deriving electromagnetic fields from the spinor solution of the massless dirac equation

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009 2907

Deriving Electromagnetic Fields From the SpinorSolution of the Massless Dirac Equation

Tullio Rozzi, Life Fellow, IEEE, Davide Mencarelli, and Luca Pierantoni

Abstract—In this paper, we present the explicit derivation of theelectromagnetic (EM) field solution of Maxwell equations startingfrom the Dirac equation, used in describing the so-called spinorwave function of quantum particles. In particular, we show that ifthe four-component vector (spinor) solution of the Dirac equationfor zero mass is identified with the four-potential of the EM field,then, under the Lorentz gauge, fields derived from that potentialsatisfy Maxwell equations. Vice versa, the four-potential could beused to express a spinor solution, provided that the latter satisfiesthe Lorenz gauge. Some examples in the frequency domain clarifythis connection. A crucial choice is needed: the EM potential has tobe assumed as a linear combination of positive- and negative solu-tions of the spinor. This work may help to clarify the controversialrelation between Maxwell and Dirac equations, while presentingan original way to derive the EM fields, leading, perhaps, to novelconcepts in EM simulations.

Index Terms—Dirac equation, electromagnetic (EM) theory,four-vector spinor, Maxwell tensor form.

I. INTRODUCTION

I N the framework of the emerging interest in nanoelectro-magnetics and nanoelectronics, scientists are considering

the equations of charge transport holding at the nanoscale,namely, the Schrödinger and Dirac equations. The discoveryand experimental characterization of new materials, such ascarbon nanotubes, nanoribbons, and graphene, that featurelow-dimensional systems, seem to open new possibilities[1]–[8]. In fact, the above nanostructures, owing to their typicalabsence of crystal defects, behave as ideal transport channels.These in turn require understanding and accurate simulation oftheir behavior under RF and optical signals.

Beside the technological aspects, intriguing theoretical chal-lenges assume now novel significance. Over the last years, inparticular, renewed efforts have been made in order to establisha clear correspondence between Dirac and Maxwell equations[9]–[16]. Classically, Maxwell equations describe the evolutionof the electromagnetic (EM) fields generated by charge and cur-rent sources, and are relativistically covariant. On the other hand,theDiracequationgoverns thecoherent transportofquantumpar-ticles by the space–time evolution of a four-component vector,usually referred to as “spinor.” The state of a system is thus ex-pressed by a state vector, the spinor, in a linear space: usually,the two pairs of components of the spinor describe the two pos-siblespinstatesofaquantumparticle.TheDirac formulationcon-stitutes the relativistic and vector counterpart of the celebrated

Manuscript received February 27, 2009; revised July 23, 2009. First publishedNovember 13, 2009; current version published December 09, 2009.

The authors are with the Università Politecnica delle Marche, Ancona60131, Italy (e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TMTT.2009.2034225

scalar Schrödinger equation and can be formally derived by theKlein–Gordon equation, which stands, in quantum mechanics,for the familiarD’Alembertwaveequation.Foranexhaustivedis-cussion, see, for example, [7]. The aim of unifying the above twosystems of equations in a simple and elegant formulation, able toexplainphenomenaofsuchapparentlydifferentnature,promotedinvestigation of their possible analogies and connections. For ex-ample, in [9] and [10], a careful study of the mathematical formof the Maxwell and Dirac equations pointed out that EM fieldsand spinors are joined by many symmetry properties, invariantforms, and conservation laws. Some attempts to find other analo-gies have been made using the Clifford numbers, which can beseen just as a particular way to express the Dirac matrices (re-ported in Section II), to which the Clifford algebra is isomorphic[12], [13]. In [13], for instance, the far field radiated from a simpledipoleantennaiscalculatedfromanintegralequationbasedontheClifford algebras. Other authors [17] point out that the solutionsof the Maxwell system can be associated to the solutions of theDirac equation through some nonlinear relation. The Majoranarepresentation of the EM fields, reported, for example, in [18], isalsoworthmentioning. In this formulation, however, a significantlack of generality arises from the assumption that current–sourceterms have the form of gradients.

The above references suggest a close correlation between thezero-mass Dirac and Maxwell equations. In this context, a majorrole could be played by the EM simulators, which are usuallyemployed to solve very complex and challenging EM field prob-lems. In the literature, a few examples of application of EMsolvers to quantum problems have already been reported [4], butthese have never been actually applied in connection with theDirac equation. Although, in this work, we are not concernedwith the use of EM solvers for solving the spinor space–timedependence, it will be shown how the spinor solutions of theDirac equation can be directly used to express the solutions ofthe Maxwell equations. An equally important open question is tobe able to proceed the opposite way, i.e., to pass from Maxwellto Dirac solutions. Eventually, our analysis may be the first stepin the direction of a profitable use of EM solvers for a rigorousdescription of quantum problems, as well as the development ofEM simulators based on the Dirac equation.

II. THEORY

A. Tensor Form of Maxwell’s Equations

In order to expedite further analysis, we will briefly recall theordinary tensor form of the Maxwell equations. It is convenientto introduce indexed coordinates , , definedby [19]

(1)

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2908 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 12, DECEMBER 2009

which are the so-called contravariant components of the positionvector, whereas the covariant components read, by definition

(2)

In a similar fashion, it is possible to define the componentsand of the contravariant and covariant four-vector potentials,respectively, in terms of the usual three-vector potential andthe scalar potential . The contravariant potential is

(3)

whereas the covariant form is

(4)

Now, it is quite usual to express the partial derivatives withrespect to the space–time components defined in (1) and (2),respectively

and (5)

In the following, in order to recover a compact and elegantanalytical derivation of the EM fields, we will make use of theEM tensor, defined as

(6)

where the subscripts (superscript) after commas indicate partialderivatives in terms of the contravariant (covariant) components.The six components of the EM field are easily related to the sixnonzero elements of the antisymmetric EM tensor, which reads

(7)

In the absence of sources, it can be shown that the fourMaxwell equations are equivalent to just two equations for[19]

(8)

In (8), and throughout the following, the standard Einsteinrepeated index summation rule is used. The components of thefour-potential are assumed to satisfy the Lorentz gauge condi-tion that can be written as

(9)

Under the Lorentz condition, the four-potential satisfies therelation

(10)

which is the usual wave equation.

B. Dirac Equation

The Dirac system of equations is briefly described in this sec-tion. The Dirac matrices are defined as

(11)

where is the 2-D unit and zero matrices, and are the Paulimatrices given by

(12)

The fundamental algebraic structure of the Dirac matricesthat is the same as the Clifford algebras is described by the re-lation

(13)

where

(14)

is the metric tensor. In the zero-mass case [7], the four-spinorsatisfies the equation

(15)

which is the zero-mass Dirac equation in the standard represen-tation. The representation is not unique, for, the same relationis satisfied, together with (13), if is replaced by ,where is an arbitrary nonsingular matrix [7]. Writing out (15)explicitly, the Dirac equation (15) reads

(16)

By simple matrix product, one can see that the repeated ap-plication of the operator on the left-hand side of (16) yields

(17)

which is usually referred to as the Klein–Gordon equation. Asa matter of fact, the Dirac equation has been historically de-rived trying to find an expression for the “square root” of theKlein–Gordon equation [7]. Note that the latter equation can befactorized as is done for a difference of perfect square, using theDirac matrices: in part B of Appendix I, we will show how thepositive and negative spinor solutions can be used to express thefour-potential.

ROZZI et al.: DERIVING ELECTROMAGNETIC FIELDS FROM THE SPINOR SOLUTION OF THE MASSLESS DIRAC EQUATION 2909

III. ELECTROMAGNETIC FIELDS FROM DIRAC SPINORS

For the sake of readability, we report in Appendix I the proofof the basic result, mentioned earlier, highlighting the close rela-tion between the solutions of the Maxwell and Dirac equations.

A. Solution of the Dirac Equation

In this section, we will establish the method for the derivationof the EM field solutions, in the source-free case, starting from(15). This, in fact, constitutes the main conceptual result of thiswork, i.e., associating an EM field to a spinor, solution of theDirac equation. To this aim, we let, in (15), the four-potential

play the role of the four-spinor , and subsequently, we willderive the resulting electric and magnetic field components inthe frequency domain

(18)

Since our task is to extract the Maxwell–Dirac connection, wewill make use of instead of , thus restoring the conventionalEM symbol for the frequency. For any fixed frequency ,(18) will be solved in its original form, explicitly shown in (16),as well as with the reversed sign of the time derivative: in fact,in connection with what was discussed at the end of Section II,the general EM solution is formed by a combination of positiveand negative spinor solutions that we indicate with and ,respectively.

First, we assume a “plane wave” form for the space–time de-pendence of the EM potential that may be conveniently rewrittenas

(19)

where

(20)

and and satisfy

(21)

In the above, we made use of position (19). In order to obtainnontrivial solutions of (21), the following relation, expressingthe wave-number conservation, should hold

(22)

Restricting the analysis, for the moment, to a 2-D domain,and thus setting, for instance, , a general solution of (21)is found

(23)

Forming the sum , one obtains for the four-vector po-tential

(24)

The components in (24) are not independent, since theLorentz condition (9) still remains to be applied: the centralpoint to be stressed is that in order to retain the most generalsolution the Lorentz gauge (9) has to be applied to the combi-nation of positive and negative solutions, and not separately toeach of them. This yields

(25)Using (6), (7), and (24), and omitting the common exponen-

tial term of (19), the following expressions for the electric fieldcomponents are derived:

(26)and the associated magnetic field components are given by

(27)


B. Example I: Uniform Plane Wave

As a first example, the derivation of the plane-wave solutionis shown. By setting in (26) and (27) and applying theconditions (25), we obtain the usual expression of EM planewaves propagating along the -direction

(28)

Imposing in (28)

or

allows to distinguish between the - and -polarized electricfield solutions, respectively. The case of , with propa-gation along the -direction, may be recovered in a similar way.

C. Example II: 2-D Homogeneous Waveguides

We will now illustrate the more significant case of wave-guide modes. As an example, let us search for transverse elec-tric (TE) modes propagating along the -direction and confinedby metallic parallel plates normal to the -direction. Beside theLorentz constraint (9), the following conditions are to be im-posed on the field components (26) and (27):

yielding

yielding

yielding

(29)

As a matter of fact, two of the four conditions (25) and (29)are independent. Thus, the electric and magnetic fields can berewritten in terms of and only, and theirexpressions are given by

(30)

As in the case of the plane wave, the above expressions canbe simplified by an appropriate choice of the arbitrary constants.We may set, for example, , so that (30) becomes

(31)

Imposing in (30) would produce the same final re-sult, apart from a multiplication constant. In order to recover thecorrect standing-wave behavior of TE fields confined in the -di-rection, we just need to subtract expression (31) from the sameexpression where is replaced by . Then, imposing tovanish on the parallel metallic plates yields the usual transversewave-number quantization. For the sake of brevity, these simplefinal passages are omitted. Nonetheless, it is easy to show thatthe ratio between the transverse field components equals, as ex-pected, the characteristic impedance of TE modes

(32)

The analysis of transverse magnetic (TM) modes may be car-ried out by analogy. The general case of 3-D waveguides can bedeveloped through the same steps as above, and will be reportedin Appendix II.

D. Example III: Dielectric Waveguides

The analysis of the discrete and continuous spectrum of inho-mogeneous waveguides, such as, for example, dielectric slabs ormultilayers, can be carried out starting from the results of Sec-tion III-C. As an example, let us consider the even TE modes of asymmetric dielectric slab propagating in the -direction, shownin Fig. 1, where a magnetic-wall boundary condition holds at

.Indicating by and (with ) the phase

velocities in the cladding region and in the underlying region, respectively, (21) can be rewritten separately for each region

(33)The same equation with appropriate sign, as in (21), can bewritten for . In the above, is the transverse wave numbersin region and is the longitudinal wave number: theyare related by the wave-number conservation holding in bothregions. In order to recover TE modes from the four-spinor so-lution of (21), (24) is rewritten separately for cladding and core


Fig. 1. Dielectric slab.

Fig. 2. Usual metallic rectangular waveguide.

By following the same steps of the previous section, we derivethe counterpart of (30), showing the components of the EM fieldin the region

(34)where the -dependence is shown explicitly and an exponentialdecay is imposed: . Also, we derive the standing wavesolution in region as indicated at the end of the previoussection

(35)

The above expressions can be simplified by an appropriatechoice of the arbitrary constants. Setting, for instance, in(34), and imposing the field continuity at the dielectric interface,one obtains the usual dispersion relation of TE modes and theexpression of the field components as a function of the four-potential, a solution of (33).

IV. CONCLUSION

In conclusion, we consider the relation between the Maxwelland Dirac equations. First, we look at the connection betweena four-spinor and a four-potential, instead of the EM field: thischoice appears to be a more natural bridge between the abovetwo equations. The central point is that the spinor has to be as-sumed as a combination of positive and negative solutions ofthe Dirac equation, satisfying the Lorentz condition. Second,we report some practical examples of how EM fields can be re-constructed, through the four-potential, from the knowledge ofthe spinor. In particular, we demonstrate the case of EM fieldspropagating in standard waveguides, such as dielectric slab andrectangular waveguide.

In the light of the present results, we may consider proceedingthe other way around, that is, from Maxwell to Dirac and ap-plying EM simulators to the Dirac equation. In fact, the spinoris constrained by the Lorentz condition, and this seems to bethe actual limit of the above correspondence. This aspect andthe practical application of the EM simulators warrant furtherinvestigation.

APPENDIX IBASIC PROOFS

A. From Dirac to Maxwell

In this section, the EM tensor is shown to satisfy Maxwellequations, provided that the four-potential, from which isdefined in (6), satisfies the Dirac equation and the Lorentz gaugecondition. Hence, the hypotheses are

(A1)

and the thesis is given by (8). Staring from hypotheses (A1), wecan write

Then, each component of satisfies the above equation

By subtracting in the last equation the following quantity:

which vanishes due to the second hypothesis of (A1), we obtainthe final result

(A2)

which is one of the two tensor Maxwell’s equations (8).The second one follows directly from (6); in fact

(A3)

B. Reconstruction of the Four-Potential From the Dirac Spinor

Starting from Maxwell equation, we can follow back the samepassages of the previous section, obtaining

The four-potential expressing the EM tensor is a solution of thewave equation, equivalent to a repeated application of the Diracequation, and not to the Dirac equation only. However, for thecase of zero mass, the latter can be based on three anticom-muting matrices

(A4)


In fact, defining the operator in terms of (A4)

(A5)

which satisfies the equation

(A6)

We obtain the following factorization of the wave equation:

(A7)

The general form for is given by linear combination of

and

APPENDIX IIRECTANGULAR WAVEGUIDE

In order to carry out the analysis of a uniform rectangularwaveguide, shown in Fig. 2, we follow the same line of rea-soning of Section III-C, having restored the -dependence in(19).

The, the conservation of the wave number (22) becomes

(A8)

The general solution of (1) is now given by

(A9)

Forming the sum yields

(A10)The Lorentz condition (9) leads to the following expression:

(A11)

Again, we make use of (6), (7), and (A10) to derive the electricand magnetic field components shown in (A12) and (A13) at thebottom of the page. In order to focus on a particular example,let us search for a TE modes propagating along the -direction,which requires that

yielding: (A14)

By using, in (A12) and (A13), the condition (A14), and im-posing the Lorentz constrain (A11) that can be slightly simpli-fied as

(A15)

one obtains the general expression for TE modes. In (A16),shown at the bottom of the page, we report the electric fieldcomponent. The magnetic fields are not shown because, as forthe 2-D case, the ratio between the transverse field componentssimply returns the expected characteristic impedance

(A17)

(A12)

(A13)

(A16)


In order to recover the standing wave dependence of the field inthe transverse direction, we need to sum (A16) to the expressionitself, where the following substitutions have been made:

Finally, an appropriate choice of the arbitrary constants of (A16)is needed to obtain the correct spatial dependence of the fieldswith respect to the - and -directions

(A18)

Hence, employing (A18) in (A16) yields the electric fields interms of the negative and positive solutions of the Dirac equation

(A19)Of course, imposing and to vanish, respectively, at

and yields the usual transverse wave-number quanti-zation. The magnetic field may be recovered by using (A17).

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Tullio Rozzi (M’66–SM’74–F’90–LF’07) receivedthe Laurea degree in physics, University of Pisa,Pisa, Italy, in 1965 and the Ph.D. degree in electronicengineering from Leeds University, Leeds, U.K.

From 1968 to 1978, he was a Research Scientist atthe Philips Research Laboratories, The Netherlands.In 1978, he obtained the Chair of Electrical Engi-neering at the University of Liverpool, U.K., and in1981, the Head of the Electronics Group at the Uni-versity of Bath, U.K. Currently, he is the Head of theDepartment of Electromagnetics, Università Politec-

nica delle Marche, Ancona, Italy.Dr. Rozzi was awarded the Microwave Prize by the IEEE Microwave Theory

and Techniques Society in 1975.

Davide Mencarelli received the Laurea degree inelectronics engineering from the Università Politec-nica delle Marche, Ancona, Italy, in 2002.

Currently, he is a Postdoctoral Researcher at Uni-versità Politecnica delle Marche. His research inter-ests are in analysis and modeling of integrated opticaldevices. He is currently involved with analytical/nu-merical techniques for the modeling of nanodevices.

Mr. Mencarelli has been a member of the ItalianNational Institute for the Physics of Matter (INFM)since 2007.

Luca Pierantoni received the Laurea degree in elec-tronics engineering and the Ph.D. degree in computa-tional electromagnetics from the Department of Elec-tronics and Automatics, University of Ancona, An-cona, Italy, in 1988 and 1993, respectively.

From 1989 to 1995, he was with the Department ofElectronics and Automatics, University of Ancona,as a Research Fellow. From 1996 to 1998, he workedat the Institute of High-Frequency Engineering, Tech-nical University of Munich, Germany. In 1999, hejoined the Department of Electromagnetics, Univer-

sità Politecnica delle Marche, Ancona, Italy, as an Assistant Professor. His cur-rent research interests are in the multiphysics modeling of nanodevices.

deriving electromagnetic fields from the spinor solution of the massless dirac equation

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