Ground state and the spin precession of the Dirac electron in counterpropagating plane electromagnetic waves
aa r X i v : . [ qu a n t - ph ] J un Ground state and the spin precession of the Dirac electron in counterpropagatingplane electromagnetic waves
G. N. Borzdov ∗ Department of Theoretical Physics and Astrophysics,Belarusian State University, 4 Nezavisimosti Av., 220030 Minsk, Belarus
The fundamental solution of the Dirac equation for an electron in an electromagnetic field withharmonic dependence on space-time coordinates is obtained. The field is composed of three stand-ing plane harmonic waves with mutually orthogonal phase planes and the same frequency. Eachstanding wave consists of two eigenwaves with different complex amplitudes and opposite directionsof propagation. The fundamental solution is obtained in the form of the projection operator definingthe subspace of solutions to the Dirac equation. It is illustrated by the analysis of the ground stateand the spin precession of the Dirac electron in the field of two counterpropagating plane waves withleft and right circular polarizations. Interrelations between the fundamental solution and approxi-mate partial solutions is discussed and a criterion for evaluating accuracy of approximate solutionsis suggested.
PACS numbers: 03.65.Pm, 03.30.+p, 02.30.Nw, 02.30.Tb
I. INTRODUCTION
Considerable recent attension has been focussed on thepossibility of time and space-time crystals [1–4], analo-gous to ordinary crystals in space. The papers [1, 2]provide the affirmative answer to the question, whethertime-translation symmetry might be spontaneously bro-ken in a closed quantum-mechanical system [1] and atime-independent, conservative classical system [2]. Aspace-time crystal of trapped ions and a method to real-ize it experimentally by confining ions in a ring-shapedtrapping potential with a static magnetic field is pro-posed in [3]. Standing electromagnetic waves comprizeanother type of space-time crystals. It was shown [4]that one can treat the space-time lattice, created by astanding plane electromagnetic wave, by analogy withthe crystals of nonrelativistic solid state physics. In par-ticular, the wave functions, calculated within this frame-work by using the first-order perturbation theory for theSchr¨odinger-Stuekelberg equation, are Bloch waves withenergy gaps [4]. The analylitical solution for the Kein-Gordon equation in the case of a field composed of twocounterpropagating laser waves is obtained in [5].Standing electromagnetic waves constitute an interest-ing family of localized fields which may have importantpractical applications. In particular, optical standingwaves can be used to focus atoms and ions onto a surfacein a controlled manner, nondiffracting Bessel beams canbe used as optical tweezers which are noninvasive toolsgenerating forces powerful enough to manipulate micro-scopic particles. Superpositions of homogeneous planewaves propagating in opposite directions, the so-calledWhittaker expansions, play a very important role in an-alyzing and designing localized solutions to various ho-mogeneous partial differential equations [6]. ∗ [email protected]
In this article we treat the motion of the Dirac electronin an electromagnetic field with four-dimensional period-icity, i.e., with periodic dependence on all four space-timecoordinates. In terms of the three-dimensional descrip-tion, such electromagnetic space-time crystal (ESTC)can be treated as a time-harmonic 3D standing wave.In solid state physics, the motion of electrons in naturalcrystals is described by the Schr¨odinger equation with aperiodic electrostatic scalar potential. The descriptionof the motion of electrons in ESTCs by the Dirac equa-tion takes into account both the space-time periodicity ofthe vector potential and the intrinsic electron properties(charge, spin, and magnetic moment). In this case, theDirac equation reduces to an infinite system of matrixequations. To solve it, we generalize the operator meth-ods developed in [7] to the cases of infinite-dimensionalspaces and finite-dimensional spaces with any number ofspace dimensions. The evolution, projection and pseu-doinverse operators are of major importance in this ap-proach. The evolution operator (the fundamental solu-tion of a wave equation) describes the field dependenceon the space-time coordinates for the whole family of par-tial solutions. The method of projection operators is veryuseful at problem solving in classical and quantum fieldtheory [8–10]. It was developed by Fedorov [8, 9] to treatfinite systems of linear homogeneous equations. In theframe of Fedorov’s approach, it is necessary first to findprojection operators which define subspaces of solutionsfor two subsystems (constituent parts) of the system tosolve, and then to find its fundamental solution, i.e., theprojection operator defining the intersection of these sub-spaces, by calculating the minimal polynomial for someHermitian matrix of finite dimensions. We present a dif-ferent approach, based on the use of pseudoinverse op-erators, which is applicable to both finite and infinitesystems of equations and has no need of minimal poly-nomials.The fundamental solution of the Dirac equation for thefield composed of three standing plane harmonic waveswith mutually orthogonal phase planes and the same fre-quency is presented in Sec. II. The case of two counter-propagating plane waves with left and right circular po-larizations is treated in Sec. III. Additional informationon the numerical implementation of the presented ap-proach and some results of its computer simulation canbe found in [11–13].
II. BASIC RELATIONSA. Matrix form
An electron in an electromagnetic field with the four-dimensional potential A = ( A , iϕ ) is described by theDirac equation (cid:20) γ k (cid:18) ∂∂x k − iA k ec ~ (cid:19) + κ e (cid:21) Ψ = 0 , (1)where κ e = m e c/ ~ , c is the speed of light in vacuum, ~ isthe Planck constant, e is the electron charge, m e is theelectron rest mass, γ k are the Dirac matrices, Ψ is thebispinor, x , x and x are the Cartesian coordinates, x = ict , and summation over repeated indices is carriedout from 1 to 4. In [11–13] we have treated the field with A ≡ iϕ = 0 and A ′ ≡ em e c A = X j =1 (cid:0) A j e i K j · x + A ∗ j e − i K j · x (cid:1) , (2)which is composed of six plane waves with unit wave nor-mals ± e α , where e α are the orthonormal basis vectors, α = 1 , , x = ( r , ict ), r = x e + x e + x e . All sixwaves have the same frequency ω and K j = k N j , j = 1 , , ..., , k = ω c = 2 πλ , N j = ( e j , i ) , N j +3 = ( − e j , i ) , j = 1 , , . (3)They may have any polarization, so that their complexamplitudes are specified by dimensionless real constants a jk and b jk as follows A j = X k =1 ( a jk + ib jk ) e k , j = 1 , , ..., , (4)where a jj = b jj = a j +3 j = b j +3 j = 0 , j = 1 , , . For the electromagnetic lattice under consideration,the solution of Eq. (1) can be found in the form of aFourier seriesΨ = Ψ e i x · K , Ψ = X n ∈L c ( n ) e i x · G ( n ) , (5)where K = ( k , iω/c ) is the four-dimensional wave vec-tor, k = k e + k e + k e , n = ( n , n , n , n ) isthe multi-index specifying n = n e + n e + n e and G ( n ) = ( k n , ik n ). Here, c ( n ) are the Fourier ampli-tudes (bispinors), and L is the infinite set of all multi-indices n with an even value of the sum n + n + n + n .Substitution of A (2) and Ψ (5) in Eq. (1) results in theinfinite system of matrix equations X s ∈ S V ( n, s ) c ( n + s ) = 0 , n ∈ L , (6)where S = { s h ( i ) , i = 0 , , ..., } (7)= { (0 , , , , (0 , , − , − , (0 , − , , − , ( − , , , − , (1 , , , − , (0 , , , − , (0 , , , − , (0 , , − , , (0 , − , , , ( − , , , , (1 , , , , (0 , , , , (0 , , , } is the set of 13 values of function s h = s h ( i ), where s h (0) = (0 , , ,
0) is the null shift. At i = 1 , ...,
12, thisfunction specifies the shifts s = ( s , s , s , s ) = s h ( i ) ofmulti-indices n , defined by the Fourier spectrum of thefield A (2), which satisfy the condition | s | + | s | + | s | = | s | = 1. Because of this, they will be denoted theshifts of the first generation [ g d ( s ) = 1]. By the def-inition, g d ( s , s , s , s ) = max {| s | + | s | + | s | , | s |} .Thus, each equation of the system relates 13 Fourieramplitudes (bispinors), in other words, each amplitudeenters in 13 different matrix equations. We intensivelyuse indexing of various mathematical objects by points n = ( n , n , n , n ) of the integer lattice L . The sequen-tial numbering of these points, based on the use of g d ( n ),drastically simplifies both numerical implementation ofthe presented techniques and analysis of solutions, be-cause it takes into account the specific Fourier spectra ofthe electromagnetic lattice and the wave function, as wellthe structure of the finite models described below and inmore detail in [12].It is well known, e.g., see Ref. [14], that 16 Dirac ma-trices form a basis in the space of 4 × k , k = 0 , ...,
15, which makes it possible, inparticular, to reconstruct any matrix Γ k from its number k , see Ref. [11]. Any 4 × V = P k =0 V k Γ k isuniquely defined by the set of its components D s ( V ) = { V k } in the Dirac basis [Dirac set of matrix V , briefly, D-set of V ]. Due to the structure of the Dirac equation, suchexpansions yield a convenient way to represent derivedmatrix expressions in a concise form, accelerate numeri-cal calculations, and reduces data files. This approach isof particular assistance in solving the system of Eqs. (6),see Ref. [11–13]. D-sets of matrices V ( n, s ) are presentedin [11] as functions of the dimensionless parameters Q = ( q , iq ) = K /κ e , Ω = ~ ω m e c , (8) q = q e + q e + q e = ~ k m e c , q = ~ ωm e c . (9) B. Operator form
Let us treat the infinite set C = { c ( n ) , n ∈ L} of theFourier amplitudes c ( n ) of the wave function Ψ (5) as anelement of an infinite dimensional linear space V C . Since,for any n ∈ L , c ( n ) = c ( n ) c ( n ) c ( n ) c ( n ) ≡ c c c c n (10)is the bispinor, C ∈ V C will be denoted the multispinor.Let us define a basis e j ( n ) in V C and the dual basis θ j ( n ) = e † j ( n ) in the space of one-forms V ∗ C ( n ∈ L ): e ( n ) = n , e ( n ) = n ,e ( n ) = n , e ( n ) = n , (11) θ ( n ) = (cid:0) (cid:1) n , θ ( n ) = (cid:0) (cid:1) n ,θ ( n ) = (cid:0) (cid:1) n , θ ( n ) = (cid:0) (cid:1) n . (12)In this notation, the system of equations (6) takes theform h f j ( n ) , C i ≡ X s ∈ S V jk ( n, s ) c k ( n + s ) = 0 , (13)where j = 1 , , , n ∈ L , and f j ( n ) = X s ∈ S V jk ( n, s ) θ k ( n + s ) , h f j ( n ) , e k ( n + s ) i = V jk ( n, s ) . (14)These relations can be rearranged to the basic system ofequations P ( n ) C = 0 , n ∈ L , (15)where P ( n ) = [ f α ( n )] † ⊗ a αβ ( n ) f β ( n ) (16)is the Hermitian projection operator with the trace tr [ P ( n )] = 4 and following properties:[ P ( n )] = [ P ( n )] † = P ( n ) , (17) a ( n ) = [ L ( n )] − , L αβ ( n ) = D f α ( n ) , (cid:2) f β ( n ) (cid:3) † E , (18)where α, β = 1 , , ,
4. The Hermitian 4 × L ( n ) and a ( n ) at n = ( n , n , n , n ) are defined by thefollowing D-sets: D s [ L ( n )] = (cid:8) I A + w + w + w + w , , , , − w , , , , , w w , w w , w w , , , , } , (19) D s [ a ( n )] = 1 | L ( n ) | (cid:8) I A + w + w + w + w , , , , w , , , , , − w w , − w w , − w w , , , , } , (20)where I A =2 X j =1 | A j | = 2 (cid:0) a + b + a + b + a + b + a + b + a + b + a + b + a + b + a + b + a + b + a + b + a + b + a + b (cid:1) , (21) | L ( n ) | = I A + 2 I A (cid:0) w + w + w + w (cid:1) + (cid:0) w + w + w − w (cid:1) , (22)and w j = q j + n j Ω. It is significant that, for a nonva-nishing electromagnetic field ( I A = 0), the determinant | L ( n ) | > n ∈ L . C. Fundamental solution
The fundamental solution S , i.e., the operator of pro-jection onto the solution subspace of the multispinorspace V C , and the projection operator P of the infinitesystem of equations (15) are defined as follows [11] S = U − P , P = + ∞ X k =0 X n ∈F k ρ k ( n ) , (23) + ∞ [ k =0 F k = L , F j \ F k = ∅ , j = k, (24)where ρ k ( n ) are Hermitian projection operators with thetrace tr [ ρ k ( n )] = 4, and U is the unit operator in V C ,which can be written as U = X n ∈L I ( n ) , I ( n ) = e j ( n ) ⊗ θ j ( n ) , tr [ I ( n )] = 4 . (25)For any C ∈ V C , C = S C is a partial solution ofEq. (15), i.e., the function Ψ (5) with the set of Fourieramplitudes { c ( n ) , n ∈ L} = S C satisfies the Dirac equa-tion (1) for the problem under consideration.The Hermitian operator P of the system of equa-tions (15), by definition (see appendix), has the followingproperties P † = P = P , P ( n ) P = P P ( n ) = P ( n ) (26)for any n ∈ L , and ρ k ( n ) satisfy the relations ρ † k ( n ) = ρ k ( n ) = ρ k ( n ) , tr [ ρ k ( n )] = 4 , n ∈ L , (27) ρ k ( m ) ρ l ( n ) = 0 if k = l or (and) m = n, (28) ρ ( n ) = P ( n ) , n ∈ F . (29)There exist various ways to split the lattice L into sub-lattices F k to fulfil conditions (24) and (28), one of themis described in Ref. [12]. Providing these conditions aremet, substitution of α = k − X j =0 X n ∈F j ρ j ( n ) ≡ P k − , β = P ( m ) , m ∈ P k (30)into Eqs. (A.9) and (A.10) results in ρ k ( m ) = δ (A.9).It follows from Eq. (16) that P ( m ) P ( n ) = (cid:2) f i ( m ) (cid:3) † ⊗ [ a ( m ) N ( m, n ) a ( n )] i j f j ( n ) , (31)where N ij ( m, n ) = D f i ( m ) , (cid:2) f j ( n ) (cid:3) † E , i, j = 1 , , , , (32) N ( n, n ) ≡ L ( n ) (18). At any given n, Eq. (6) relates theFourier amplitude c ( n ) only with 12 amplitudes c ( n + s ),where g d ( s ) = 1. In consequence of this, N ( m, n ) ≡ g d ( n − m ) >
2. Substitution of (14) in (32) at n = m + s gives N † ( n, m ) = N ( m, n ) = L ( m ) for n = m, = N ( m, s ) for g d ( s ) = 1 , = N ( s )Γ for g d ( s ) = 2 . (33)The D-sets of 12 matrices N ( m, s ) and the table of 56scalar coefficients N ( s ) are presented in Ref. [13]. Thesemajor structural parameters of the electromagnetic lat-tice specify interrelations in the system of equations (6).The relations presented in this section and appendixprovide convenient means to find operators ρ k ( n ) bymaking use the recurrent algorithm devised to minimizevolumes of computations and data files [11, 12]. It be-gins with the selection of an infinite subsystem consist-ing from independent equations and the calculation ofthe projection operators ρ ( n ) = P ( n ) , n ∈ F ⊂ L ,which uniquely define the fundamental solutions of theseequations. At each new step of the recurrent process, weadd another infinite set of mutually independent equa-tions which, however, are related with some of the equa-tions introduced at the previous steps. Consequently, weobtain an infinite set of independent finite systems of in-terrelated equations [fractal clusters of equations]. It canbe described as a 4d lattice of such clusters. Each stepof the recurrent procedure expands clusters for which itprovides the exact fundamental solutions. D. Approximate solutions
Numerical implementation of the obtained solution im-plies the replacement of the projection operator P (23)of the infinite system (15) by the projection operator P ′ = X k ∈ k L X n ∈ n L ( k ) ρ k ( n ) (34) of its finite subsystem P ( n ) C = 0 , n ∈ L ′ = [ k ∈ k L n L ( k ) ⊂ L . (35)Here, k L is an ordered finite list of integers, and n L ( k ) isa finite list of points n ∈ F k , specifying a finite model ofthe infinite lattice. The projection operator S ′ = U − P ′ (36)defines the exact fundamental solution of Eq. (35), whichis also the approximate solution of Eq. (15), provided bythis finite model.In this article, we restrict our consideration to the casewhen the amplitude C specifying a partial solution isgiven by C = a j e j ( n o ), n o = (0 , , , C = { c ( n ) , n ∈ S d } = S ′ C = C − P ′ C (37)describes the four-dimensional subspace of exact solu-tions of Eq. (35). Here, S d ⊂ L is the solution do-main, i.e., the subset of L with nonzero bispinors c ( n ).Bispinors c ( n ) and a are linearly related as c ( n ) = S ( n ) a , (38)where S ( n ) is the 4 × S ij ( n ) = (cid:10) θ i ( n ) , S ′ e j ( n o ) (cid:11) are defined in [12]. Substituting c ( n ) in Eq. (5) givesΨ = X n ∈ S d c ( n ) e iϕ n ( x ) ≡ E v a , (39)where E v = X n ∈ S d e iϕ n ( x ) S ( n ) (40)is the evolution operator. In terms of the dimensionlesscoordinates r ′ = r /λ = X e + X e + X e , X = ct/λ , the phase function ϕ n ( x ) can be written as ϕ n ( x ) = ( k + k n ) · r − ( ω + ω n ) t = 2 π [( n + q / Ω) · r ′ − ( n + q / Ω) X ] . (41)The evolution operator E v is the major characteristicof the whole family of partial solutions Ψ (39). In partic-ular, it provides a convenient way to calculate the meanvalue h A i = a † A E a a † U E a (42)of an operator A with respect to function Ψ, where A E = I ∆ X ( E † v AE v ), U E = I ∆ X ( E † v E v ) = X n ∈ Sd S † ( n ) S ( n ) , (43) I ∆ X ( f ) ≡ Z ∆ X f dX dX dX dX , (44)and∆ X is the domain given by intervals [ X k , X k +1] , k =1 , , , . E. Evaluating accuracy of solutions
The distinguishing feature of the presented techniqueis that each step of the recurrent procedure expands thesubsystem of equations for which it provides the exactfundamental solution. One can check the calculation foraccuracy by using relations (27) and (28). Substitutionof c ( n ) (38) into the left side of Eq. (6) reduces it to theform V S ( n ) a , where V S ( n ) = X s ∈ S V ( n, s ) S ( n + s ) . (45)At n ∈ L ′ , the equation V S ( n ) a = 0 is satisfied at any a , because in this domain V S ( n ) ≡
0. This providesmeans for final numerical checking of the fundamentalsolution S ′ of the system (35) and the evolution operator E v ( x ) (40) for accuracy [12].Let D be a differential operator in a space V Ψ of scalar,vector, spinor, or bispinor functions, and k Ψ k be thenorm of Ψ on V Ψ . The functional R : Ψ
7→ R [Ψ] = k Ψ D kk Ψ k (46)where Ψ D = D Ψ, evaluates the relative residual at thesubstitution of Ψ into the differential equation D Ψ = 0.It provides a fitness criterion to compare in accuracy var-ious approximate solutions of this equation. For an exactsolution Ψ, the residual Ψ D vanishes, i.e., R [Ψ] = 0. IfΨ D = 0, but R [Ψ] ≪
1, the function Ψ may be treatedas a reasonable approximation to the exact solution, andthe smaller is R [Ψ], the more accurate is the approxima-tion. In terms of distances d = k Ψ k and d D = k Ψ D k ofΨ and Ψ D to the origin of V Ψ (the zero function), onecan graphically describe R [Ψ] as shrinkage in distance R [Ψ] = d D /d . The functional R , as applied to a familyof functions Ψ( x , λ ) with members specified by a param-eter λ , results in function R [Ψ( x , λ )] of λ , denoted below R ( λ ) for short.To introduce this criterion in the problem under con-sideration, we first transform Eq. (1) to the equivalentequation D Ψ = 0 with the dimensionless operator D = X k =1 α k (cid:18) − i ~ m e c ∂∂x k − A ′ k (cid:19) − i ~ m e c ∂∂t + α . (47)From Eqs. (39) and (47) followsΨ D = D Ψ = D v a , (48)where D v = D E v is the evolution operator describing thefamily of remainder functions Ψ D [12]. The norm of Ψ D (48) can be written as k Ψ D k = q a † U D a , (49)where the matrix U D is presented in [12]. Thus, for thefunction Ψ (39), from the definition (46) follows R = s a † U D a a † U E a . (50) F. Orthogonality relation
Let Ψ a = Ψ a e i x · K a and Ψ b = Ψ b e i x · K b be solutionsof the Dirac equation, i.e., D Ψ a ≡ , D Ψ b ≡
0, where K a = ( k , iω a /c ) , K b = ( k , iω b /c ) , ω a = ω b , andΨ a = X n ∈L a ( n ) e i x · G ( n ) , Ψ b = X n ∈L b ( n ) e i x · G ( n ) . (51)Upon integrating the identity Ψ † b D Ψ a − (Ψ † a D Ψ b ) ∗ ≡ I ∆ X (Ψ † b Ψ a ) = 0,which can be also written as X n ∈L b † ( n ) a ( n ) = 0 . (52) G. Dispersion relation
It should be emphasized that the analytical fundamen-tal solution S (23) is obtained without recourse to anydispersion relation, i.e., for any vector Q (8). Let us ex-plain this on the example of the exact Volkov solutionfor an electron in the field of a plane wave. There ex-ist different representations of this solution [9, 15]. Wepresent below another one which is more straightforwardand convenient for our purposes. In this particular case,there is only one wave of six waves in Eq. (2), namely,the wave with amplitude A = a e + ib e . Substi-tuting Ψ( x ) = Ψ( ζ ) e iκ e Q · x with ζ = N · x = x − ct inEq. (1) gives an ordinary differential equation which hasthe exact solution Ψ( x ) = E v ( x ) a , where E v ( x ) = e i Φ( x ) J ( ζ ) (53)is the evolution operator (the fundamental solution ofthis equation), J = J ( ζ ) is the 4 × J = J, trJ = 2) defined by D s ( J ) = { / , , − iJ , iJ , J , , , , , − / , J , J , , − iJ , , } , (54) J = [2( q − q )] − , J = J ( q − a cos k ζ ) ,J = J ( q + 2 b sin k ζ ) . (55)At any given ζ , the bispinor Ψ( ζ ) belongs to the two-dimensional subspace defined by J ( ζ ). The phase func-tion Φ consists of two parts which are linear in x andperiodic in ζ , respectively, as followsΦ = κ e Q ′ · x + J Ω [4 b q (1 − cos k ζ ) − a q sin k ζ + ( a − b ) sin 2 k ζ (cid:3) , (56) Q ′ = Q − Q + I A Q · N N , I A = 2( a + b ) . (57)It is easy to verify that Q ′ satisfies the dispersion re-lation 1 + Q ′ + I A = 0 at any Q . In other words, thefundamental solution has the build-in dispersion relation.Similarly, in optics of plane-stratified complex mediums,fundamental solutions (exponential evolution operators)define both wave vectors and polarizations of eigenwavesin an anisotropic or bianisotropic slab [7]. It is conve-nient to preset Q satisfying the dispersion relation, then Q ′ ≡ Q and the parameter ξ V = q − p q speci-fies the deviation from the free-space dispersion relation1 + q = q as follows ξ V = p q + I A − p q (58)for any given q .In the general problem under study, the dispersionrelation manifests itself in the spectral distribution ofFourier components c ( n ) (5). In numerical calculationsfor a finite model with a localized Fourier spectrum, when g d ( n ) ≤ g max for all n in Eq. (34), it has a pictorial pre-sentation in the form of spectral curves of approximatesolutions R j = R j ( ξ ), where ξ = q − p q = ~ ωm e c − s (cid:18) ~ k m e c (cid:19) , (59)and R j = p λ j is given by Eq. (50) at a = c j . Thegeneralized eigenvalues λ j and eigenvectors c j are de-fined by the equation U D c j = λ j U E c j with the Hermi-tian 4 × U E and U D , and the quartic equationdet( U D − λU E ) = 0 has real coefficients and positiveroots λ j indexed below in increasing order of magnitude.The minimum { ξ , R = R ( ξ ) } of the spectral curve R = R ( ξ ) specifies the most accurate approximate so-lution. It follows from the results of computer simula-tions [13] that ξ converges to a positive limit and R ( ξ )tends to zero with increasing g max . In the limit, Ψ (39)converges to a family of exact solutions with the disper-sion relation ~ ωm e c = ξ + s (cid:18) ~ k m e c (cid:19) . (60) III. TWO COUNTERPROPAGATING WAVESA. Dispersion relation
In this section we apply the presented technique to findthe ground state of the Dirac electron with, by definition,the quasi-momentum p = ~ k = m e c q = 0, in the field oftwo counterpropagating circularly polarized waves withthe same amplitude A = A = A m ( e + i e ) / √ . (61)The other four amplitudes in Eq. (2) are equal to zeroand hence I A = 4 A m . In this case, most of the structuralparameters in Eq. (33) are vanishing, only N ( m, s ) for s ∈ { ( − , , , − , ( − , , , , (1 , , , − , (1 , , , } a b1.986 1.988 1.990 1.992 ´ - Ξ R FIG. 1. Spectral curve of approximate solutions R = R ( ξ )and its models R = R ap ( ξ ) (dashed curves) for the spec-tral lines (a) ξ = ξ oa = 0 . R = 1 . × − , β = 1 . × , δξ ( R av ) = 1 . × − and (b) ξ = ξ ob = 0 . R = 1 . × − , β = 2 . × , δξ ( R av ) = 9 . × − at Ω = 0 . R av = √ I A = 0 . g max = 4. and N ( s ) for s ∈ { ( − , , , , (2 , , , } are not zero,therefore Ψ (39) contains only Fourier components with n = ( n , , , n ), where | n | ≤ g max , whereas | n | = 0 ,
1. Figure 1 shows the corresponding spectralcurve of approximate solutions, which reveals that theground state has two different frequency levels specifiedby minimums of spectral lines a and b . Their bottomparts (see dash curves in Fig. 1) can be closely approxi-mated as follows R ap ( ξ ) = q R + β ( ξ − ξ ) . (62)The half-width δξ ( R av ) of the solution line, i.e., the half-width of ξ domain, where R ≤ R ≤ R av , can be esti-mated from Eq. (62) as δξ ( R av ) = 1 β q R av − R . (63)This half-width is a rapidly decreasing function of g max .The condition R ≪ ξ values, whereas R , , ≫ R and theydo not satisfy the similar condition at any value of ξ , for example, {R j , j = 2 , , } = { . , . , . } and { . , . , . } at ξ = ξ a and ξ = ξ b , respec-tively. Thus the amplitude subspaces in Eq. (39) forboth of levels are one-dimensional, they are specified bythe generalized eigenvectors a a = c ( ξ a ) = a + and a b = c ( ξ b ) = a − or, in other words, by the projec-tion matrices P a = P + and P b = P − , where a ± = 1 √ ± , P ± = 12 ± ± . (64)It is convenient to describe the closely spaced levelsof the Dirac electron, i.e., the normalized frequencies ξ a AB - - - - W Ξ m FIG. 2. Plot of ξ m against log Ω at (A) I A = 0 . I A = 0 . A B - - - - W ´ - DΞ FIG. 3. Plot of ∆ ξ against log Ω at (A) I A = 0 . I A = 0 . and ξ b , in terms of the mean value ξ m = ( ξ a + ξ b )and the difference of levels ∆ ξ = ξ b − ξ a . The de-pendence of ξ m and ∆ ξ on the normalized frequency Ωof the electromagnetic lattice is shown in Fiq. 2 andFiq. 3, respectively. The dots represent calculations,while the curves are obtained by the linear interpola-tion, for the range of Ω from 1/1280 to 1/10, i.e., forthe X-ray standing waves with the wavelength λ from0.024 nm to 3.1 nm. In the central band of this range∆ ξ has the maximum at λ = λ A = 0 . A = 0 . λ = λ B = 0 . B = 0 . ξ m weakly de-pends on Ω, see Fiq. 2. The dependence ξ m on I A canbe approximated by ξ V = √ I A − I A , see Fiq. 4. Figure 5 illustrates the dependence of∆ ξ on I A in this range. The smaller is Ω or the greateris I A or both, the greater is g max which provides reason-ably small values of R , because the Fourier spectrumof the wave function expands with such variations of Ωand I A . For I A = 0 . g max ≥ . R ≤ . × − , whereas g max ≥
16 at Ω = 1 / R ≤ . × − . For Ω = Ω A , g max ≥ I A = 3 . × − provides R ≤ . × − , whereas g max ≥
10 at I A = 0 . R ≤ . × − .Eq. (64) is valid for the whole domain of study. BA - - - - - - I A Ξ m (cid:144) Ξ V FIG. 4. Ratio ξ m /ξ V against log I A at (A) Ω = Ω A , and(B) Ω = Ω B . B A - - - - - - I A - - - - - - DΞ FIG. 5. Plot of log ∆ ξ against log I A at (A) Ω = Ω A , and(B) Ω = Ω B . B. Doublet structure of the ground state
Let us now compare the ground state wave functionsspecified by { ξ a , a a } and { ξ b , a b } in terms of the cor-responding mean values of Hamiltonian H = c X k =1 α k p k + m e c α , (65)operators of kinetic momentum p k = − i ~ ∂∂x k − ec A k , (66)probability current density j k = cα k , and spin S k = ~ Σ k , k = 1 , ,
3. Both of these functions provide mean values: h j k i = 0, h p k i = 0, k = 1 , ,
3, and h S i = h S i = 0.The mean values h S i a = ~ h Σ i a and h S i b = ~ h Σ i b for the doublet lines a and b , respectively, are equal inmagnitude but opposite in sign. They depend on I A andcan be approximated as follows h Σ i a = −h Σ i b ≈ − I A + 32 I A . (67)The normalized energy levels E a and E b of the doubletare different and depend on both Ω and I A as shown inFig. 6, Fig. 7, and Fig. 8, where E = h H i / ( m e c ). BA DC - - - - W FIG. 6. Normalized energy E against log Ω at (A) I A =0 . ξ = ξ a , (B) I A = 0 . ξ = ξ b , (C) I A = 0 . ξ = ξ a , (D) I A = 0 . ξ = ξ b . - - - - W- - - - - D E FIG. 7. Logarithm of ∆ E = E b − E a against log Ω at I A =0 . I A = 0 . C. Spin precession
The whole family of the ground state wave functionsis defined the evolution operator (see Eqs. (40) and (41)) E v ( x ) ≡ E v ( X , X ) = (68) e iϕ b X n ∈ S da S a ( n ) P a e i ( ϕ ′ n + ϕ ab ) + X n ∈ S db S b ( n ) P b e iϕ ′ n ! , where ϕ b = − π (1 + ξ b ) X / Ω, ϕ ab = 2 π ∆ ξX / Ω, and ϕ ′ n = 2 π ( n X − n X ), S da and S db are the solution AB CD - - - - - - I A - - - - H E a - L , log D E FIG. 8. Plot of log ( E a −
1) against log I A (solid curves)at (A) Ω = Ω A , and (C) Ω = Ω B . Plot of log ∆ E againstlog I A (dashed curves) at (B) Ω = Ω A , and (D) Ω = Ω B . domains of the doublet lines a and b , respectively. In thiscase, the set S da contains only points n = ( n , , , n ) ∈L with n = − ,
0, whereas the set S db contains onlypoints n with n = 0 ,
1. The Fourier amplitudes a = a ( n )and b = b ( n ) [see Eq. (51)] have the following symmetryproperties: a ∗ = ( − n a, Σ a = ( − n a, n ∈ S da , (69) b ∗ = ( − n b, Σ b = − ( − n b, n ∈ S db . (70)Each member Ψ = E v ( x ) a of this family is specified bythe amplitude a which can be written without loss ofgenerality as a = a a e iδ cos α + a b sin α, (71)where α ∈ [0 , π/
2] and δ ∈ [0 , π ].The matrix function E v ( X , X ) is periodic in X . Itis not periodic in X , but ∆ ξ/ Ω ≪
1, so that variationsof ϕ ab at any unit interval of the X axis are negligi-bly small, for example, in calculation of norms and meanvalues using Eqs. (42), (43) and (44). In this approxi-mation, for the normalized energy E and the mean value h S i = ~ h Σ i of the spin operator one can readily obtainthe relations: E = h H i / ( m e c ) = E a + ∆ E sin α − u cos α , (72) h Σ i = e h Σ i a (cos 2 α − u cos α ) + e ρ ( v −
1) sin 2 α − u cos α , (73)where e ρ = e cos ϕ + e sin ϕ, ϕ = δ + 2 π ∆ ξX / Ω = δ + 2 πν pr t , δ specifies the initial precession phase, ν pr =∆ ξm e c /h is the precession frequency, and u = 1 − u a u b , u a = X n ∈ S da | a ( n ) | , u b = X n ∈ S db | b ( n ) | , (74) h Σ i a = 1 u a X n ∈ S da ( − n | a ( n ) | , (75) v − u b X n ∈ S da T S db a † ( n )Σ b ( n )= 2 u b X n ∈ S da T S db [ a ( n ) b ( n ) + a ( n ) b ( n )] . (76)The mean values h j k i and h p k i ( k = 1 , ,
3) of the prob-ability current density operators j k and the kinetic mo-mentum operators p k are equal to zero for any groundstate wave function Ψ. The mean value h Σ i a dependon I A and can be approximated by Eq. (67), parameters u and v depend on Ω and I A as shown in Fig. 9 andFig 10, respectively.Since u ≪ v ≪
1, the ground state wavefunctions specified by 0 ≤ α ≤ π describe various spin AC BD - - - - W u , v FIG. 9. Plot of u against log Ω (solid curves) at (A) I A =0 . I A = 0 . v against log Ω (dashedcurves) at (B) I A = 0 . I A = 0 . A BCD - - - - - - I A - - - - - u , log v FIG. 10. Plot of log u against log I A (solid curves) at(A) Ω = Ω A , and (C) Ω = Ω B . Plot log v against log I A (dashed curves) at (B) Ω = Ω A , and (D) Ω = Ω B . states of the Dirac electron, including the spin precessionwith the frequency ν pr at 0 < α < π . The correspond-ing normalized energy levels E fill the band from E a to E b = E a + ∆ E , see Fig. 6, Fig. 7, and Fig. 8. Thefrequency ν pr is defined by ∆ ξ = ξ b − ξ a , see Fig. 3and Fig. 5, in particular, ν pr = 3 . × Hz atΩ = Ω A , I A = 0 . ν pr = 1 . × Hz atΩ = Ω B , I A = 0 . A = A (61) by A = A = A m ( e − i e ) / √ h Σ i a and h Σ i b and reverses the precession direction, i.e., e ρ inEq. (73) takes the form e ρ = e cos ϕ − e sin ϕ . In thecase of counterpropagating waves with the same circu-lar polarization ( A = A ∗ = A m ( e ± i e ) / √
2) or thesame linear polarization ( A = A = A m e ), the spinprecession is absent, because ∆ ξ ≡ IV. CONCLUSION
The fundamental solution of the Dirac equation foran electron in the electromagnetic field with four–dimensional periodicity is obtained. The projection op-erator S ′ (36) defines the exact fundamental solution ofthe finite subsystem (35) which expands with each new step of the recurrent process. The relations, presentedabove and in [11, 12], form the complete set which issufficient for the fractal expansion of this subsystem toa finite model of ESTC of any desired size. A criterionfor evaluating accuracy of the approximate solutions, ob-tained by the use of such model, is suggested. It plays aleading role in search for the best approximate solutionsin the framework of the selected model. The presentedtechniques are illustrated by analyzing the ground stateof the Dirac electron in the field of counterpropagatingplane waves. It is shown that in the electromagnetic lat-tice, composed by the left and right circularly polarizedwaves, the ground state is described by the family of wavefunctions with zero mean values of the probability cur-rent density operators and kinetic momentum operators,but with different energy levels and various spin states,including the spin precession. Appendix1. Dirac basis for the linear space of × matrices Let us enumerate 16 Dirac matrices, forming a basis forthe linear space of 4 × × × = = U, Γ = − − = Σ , Γ = = Σ , Γ = − i i − i i = Σ , Γ = − − = γ = α , Γ = − − = τ , = −
10 0 − = τ , Γ = − i i i − i = τ , Γ = − − − − = γ , Γ = −
11 0 0 00 − = α Γ = = α , Γ = − i i − i i = α , Γ = i
00 0 0 i − i − i = τ , Γ = − i
00 0 0 ii − i = γ , Γ = − i − i i i = γ , Γ = −
10 0 1 00 1 0 0 − = γ . Commonly used notation to the right of each matrix isgiven for convenience. At the presented numeration or-der, the structural information on each matrix Γ ν is en-closed in its number ν , i.e., one can reconstruct Γ ν from ν , and the multiplication rule for Γ λ Γ µ can be written asa function of λ and µ [11].Any 4 × A can be written A = X ν =0 A ν Γ ν , where A ν = tr ( A Γ ν ), and tr A = 4 A . To single outthe specific basis used in this expansion, the set of co-efficients { A ν } is called in this article the Dirac set ofmatrix A , briefly, D-set of A , and it is denoted D s ( A ).This approach is of particular assistance in solving thesystem of Eqs. (6). It is best suited to the structure ofits matrix coefficients, accelerates numerical calculationsand reduces data files. It should be emphasized that allmajor matrix operations (summation, multiplication, in-version, etc.) can be performed directly with D-sets, i.e.,without matrix form retrieval [11].
2. Projection operator of a system of homogeneouslinear equations
Let V and V ∗ be a linear space (finite or infinite di-mensional) and its dual. At given ω ∈ V ∗ , the linearhomogeneous equation in x ∈ Vh ω, x i = 0 (A.1)can be transformed to the equivalent equation α x = 0 , (A.2)where α = ω † ⊗ ω h ω, ω † i (A.3)is the Hermitian projection operator (dyad) with thetrace tr α = 1, and ω † ∈ V . Let U be the unit oper-ator, i.e., U x = x for any x ∈ V and ωU = ω for any ω ∈ V ∗ . The Hermitian projection operator S = U − α is the fundamental solution of (A.2), i.e., for any given x ∈ V , x = S x is a partial solution of (A.1) and (A.2).Let now α and β be Hermitian projection operators( α † = α = α, β † = β = β ) in V . Providing the series A = α + β + + ∞ X k =1 (cid:2) ( αβ ) k α − ( αβ ) k + ( βα ) k β − ( βα ) k (cid:3) (A.4)is convergent, it defines the Hermitian projection opera-tor with the following properties A † = A = A, αA = Aα = α,βA = Aβ = β, tr A = tr α + tr β. (A.5)Hence, the system of equations in x ∈ V α x = 0 , β x = 0 (A.6)1reduces to one equation A x = 0 and has the fundamentalsolution S = U − A . The operator A will be designatedthe projection operator of the system (A.6). The trace tr α of the projection operator α specifies the dimensionof the image α ( V ) of V under the mapping α . It is sig-nificant that the relations (A.4) and (A.5) are valid forany values of integers tr α and tr β . This enables us toextend this approach to systems with any (finite or infi-nite) number of homogeneous linear equations. To thisend, we transform (A.4) to the following expression [16] A = ( α − αβα ) − ( U − β ) + ( β − βαβ ) − ( U − α ) , (A.7)where ( α − αβα ) − is the pseudoinverse operator with thefollowing properties( α − αβα ) − ( α − αβα ) = ( α − αβα )( α − αβα ) − = α,α ( α − αβα ) − = ( α − αβα ) − α = ( α − αβα ) − , + ∞ X k =1 ( αβ ) k = ( α − αβα ) − β. (A.8)The similar relations for ( β − βαβ ) − can be obtained from(A.8) by the replacement α ↔ β . Numerical implemen-tation of the pseudoinversion reduces to the inversion of( tr α ) × ( tr α ) matrix for ( α − αβα ) − and ( tr β ) × ( tr β )matrix for ( β − βαβ ) − . In [16], we have proposed a technique based on theuse of (A.7) to find the fundamental solution of the sys-tem (15). Here, we present the advanced version of thistechnique based on a fractal expansion of the system ofequations taking into account and on the use of A (A.4)expressed as A = α + δ, δ = ( β − α ) γ ( β − α ) , (A.9)where γ = β + + ∞ X k =1 ( βαβ ) k = ( β − βαβ ) − , (A.10) α, β, δ , and A are projection operators, α, β, γ, δ , and A are Hermitian operators interrelated as βγ = γβ = γ, βαγ = γαβ = γ − β,αδ = δα = 0 , βδ = β − βα, δβ = β − αβ,αA = Aα = α, βA = Aβ = β, δA = Aδ = δ. In the frame of this approach, calculation of all pseudoin-verse operators in use reduces to the inversion of 4 × [1] F. Wilczek, Phys. Rev. Lett. , 160401 (2012).[2] A. Shapere and F. Wilczek, Phys. Rev. Lett. , 160402(2012).[3] T. Li, Z.-X. Gong, Z.-Q. Yin, H. Quan, X. Yin, P. Zhang,L.-M. Duan, and X. Zhang, Phys. Rev. Lett. , 163001(2012).[4] L. P. Horwitz and E. Engelberg, Phys. Lett. A , 40(2009).[5] H. Hu and J. Huang, Phys. Rev. A , 062105 (2015).[6] R. Donnelly and R. W. Ziolkowski, Proc. R. Soc. LondonA , 673 (1992); , 541 (1993); A. M. Shaarawi,R. W. Ziolkowski, and I. M. Besieris, J. Math. Phys. ,5565 (1995).[7] G. N. Borzdov, Sov. Phys. Cryst. , 313 (1990); , 317(1990); , 322 (1990); Opt. Commun , 159 (1992);J. Math. Phys. , 3162 (1993); , 6328 (1997); in Electromagnetic Fields in Unconventional Materials andStructures , edited by O. N. Singh and A. Lakhtakia (Wi- ley, New York, 2000) Chap. 3, pp. 83–124.[8] F. I. Fedorov, Sov. Phys. – JETP , 339 (1959).[9] F. I. Fedorov, Lorentz Group (Nauka, Moscow, 1979).[10] A. A. Bogush and L. G. Moroz,
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