Poincar´einvariance is presumed, and all states can be obtained from a single state by a sequence of the Lorentz transformations. Some theories of Grand Unification predict proton’s decay with a lifetime greater than the currently estimated age ofUniverse. From our perspective, only antiproton can be unstable.
A. Notation and algebraic conventions.
All observables associated with the Dirac field are bilinear forms built with the aid of
Hermitian
Diracmatrices α i = ( α i ) + and β = β + , which satisfy the commutation relations α a βα b + α b βα a = 2 βη ab , α a β + βα a = 0 , β = 1 , (A.1)Throughout this paper, the Dirac matrices associated with a tetrad h µa ∈ R are numeric and are chosen inthe spinor representation, α = (cid:18) (cid:19) , α i = (cid:18) τ i − τ i (cid:19) , σ i = (cid:18) τ i τ i (cid:19) ρ = (cid:18) (cid:19) , ρ = (cid:18) − i · i · (cid:19) , ρ = (cid:18) − (cid:19) . (A.2)where τ i are the 2 × ψ = u R exp (cid:0) iφ uR (cid:1) d R exp (cid:0) iφ dR (cid:1) u L exp (cid:0) iφ uL (cid:1) d L exp (cid:0) iφ dL (cid:1) , (A.3)then, with the Dirac matrices (A.2), the scalars and the four Dirac currents have the following components, j a = u L + d L + u R + d R u L d L cos (cid:18) φ uL − φ dL (cid:19) − u R d R cos (cid:18) φ uR − φ dR (cid:19) − u L d L sin (cid:18) φ uL − φ dL (cid:19) + 2 u R d R sin (cid:18) φ uR − φ dR (cid:19) u L − d L − u R + d R , J a = u L + d L − u R − d R u L d L cos (cid:18) φ uL − φ dL (cid:19) + 2 u R d R cos (cid:18) φ uR − φ dR (cid:19) − u L d L sin (cid:18) φ uL − φ dL (cid:19) − u R d R sin (cid:18) φ uR − φ dR (cid:19) u L − d L + u R − d R , (A.4)Θ a = − u L d R cos (cid:18) φ uL + φ dR (cid:19) + 2 d L u R cos (cid:18) φ uR + φ dL (cid:19) u L u R cos (cid:18) φ uL + φ uR (cid:19) − d L d R cos (cid:18) φ dR + φ dL (cid:19) − u L u R sin (cid:18) φ uL + φ uR (cid:19) − d L d R sin (cid:18) φ dR + φ dL (cid:19) − u L d R cos (cid:18) φ uL + φ dR (cid:19) − d L u R cos (cid:18) φ uR + φ dL (cid:19) , Φ a = − u L d R sin (cid:18) φ uL + φ dR (cid:19) + 2 d L u R sin (cid:18) φ uR + φ dL (cid:19) u L u R sin (cid:18) φ uL + φ uR (cid:19) − d L d R sin (cid:18) φ dR + φ dL (cid:19) u L u R cos (cid:18) φ uL + φ uR (cid:19) + 2 d L d R cos (cid:18) φ dR + φ dL (cid:19) − u L d R sin (cid:18) φ uL + φ dR (cid:19) − d L u R sin (cid:18) φ uR + φ dL (cid:19) , (A.5)678lexander Makhlin S + i P = 2 (cid:18) u R u L e i ( φ uL − φ uR ) + d R d L e i ( φ dL − φ dR ) (cid:19) = R e i Y , , R = 4[ u R u L + d R d L + 2 u R u L d R d L cos( φ uL − φ uR − φ dL + φ dR )] ..