New solutions of the Dirac, Maxwell and Weyl equations from the fractional Fourier transform
aa r X i v : . [ qu a n t - ph ] J a n New solutions of the Dirac, Maxwell and Weyl equationsfrom the fractional Fourier transform
Iwo Bialynicki-Birula ∗ Center for Theoretical Physics, Polish Academy of SciencesAleja Lotnik´ow 32/46, 02-668 Warsaw, Poland
New solutions of relativistic wave equations are obtained in a unified manner from generatingfunctions of spinorial variables. The choice of generating functions as Gaussians leads to representa-tions in the form of generalized fractional Fourier transforms. Wave functions satisfying the Dirac,Maxwell, and Weyl equations are constructed by simple differentiations with respect to spinorialarguments. In the simplest case, one obtains Maxwell and Dirac hopfion solutions.
PACS numbers: 03.65.Pm, 03.65.Db
I. INTRODUCTION
The aim of this work is to describe a new universaltool in the study of the solutions of Weyl, Maxwell andDirac equations. This tool is used here to derive a largefamily of solutions of these equations. The essential roleis played by the generating function Υ( η, η ∗ | x ) which ina simple form encodes the information about the solu-tions of Weyl and Maxwell equations and by the functionΥ( η, η ∗ , ζ, ζ ∗ | x ) which encodes the information about thesolutions of the Dirac equation. Solutions of relativis-tic wave equations are obtained by the differentiation ofgenerating functions with respect to their spinorial argu-ments η and ζ .The generating functions have the form of generalizedfractional Fourier transforms. Fractional Fourier trans-form (sometimes also called the Fresnel transform) wasintroduced long time ago by Condon [1]. However, itsnumerous applications in optics, in signal processing, inimage compression, in computed tomography, and otherfields began after the work of Namias [2]. The gener-alization of the fractional Fourier transform to spinorialvariables extends its range of applications. It enablesone to obtain analytic localized solutions of relativisticwave equations with ever increasing degree of complex-ity. The term localized has here the following meaning.The probability to find the particle outside the sphereof a fixed radius tends to zero when the scaling param-eter a tends to zero. Of course, the localization can beachieved only around t = 0 because all solutions of wavefunctions describing particles evolving in free space un-dergo dispersion. As a result, the mean square radius isa quadratic function of time [3, 4], h r i t = A + Bt . (1)Therefore, in the remote past and future the extensionof every wave packet is arbitrarily large.The generating functions introduced here depend oncomplex parameters: the components of the relativistic ∗ [email protected] spinors and their complex conjugates. These generatingfunctions have interesting properties on their own be-cause they form a representations of the full 15 parame-ter conformal group for Weyl and Maxwell equations andof the inhomogeneous Poincar´e group for the Dirac equa-tion. Our method of generating solutions is particularlyuseful for the construction of knotted solutions with intri-cate topological properties. As special cases one obtainsthe solutions describing the Dirac [4] and Maxwell [5–7]hopfions.The main mathematical ingredients in this work aretwo-component spinors and four component bispinors ap-pearing as arguments of the functions that generate so-lutions of Weyl, Maxwell and Dirac equations. Thesesolutions are obtained as the derivatives of the generat-ing functions with respect to their spinorial arguments.Of course, spinors and bispinors also appear as the wavefunctions in spacetime. Our notation is a slight modifi-cation of that used in [8–10] and it is summarized in theAppendix A.The formalism based on spinors is particularly wellsuited to describe the solutions of relativistic wave equa-tions because it is often directly connected with topo-logical properties of these solutions. These propertiesin many cases [6, 7] involve Hopf fibration [11]. Thissubject in the case of the electromagnetic field has beenthoroughly studied (see a recent review [12]). The spino-rial representation introduced in the present work gives aunified framework to describe also the solutions of otherrelativistic wave equations. II. MASSLESS PARTICLES: WEYL ANDMAXWELL EQUATIONS
Let us consider an arbitrary complex function Υ( η, η ∗ )of a two-component spinor η A and its complex conjugate η ˙ A . The space of these functions becomes a representa-tion of the conformal group after the introduction of the15 group generators. The generators of translations P µ ,rotations and special Lorentz transformations M µν , spe-cial conformal transformations K µ , and dilation D are( ~ = 1 , c = 1): P µ = − ∂ ˙ A g ˙ ABµ ∂ B , (2a) M µν = i (cid:16) η A S BµνA ∂ B − η ˙ A S ˙ Bµν ˙ A ∂ ˙ B (cid:17) , (2b) K µ = η A g µA ˙ B η ˙ B , (2c) D = 12 i (cid:16) η A ∂ A + η ˙ A ∂ ˙ A (cid:17) , (2d)where ∂ A = ∂/∂η A and ∂ ˙ A = ∂/∂η ˙ A . This representa-tion has a close connection with twistors [13–15] but thisline of investigation will not be pursued here.The generators of translations are of special signifi- cance because with their help one can construct from thefunction Υ( η, η ∗ ) a complete field defined at all space-time points x µ . To this end, Υ( η, η ∗ ) is represented as afour-dimensional Fourier integral over the spinorial vari-ables,Υ( η, η ∗ ) = Z d κd κ ∗ exp h − i ( κ A η A + η ˙ A κ ˙ A ) i ˜Υ( κ, κ ∗ ) . (3)The integration variables are the real and imaginaryparts of both components of the spinor κ . The trans-lation operators acting on the spinorial integral (3) pro-duce an integral in the form of a multidimensional frac-tional Fourier transform since the exponent has both thequadratic and the linear part in the integration variables,Υ( η, η ∗ | x ) = exp( − iP µ x µ )Υ( η, η ∗ ) = Z d κd κ ∗ exp h − iκ ˙ A g ˙ ABµ κ B x µ i exp h − i ( κ A η A + η ˙ A κ ˙ A ) i ˜Υ( κ, κ ∗ ) . (4)This function satisfies four Schr¨odinger-like equations, i∂ µ Υ( η, η ∗ | x ) = P µ Υ( η, η ∗ | x ) . (5)For every choice of ˜Υ( κ, κ ∗ ) which guarantees the con-vergence of the integral and for all values of the spinorialparameters η and η ∗ , the generating function Υ( η, η ∗ | x )satisfies the d’Alembert equation,( ∂ t − ∆)Υ( η, η ∗ | x ) = 0 , (6)because the derivative ∂ µ produces κ ˙ A g ˙ ABµ κ B under theintegral and this is a lightlike vector. Negative energy so-lutions are obtained by the translation exp( iP µ x µ ). Thegeneral solution is a superposition of positive and nega-tive energy contributions but choosing only one sign at atime simplifies the formulas.The generation of the solutions of Maxwell equationsfrom a solution of the d’Alembert equation has beenknown already to Whittaker [16]. This method was for-mulated in the spinorial framework by Penrose [17]. Inboth these constructions, the Maxwell field is built fromsecond derivatives with respect to spacetime variables .Therefore, these methods are closely related to Hertz po-tentials. In contrast, the generation of the solutions ofWeyl and Maxwell equations from our spinorial gener-ating function Υ( η, η ∗ | x ) is quite different because it in-volves the derivatives with respect to the auxiliary spino-rial argument η A . Derivatives with respect to the com-ponents of spinors also appear but in an entirely differentrole in the solutions of the massless wave equations ex-pressed in terms of Penrose transforms [18].The first derivative is a solution of the Weyl equation, φ C ( x ) = i∂ C Υ( η, η ∗ | x ) , g µ ˙ EC ∂ µ φ C ( x ) = 0 . (7) The second derivative is a solution of the Maxwell equa-tions, φ CD ( x ) = − ∂ C ∂ D Υ( η, η ∗ | x ) , g µ ˙ EC ∂ µ φ CD ( x ) = 0 . (8)Both equations follow from the algebraic relation: g µ ˙ EC κ ˙ A g ˙ ABµ κ B κ C = 2 κ ˙ A ǫ ˙ A ˙ E ǫ BC κ B κ C = 0 , (9)applied to the derivatives of the spinorial transform (4).The wave equation (8) is equivalent to Maxwell equationsupon the following identification of the components of φ CD ( x ) with the components of the Riemann-Silberstein(RS) vector [3]: F x = φ − φ , F y = − i ( φ + φ ) , F z = 2 φ = 2 φ . (10)It follows from these relations that positive/negative fre-quency solutions of Maxwell equations F ± ( x ) have thefollowing spinorial representation: F ± ( r , t ) = Z d κd κ ∗ κ − κ − iκ − iκ κ κ × exp h ∓ iκ ˙ A g ˙ ABµ κ B x µ i ˜Υ ± ( κ, κ ∗ ) . (11)General solutions are sums of positive and negative fre-quency solutions. It is shown in the next Section thatthe spinorial representation through the Hopf fibrationis directly related to the Fourier representation that iscommonly used in physical applications. The alternativerepresentation based on the Penrose transform [18] doesnot have this property. III. SPINORIAL REPRESENTATION ANDTHE HOPF FIBRATION
The Hopf fibration [11] is a decomposition of the three-dimensional sphere into linked circles: the fibers. Everyfiber corresponds to one point on the two-dimensionalsphere. In the Hopf construction the spheres areparametrized in terms of the Cartesian coordinates sub-jected to the condition that the radius of the sphereis fixed. The mapping of the points in the three-dimensional sphere ξ i onto the points in the two-dimensional sphere k i was defined by Hopf as follows: k x = 2( ξ ξ + ξ ξ ) , k y = 2( ξ ξ − ξ ξ ) ,k z = ξ + ξ − ξ − ξ . (12)In order to introduce the physical interpretation, thesymbols in these formulas are different than those usedby Hopf. The relations invented by Hopf preserve thelength. The 3D sphere of unit radius is mapped ontothe 2D sphere of unit radius since k x + k y + k z =( ξ + ξ + ξ + ξ ) . The connection of the Hopf fibra-tion with spinors is revealed when the parameters ξ i areidentified with the real and imaginary parts of the spinorcomponents κ = ξ + iξ , κ = ξ + iξ .The physical content of the Hopf formula (12) is bestdescribed with the help of Pauli matrices and it has theform of the relation between spinors κ and the lightlikewave vectors: k µ = κ ˙ A g µ ˙ AB κ B . The space componentsof the wave vector are the Hopf parameters k x , k y , k z .The fibers are formed by those spinors that differ onlyby an overall phase factor e iϕ . This phase factor isnot uniquely defined. One may choose, for example,the phase of the upper spinor component. In this case ϕ = arctan( ξ /ξ ). With this choice, the relations (12)can be inverted, ξ = √ k z + k cos ϕ √ , ξ = √ k z + k sin ϕ √ ,ξ = k x cos ϕ − k y sin ϕ √ √ k + k z , ξ = k y cos ϕ + k x sin ϕ √ √ k + k z , (13)where k = q k x + k y + k z . The relation between thespinorial representation (11) and the Fourier representa-tion of the electromagnetic field represented by the RSvector [3, 19, 20], F ( r , t ) = Z d k (2 π ) / − k x k z + ikk y − k y k z − ikk x k x + k y × (cid:2) f + ( k ) e i k · r − iωt + f ∗− ( k ) e − i k · r + iωt (cid:3) , (14)is obtained by the change of variables. The integrationwith respect to two components of the spinor in (11) isequivalent to the integration over the three componentsof the wave vector and an additional integration with re-spect to the phase ϕ . The Jacobian of the transformation ξ i → { k x , k y , k z , ϕ } is equal to 1 / k and the integral (11)expressed in terms of new variables coincides with (14)provided we make the following identification: f + ( k ) = (cid:16) π (cid:17) / Z π dϕ e iϕ ˜Υ + ( k , ϕ ) k ( k x − ik y ) , (15a) f ∗− ( k ) = (cid:16) π (cid:17) / Z π dϕ e iϕ ˜Υ − ( k , ϕ ) k ( k x − ik y ) . (15b)Even though the spinorial representation and the Fourierrepresentation are equivalent, the spinorial representa-tion is much easier to use in the derivation of varioushopfion-like solutions. IV. THE HOPFION FAMILY OF SOLUTIONSOF WEYL AND MAXWELL EQUATIONS
A simple choice of ˜Υ( κ, κ ∗ ) that leads to the analyticsolution is a Gaussian,˜Υ( κ, κ ∗ ) = exp (cid:16) − κ ˙ A g ˙ ABµ κ B a µ (cid:17) , (16)where a µ is any complex vector. Without loss of general-ity, one may assume that a µ is a real vector because theimaginary part of a µ can be eliminated by a shift of theorigin of coordinate system. From now on it is assumedthat the coordinate system is chosen in such a way that a µ = { a, , , } although in some formulas the full vec-tor a µ will appear. In order to guarantee the convergenceof the integrals, it is assumed that a >
0. In this casethe integration can be easily done and after dropping theirrelevant factor π , one obtains,Υ( η, η ∗ | x ) = D ( x ) exp (cid:16) − iD ( x ) η A g µA ˙ B ( x µ − ia µ ) η ˙ B (cid:17) , (17)where D ( x ) = (( a + it ) + x + y + z ) − . This sim-ple calculation shows the advantage of using the spino-rial representation. The evaluation of the correspondingintegrals in the Fourier representation would have beenmuch more complicated. This method of generating solu-tions is applicable also to wave equations for higher spins.In particular, the gravitational waves in linearized grav-ity are described by fourth derivatives of the generatingfunction.The solutions of Weyl and Maxwell equations obtainedfrom the function (17) by differentiation, according to theformulas (7) and (8), have the form: φ C ( x ) = D ( x ) exp (cid:0) − η A ψ A ( x ) (cid:1) ψ C ( x ) , (18) φ CD ( x ) = D ( x ) exp (cid:0) − η A ψ A ( x ) (cid:1) ψ C ( x ) ψ D ( x ) , (19)where ψ A ( x ) = D ( x ) g µA ˙ B η ˙ B ( x µ − ia µ ) . (20)These formulas contain arbitrary parameters η A and η ˙ A .By differentiating (18) and (19) (or integrating with anyfunction of these parameters) with respect to η A and/or η ˙ A one still obtains solutions of the wave equations.One may check by a direct calculation that not onlyfor the exponential functions appearing in (18) and (19)but for all functions h ( ψ ( x )) the spinors φ C ( x ) = D ( x ) h ( ψ ( x )) ψ C ( x ) , (21) φ CD ( x ) = D ( x ) h ( ψ ( x )) ψ C ( x ) ψ D ( x ) (22)are solutions of the wave equations (7) and (8). Oneobtains in this way a large class of solutions of Weyl andMaxwell equations controlled by an arbitrary complexfunction of the two components of ψ A ( x ).Incidentally, one obtains a realization of deBroglie idea of fusion [21] by choosing h ( ψ ( x )) = h ( ψ ( x )) h ( ψ ( x )). Indeed, it looks like “photonsare made of two neutrinos” because the photon wavefunction (apart from the factor D ( x )) is a product ofneutrino wave functions, φ AB = D ( x ) − φ A φ B .The simplest hopfion solutions of the Weyl equationare obtained from (21) when h ( ψ ) = − i and by choosingeither η ˙ A = { , } or η ˙ A = { , } . φ A ( x ) = [ D ( x )] (cid:20) t + x + (cid:21) , φ A ( x ) = [ D ( x )] (cid:20) x − t − (cid:21) , (23)where t ± = t ± z − ia and x ± = x ± iy .Owing to the appearance of the product of spinors in(22), the electromagnetic field given by this formula isnull; both field invariants vanish, i.e., E − B = 0 and E · B = 0. Null fields play a special role in electromag-netism. They possess intriguing topological properties[7]. The simplest solutions are obtained from the for-mula (22) by choosing the same spinors ψ A as for theWeyl equation. The two closely related RS vectors con-structed from these spinors according to the formulas(10) and (22) are: F H = [ D ( x )] t − x i ( t + x ) − t + x + , (24a) F H = [ D ( x )] − t − + x − i ( t − + x − ) − x − t − . (24b)The first formula coincides with Eq. (23) of [20]. Theelectric and magnetic field vectors (i.e., the real and imag-inary parts of F H ) describe the simplest knotted solu-tions of Maxwell equations: the hopfion. It was discov-ered by Synge [5] who interpreted it as “an electromag-netic model of a material particle”. Its intricate topolog-ical properties were discovered by Ra˜nada [6] who foundthe connection with Hopf fibration. The hopfion solu-tion can be obtained in many different ways, even froma simple Fourier integral [20]. However, a method ofchoice to obtain also other solutions with even more intri-cate topological properties is the Bateman construction [7, 22]. Bateman discovered that if two complex func-tions of spacetime variables α ( x, y, z, t ) and β ( x, y, z, t )obey the condition; ∇ α × ∇ β = i ( ∂ t α ∇ β − ∂ t β ∇ α ) , (25)then the vector F B = ∇ α × ∇ β is a (null) solution ofMaxwell equations.The Bateman construction is mentioned here becausethere is a direct connection between the spinorial methodand this construction. Namely, the two components ofthe spinor (20) can be used as α and β in the Batemanconstruction because they obey the condition (25). TheRS vector obtained from the Bateman construction dif-fers only by the factor − i from the one obtained from(10) and (24a). All solutions with intricate topologicalproperties, analyzed in [7], can be obtained from the for-mula (22) by choosing the function h ( ψ A ) in the form h ( ψ A ) = ( ψ ( x )) p ( ψ ( x )) q , where p and q are relativelyprime integers.The solutions of Weyl and Maxwell equations (21) and(22) have one common feature. In both cases we find forall choices of the function h the same lightlike four-vector l µ = ψ ˙ A g µ ˙ AB ψ B characterizing the solution. The current j µ for the solutions of the Weyl equation and the energy-momentum tensor T µν for the solutions of the Maxwellequations are built from l µ , j µ = | D ( x ) h ( ψ ( x )) | l µ ( x ) , (26a) T µν = | D ( x ) h ( ψ ( x )) | l µ ( x ) l ν ( x ) . (26b)The properties of the vector l µ ( x ) underscore the con-nection with Hopf fibration. Namely, integral lines ofvelocity v = l /l form linked circles, as shown in Fig. 1,which is a characteristic feature of Hopf fibration. V. MASSIVE PARTICLES: DIRAC EQUATION
The standard Dirac equation,( iγ µ ∂ µ − m ) Ψ( x ) = 0 , (27) FIG. 1. Linked circles: the trademark of the Hopf fibration.The lines of velocity plotted here are obtained by solving theset of differential equations d r ( λ ) /dλ = v ( r ( λ )) for differentinitial conditions. is converted in the Weyl representation [23] of γ matrices, γ µ = (cid:20) g µA ˙ B g µ ˙ AB (cid:21) , Ψ( x ) = (cid:20) φ A ( x ) χ ˙ A ( x ) (cid:21) , (28) into the following set of two equations for two-componentspinors: ig µ ˙ AB ∂ µ φ B ( x ) = mχ ˙ A ( x ) , ig µA ˙ B ∂ µ χ ˙ B ( x ) = mφ A ( x ) . (29)Due to the presence of two spinors in these equations,the generating function Υ should have two spinorial argu-ments. The appearance of mass calls for some modifica-tions of the spinorial formalism. The following spinorialintegral, patterned after the integral (4) for the solutionsof d’Alembert equation, produces positive energy solu-tions of the Klein-Gordon equation,Υ( η, η ∗ , ζ, ζ ∗ | x ) = Z d κd κ ∗ d λd λ ∗ exp h − i ( κ A η A + η ˙ A κ ˙ A + λ A ζ A + ζ ˙ A λ ˙ A ) i exp h − i (cid:16) κ ˙ A g ˙ ABµ κ B + λ A g µ ˙ B λ ˙ B (cid:17) x µ i × δ (cid:16) κ A λ A + κ ˙ A λ ˙ A − m (cid:17) δ (cid:16) i ( κ ˙ A λ ˙ A − κ A λ A ) (cid:17) ˜Υ( κ, κ ∗ , λ, λ ∗ ) , (30)because the action of the d’Alembertian produces underthe integral sign the following expression: κ ˙ A g µ ˙ AC κ C λ A g µA ˙ B λ ˙ B = 4 κ A λ A κ ˙ A λ ˙ A . (31)The presence of the δ -functions enables one to replacethis expression by m , resulting in the Klein-Gordonequation, (cid:0) ∂ t − ∆ + m (cid:1) Υ( η, η ∗ , ζ, ζ ∗ | x ) = 0 . (32)Negative energy solutions are obtained by reversing thesign of x µ . In full analogy with the massless case, thesolutions of the Dirac equation are obtained from (30)by differentiation with respect to spinorial parameters η C and ζ ˙ C . The Dirac bispinor Ψ( x ) has the form:Ψ( x ) = (cid:20) i∂ A Υ( η, η ∗ , ζ, ζ ∗ | x ) i ð ˙ A Υ( η, η ∗ , ζ, ζ ∗ | x ) (cid:21) , (33)where ∂ A = ∂/∂η A and ð ˙ A = ∂/∂ζ ˙ A . In order to obtain explicit formulas for the solutions,we must choose the function ˜Υ( κ, κ ∗ , λ, λ ∗ ) in such a waythat the integrations can be performed. VI. THE HOPFION FAMILY OF SOLUTIONSOF THE DIRAC EQUATION
The existence of similar representations of the solu-tions of Maxwell and Dirac equations enables one to de-fine a map between these solutions. Namely, by choosing˜Υ( κ, κ ∗ , λ, λ ∗ ) as a product of functions appearing in (4),˜Υ( κ, κ ∗ , λ, λ ∗ ) = ˜Υ ( κ, κ ∗ ) ˜Υ ( λ, λ ∗ ) , (34)one establishes a direct relation between solutions ofMaxwell and Dirac equations. To every pair of solutionsof Maxwell equations there corresponds a solution of theDirac equation. In particular, it is tempting to choose˜Υ( κ, κ ∗ ) in the Gaussian form because this choice corre-sponds to the Maxwell hopfion.˜Υ ( κ, κ ∗ ) = exp h − κ ˙ A g ˙ ABµ κ B a µ i , (35a)˜Υ ( λ, λ ∗ ) = exp h − λ A g µA ˙ B λ ˙ B a µ i . (35b)In order to do the calculations, the integral represen-tations of the δ -functions are introduced into the formula(30),Υ( η, η ∗ , ζ, ζ ∗ | x ) = Z dudv Z d κd κ ∗ d λd λ ∗ exp h − i ( κ A η A + η ˙ A κ ˙ A + λ A ζ A + ζ ˙ A λ ˙ A ) i × exp h − (cid:16) κ ˙ A g ˙ ABµ κ B + λ A g µA ˙ B λ ˙ B (cid:17) ( a µ + ix µ ) i exp h − iu ( κ A λ A + λ ˙ A κ ˙ A − m ) i exp h − v ( κ A λ A − λ ˙ A κ ˙ A ) i , (36) FIG. 2. The knotted lines of the current j = ¯Ψ γ Ψ for the solutions of the Dirac equation Ψ , Ψ , Ψ and Ψ listed in theAppendix B. The initial conditions are the same in all cases: x = 1 , y = 0 . , z = 0 and a = 1. The distances are measuredin electron Compton wave length and the size of the box is 3.FIG. 3. Strong dependence on the initial conditions. The lines of current are plotted for the same solution Ψ as in Fig. 2 butfor different initial conditions for one of the coordinates, x = { . , . , . , . } . where ˜Υ( κ, κ ∗ , λ, λ ∗ ) was replaced by the product of func-tions (34) and the irrelevant factor (2 π ) was omitted.The integral with respect to the spinorial variables again has the form of the fractional Fourier transform. Thisintegral can be evaluated and we are left with an integralwith respect to u and v ,Υ( η, η ∗ , ζ, ζ ∗ | x ) = Z du dv e imu [ D ( u, v | x )] exp h − D ( u, v | x ) (cid:16) ( η A g µA ˙ B η ˙ B + ζ ˙ A g ˙ ABµ ζ B )( a µ + ix µ ) (cid:17)i × exp h − D ( u, v | x ) (cid:16) − iu ( η A ζ A + ζ ˙ A η ˙ A ) − v ( η A ζ A − ζ ˙ A η ˙ A ) (cid:17)i , (37)where D ( u, v | x ) = (cid:2) u + v + ( a µ + ix µ )( a µ + ix µ ) (cid:3) − .This integral cannot be evaluated for arbitrary valuesof the spinorial parameters η and ζ . However, the ex-pansion in powers of these parameters leads to integrals that can be explicitly evaluated. The simplest examplesare the Dirac hopfions obtained by a different method in[4]. The corresponding bispinors Ψ H are obtained fromthe formula (37) by evaluating second derivatives of Υat the origin. The following four solutions of the Diracequation are: ð B (cid:20) ∂ A Υ ð ˙ A Υ (cid:21) = Z du dv e imu [ D ( u, v | x )] (cid:20) − ( v + iu ) δ BA g ˙ ABµ ( a µ + ix µ ) (cid:21) = πm (cid:20) δ BA K g ˙ ABµ ( a µ + ix µ ) K (cid:21) , (38a) ∂ ˙ B (cid:20) ∂ A Υ ð ˙ A Υ (cid:21) = Z du dv e imu [ D ( u, v | x )] " g µA ˙ B ( a µ + ix µ )( v − iu ) δ ˙ A ˙ B = πm " g µA ˙ B ( a µ + ix µ ) K δ ˙ A ˙ B K , (38b)where K n are expressed in terms of the Mac-donald functions K n = K n ( ms ) /s n , and s = p ( a + it ) + x + y + z . The arguments of Υ and K n are omitted.Choosing both values of the index ˙ B and both values ofthe index B we obtain four Dirac hopfions correspondingto the formulas (7) and (8) of [4] taken for the lowest val-ues of the index l . Higher order derivatives evaluated atthe origin all give analytic solutions of the Dirac equationexpressed in terms of Macdonald functions of increasingorder. The integral lines of the current shown in Fig. 2and Fig. 3 represent the solutions of the following threecoupled differential equations: d r ( λ ) dλ = j ( r ( λ ) , t ) . (39)These figures were generated from the formulas in theAppendix B in the simplest case, when t = 0. The linesof current depend strongly on the initial conditions, asshown in Fig. 3. All analytical calculations and plotswere done with Mathematica [24]. VII. CONCLUSIONS
Maxwell, Weyl, and Dirac equations play a fundamen-tal role in relativistic quantum mechanics. The prac-tically unlimited collection of new analytic solutions ofthese equations described here may help to understandbetter the intricate quantum properties of relativisticparticles. The spinorial representation described here isparticularly well suited in the analysis of solutions withintricate topological properties connected with Hopf fi-bration. The representation of the wave functions asderivatives with respect to spinorial variables makes theirtransformation properties transparent. Of course, itwould also be possible to generate new solutions by eval-uating consecutive derivatives of some simple solutionwith respect to spacetime variables. However, this leadsto highly complicated expressions. Already the secondderivative of the simplest solution (B2) produces a for-mula that is difficult to analyze because it is much morecomplicated than the expression (B5) obtained by eval-uating the seventh derivative with respect to spinorialvariables.
VIII. ACKNOWLEDGMENTS
I am thankful to Zofia Bialynicka-Birula for her veryhelpful criticism.
Appendix A
The two components of the spinor are labeled [9] with 0and 1. Spinors with upper and lower index are connectedby the spinorial metric tensor ǫ , φ A = ǫ AB φ B , φ A = ψ B ǫ BA , ǫ AB = ǫ AB = (cid:20) − (cid:21) . (A1)Repeated indices imply summation over two values ofthe index. Under rotations and Lorentz transformationsspinors are transformed by the unimodular matrices S BA ′ φ A = S BA φ B , ′ φ A = − S AB φ B . (A2)The minus sign in the second formula is a consequenceof the antisymmetry of the metric tensor and it impliesthe invariance of the scalar product ′ φ A ′ ψ A = φ A ψ A .There are two inequivalent two-dimensional repre-sentations of the Lorentz group: the spinors φ A andcomplex-conjugate spinors φ ˙ A . Dotted indices signifycomplex conjugation, φ ˙ A = ( φ A ) ∗ . The dotted spinorsare transformed with the use of complex conjugate ma-trices, ′ φ ˙ A = S ˙ B ˙ A φ ˙ B , S ˙ B ˙ A = (cid:0) S BA (cid:1) ∗ . (A3)Spinors with several indices transform as products ofspinors for example, ′ φ ˙ AB = S ˙ C ˙ A S DB φ ˙ BD . (A4)An important role is played by spin-tensors—objectswith tensorial and spinorial indices. There are four ofthem: g µ ˙ AB , g µA ˙ B , S µν BA and S µν ˙ B ˙ A . They may all beexpressed in terms of the Pauli matrices σ i and the 2 × I , g AB = I = g A ˙ B , g i ˙ AB = σ i = − g i A ˙ B , (A5a) S µν BA = 12 (cid:16) g µA ˙ C g ν ˙ CB − g νA ˙ C g µ ˙ CB (cid:17) . (A5b)Spin-tensors are invariant under the simultaneousLorentz transformations of vector and spinor indices. Appendix B
Selected solutions of the Dirac equation are obtainedby evaluating the following derivatives of the generatingfunction:Ψ = ð (cid:20) ∂ A Υ ð ˙ A Υ (cid:21) , Ψ = ð ð ˙0 ð (cid:20) ∂ A Υ ð ˙ A Υ (cid:21) , Ψ = ∂ ∂ ˙0 ∂ ∂ ˙1 ð ˙1 (cid:20) ∂ A Υ ð ˙ A Υ (cid:21) , Ψ = ∂ ˙0 ∂ ∂ ˙1 ð ð ð ˙0 ð ˙1 (cid:20) ∂ A Υ ð ˙ A Υ (cid:21) . (B1)All derivatives are to be taken at the origin. There was no special reason to choose these particular derivatives. Allnonvanishing derivatives give distinct solutions.Ψ = m π { K , , it − K , − ix + K } . (B2)Ψ = m π {− ix − K , it − K , x − t − K , (( a + it ) − x − y + z ) K } . (B3)Ψ = m π { x − t − K , − (( a + it ) − x − y + z ) K , ix − K /m − ix − ( x + y ) K , it + K /m + t − t K } . (B4)Ψ = m π { x + t − (( a + it ) − x − y + z ) K , − x t − K , ix + t − K , ix − t − K } . (B5) REFERENCES [1] E. U. Condon, Immersion of Fourier transforms in a con-tinuous group of functional transformations,
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