Covariant Majorana Formulation of Electrodynamics

Foundations of Physics, Vol . 28, No. 2, 1998

Covariant Majorana Formulation of Electrodynamics

Salvatore Esposito1

Received April 29, 1997; revised July 19, 1997

We construct an explicit covariant Majorana formulation of Maxwell electro-magnetism which does not make use of vector 4-potential. This allows us to writea `̀ Dirac’’ equation for the photon containing all the known properties of it. Inparticular, the spin and (intrinsic) boost matrices are derived and the helicityproperties of the photon are studied.

1. INTRODUCTION

Recently, many different experiments have revealed new electromagneticphenomena that, although they can be well described by Maxwell electro-magnetism, sound quite unusual with respect to our traditional view ofelectrodynamics.

An example is the observation of tunneling photons with groupvelocities also greater than c; this effect does not violate Einstein’s causalityprinciple and can be described in terms of Maxwell equations (see Refs. 1and 2 and references therein).

Another long discussed example is the experimental measurement ofthe Evans± Vigier B3 field in nonlinear optical experiments ( such as in theinverse Faraday effect, etc.) , ( 3) the B3 field being the phase-free spin field ofMaxwell electromagnetism. In this respect, Evans has interestingly noted(4 )

that the cyclic equations for the B3 field( 3 ) are particularly evident in theMajorana formulation of electrodynamics.2

231

0015-9018/98/0200-0231$15.00/0 Ñ 1998 Plenum Publishing Corporation

1 Dipartimento di Scienze Fisiche, UniversitaÁ di Napoli, Mostra d’Oltremare Pad. 19, 80125Napoli, Italy, and INFN, Sezione di Napoli, Mostra d’Oltremare Pad. 20, I-80125 Napoli,Italy.

2 Although more indirectly, also in the theoretical description of tunneling photons theemployment of the writing of Maxwell equations as a Dirac-like equation for the photonseems quite useful; this can be seen confronting Ref. 2 and Ref. 5. The author is grateful toA. Ranfagni for bringing to his attention these two references.

Majorana’s original idea ( 6) was that if the Maxwell theory of electro-magnetism has to be viewed as the wave mechanics of the photon, then itmust be possible to write Maxwell equations as a Dirac-like equation fora probability quantum wave w , this wave function being expressable bymeans of the physical E , B fields. This can, indeed, be realized introducingthe quantity

y= E 2 iB ( 1)

since y* . y= E2+ B

2 is directly proportional to the probability densityfunction for a photon.3 In terms of y , the Maxwell equations in vacuumthen read ( 7)

Ñ . y= 0 (2)

¶ y

¶ t= i Ñ 3 y ( 3)

By making use of the correspondence principle

E ® i¶¶ t

( 4)

p ® 2 i Ñ ( 5)

these equations can effectively be cast in a Dirac-like form

(E 2 a . p ) y= 0 (6)

with the transversality condition

p . y = 0 (7)

where the 3 3 3 Hermitian matrices (ai ) lm = ieilm

0 0 0 0 0 2 i 0 i 0

a1= 0 0 0 i 1 , a2= 0 0 0 0 1 , a3= 0 2 i 0 0 1 ( 8)

0 2 i 0 i 0 0 0 0 0

232 Esposito

3 If we have a beam of n equal photons each of them with energy e (given by the Planckrelation), since 1

2 (E2+ B

2) is the energy density of the electromagnetic field, then(1/ne) 1

2 (E 2+ B 2 ) dS dt gives the probability that each photon has to be detected in the areadS in the time dt. The generalization to photons of different energies ( i.e., of differentfrequencies) is obtained with the aid of the superposition principle.

satisfying

[ai , aj] = 2 ieijk ak ( 9)

have been introduced.Note that the probabilistic interpretation is indeed possible given the

`̀ continuity equation’’ (Poynting theorem)

¶ r

¶ t+ Ñ . j= 0 (10)

where

r= 12y* . y j= 2 1

2 w * a w ( 11)

are respectively the energy and momentum density of the electromagneticfield.

The fascination of Majorana formulation of electrodynamics liesmainly in the fact that it deals with directly observable quantities, such asthe electric and the magnetic fields, without the occurrence of electro-magnetic potentials. On the other hand, we know that Maxwell equationsare Lorentz invariant, but this is not manifest in the cited formulation.Actually, the lack of a covariant Majorana formulation is due to the factthat in the known covariant formulation of electrodynamics the introduc-tion of the vector 4-potential Am is of fundamental importance.4

The main goal of this paper is to show how to obtain a genuinecovariant Majorana formulation, without the use of the 4-potential. Inthe next section we develop the appropriate formalism to this end, whilein Sec. 3 the covariant Majorana± Maxwell equations are derived andtheir properties are pointed out. In Sec. 4 we construct the MajoranaHamiltonian and deal with spin and ( intrinsic) boost matrices, and withthe photon helicity. Finally, the conclusions follow in Sec. 5.

2. COVARIANT KINEMATICS OF THE ELECTROMAGNETIC

FIELD

Covariant electromagnetism is described by the field strength tensorsFmn and its dual FÄ mn= 1

2emnabFab. The basic property of these tensors is their

233Covariant Majorana Formulation of Electrodynamics

4 The present Lagrangian formulation of electromagnetism makes use of the antisymmetricfield strength tensor Fmn which is both a Lorentz and a gauge-invariant quantity. However,in this formulation the connection of Fmn to the 4-potential rather than to the physical fieldsis very important.

antisymmetric behavior under the exchange of their indices m and n; thiscan be expressed saying that for any 4-vector um the relations

um unFmn= 0, um unFÄ mn= 0 (12)

must hold. Without loss of generality, we can assume the 4-vector um to beunitary, so that

um u m= 1 (13)

Let us now introduce the following `̀auxiliary fields’’ (um -dependent)

E m= unF mn, B m= unFÄ mn ( 14)

in terms of which the property ( 12) reads

um E m= 0, um B m= 0 (15)

i.e., the 4-vectors Em , Bm are both orthogonal to um . We are now able toexpress the tensors Fmn and FÄ mn by means of the auxiliary fields in thefollowing way:

Fmn= um En 2 unEm+ emnabBau b ( 16)

FÄ mn= um Bn 2 unBm 2 emnabE au b ( 17)

We stress that the fields Em , Bm depend on um in such a manner that Fmn

and FÄ mn are independent of um itself. We see, in fact, that the auxiliary fieldsEm , Bm are nothing than covariant linear um -combinations of the physicalelectric and magnetic field

Ei= F 0i, B

i= FÄ 0i ( 18)

Namely, using (13), ( 15) , ( 16) , ( 17), we obtain the following relationsbetween the auxiliary fields and the physical fields:

E0= u . E , E= u0 E + u 3 B ( 19)

B0= u . B , B= u0 B 2 u 3 E ( 20)

For a given um , the fields Em , Bm are univocally determined by the electricand magnetic fields. Note that in the `̀ isotropic case,’’ in which

um= (1, 0 ) ( 21)

we obtain Em= (0, E ) , B m= (0, B ).

234 Esposito

It is interesting to observe the analogy between the relations (19), ( 20)and those for the transformations of the electric and magnetic field from aninertial reference frame to another moving with relative velocity v with|v |<< 1: ( 8)

E ¢ , E + v 3 B , B ¢ , B 2 v 3 E ( 22)

This suggestive analogy allows one to interpret the 3-vector part of theauxiliary fields as the Lorentz-transformed electric and magnetic fields andu as a velocity.

3. COVARIANT MAJORANA FORM OF MAXWELL EQUATIONS

Armed with the formalism developed in the previous section, we arenow able to write Maxwell equations according to Majorana’s point ofview. Here we are interested in illustrating the method, so that we willconsider only noninteracting photons; the Maxwell equations in vacuumare then

¶ m Fmn= 0, ¶ m FÄ mn= 0 (23)

Before continuing, let us note that Eq. (23) imply

¶ m Em= 0, ¶ m Bm= 0 (24)

Now, assuming um independent of spacetime coordinates, when we sub-stitute (16), ( 17) in ( 23) we have

u m ¶ m E n+ emnab ¶ m Baub= 0 (25)

u m ¶ m Bn 2 emnab ¶ m Eaub= 0 (26)

Introducing the complex field

w m= Em 2 iBm ( 27)

satisfying

¶ m wm= 0 (28)

we rewrite Eqs. (23) as

¶ m(u m w n+ iemnabw aub )= 0 (29)


Then, defining the four 4 3 4 matrices

( c m ) ab= u mg a

b + iemabc uc ( 30)

the equations of motion of the electromagnetic field in vacuum acquire theform (using (4), (5) )

( c m ) ab pm w

b = 0 (31)

This is just the `̀Dirac equation’’ for free photons.

3.1. Properties of c -Matrices

The explicit form of the four c-matrices is the following:

c0= 0u0

000

0u0

2 iu3

iu2

0iu3

u0

2 iu1

0

2 iu2

iu1

u0 1 ( 32)

c1= 0u1

0

2 iu3

iu2

0u1

00

2 iu3

0u1

2 iu1

iu2

0iu0

u1 1 ( 33)

c2= 0u2

iu3

0

2 iu1

iu3

u2

0iu0

00u2

0

2 iu1

2 iu0

0u2 1 ( 34)

c3= 0u3

2 iu2

iu1

0

2 iu2

u3

2 iu0

0

iu1

iu0

u3

0

000u3 1 ( 35)

These are Hermitian matrices

( ( c m ) ba)*= (c m ) a

b ( 36)

236 Esposito

satisfying the relations

um cm= 1 (37)

Tr cm= 4u m ( 38)

The algebra of c-matrices is defined by

[cm, c

n]=i4

emnab(Tr ca) cb ( 39)

Explicitly, in terms of um , their commutator can be expressed as follows:

[cm, c

n] ab = ua(u mgn

b 2 ungmb )+ ( gamun 2 ganu m ) ub 2 ( g magn

b 2 gnagmb )

3.2. The ``Isotropic Case’’ u l = (1, 0)

In the special case in which um= (1, 0 ) the c-matrices reduce to

c0= 0

1000

0100

0010

0001 1 , c

1= 00000

0000

000

2 i

00i0 1 ( 41)

c2= 0

0000

000i

0000

0

2 i00 1 , c

3= 00000

00

2 i0

0i00

0000 1 ( 42)

so that (c i) lm = ieilm coincide with the a-matrices in ( 8) . The equations ofmotion then reproduce exactly the Majorana± Maxwell Eqs. (2) and (3)

p . y= 0 (43)

Ey+ ip 3 y= 0 (44)

3.3. Properties of the Equations of Motion

From (31) , ( 30) we have

(u . p) w m= iemabc paw buc ( 45)


For u . p Þ 0 ( in the limit (21) this selects solutions with E Þ 0), multiplying(45) by um and summing we obtain

um wm= 0 (46)

In an analogous way, multiplying by pm we get also

pm w m= 0 (47)

From these, we see directly that the equations of motion (31) contain theorthogonality conditions ( 15) and (28). For further applications, it is usefulto deduce also the `̀adjoint equation’’ of (31); by means of the hermiticitycondition (36), we then have

w a*( c m) ab pm= 0 (48)

(where we understand that pm= i ¶ m acts on w * ). The comparison of ( 48)and (31) makes evident the fact that photons coincide with antiphotons.( 7) 5

3.4. Probability Current: The Continuity Equation

Multiplying (31) by w * and (48) by w and summing, we easily obtainthe equation

¶ m Jm= 0 (49)

where the current Jm is given by6

Jm= 2 12 w *c

mw ( 50)

and satisfies

um Jm= 2 12 w * . w ( 51)

The `̀ probability density’’ J0 in ( 50) is a well-defined positive quantity ( asone can easily check) , and in the limit ( 21) Eq. ( 49) reduces to the Poyntingtheorem (10).

238 Esposito

5 However, we stress the possibility that a (1, 0) Å (0, 1) particle does not always coincide withits antiparticle; see, e.g., Ref. 9.

6 Note that the factor 12 can be eliminated by a redefinition of the fields w and w *, while the

minus sign can be absorbed in the definition of c-matrices.

However, in the general case (with an arbitrary um) it is more correctto consider directly the energy-momentum tensor

T mn= 2 F m r Fnr + 1

4 g mnFabF ab ( 52)

whose components T 0m are proportional to the probability current density,as explained in Sec. 1 (T 00= 1

2 (E2+ B

2 ) , T 0i= E 3 B ). Substituting (16)and (30) in (52) we have the following expression for the tensor T mn:

T mn= 2 14 ( w *{c

m, cn} w + ( w * m w n+ w * nw m) ) ( 53)

where {..., ...} denotes the anticommutator. Using (40) , T mn can be cast inthe asymmetric form

T mn= 2 12 { w *c

mc

nw + w * m

wn} (54)

Another useful form of T mn in terms of um is the following:

T mn= 0 12

gmn 2 u mun 1 w * . w 2i2

(u me

nabc+ un

em

abc ) w * aw

buc

212

( w * m w n+ w * nw m) ( 55)

Using, then, the equations of motion one can prove that T mn is conserved:

¶ nT mn= 0 (56)

The general expression for the probability current density is, finally, givenby T 0m.

The fact that we have found two conserved currents, Jm and T 0m,which coincide only in the limit ( 21) , induces us to ask ourselves why thishappens. In reality, the conservation of Jm and T 0m are two expressions ofthe same physical principle, that is, the energy-momentum conservation(or, in other words, the probability conservation).7 In fact, one can easilyprove that

T mn= u mJn+ unJm 2 (u . J) g mn+ j mn ( 57)


7 It is very interesting to note this connection between the energy-momentum conservationand the electric charge conservation. In fact, as we know, the probability current conservationin quantum mechanics is connected to electric charge conservation through the continuityequation. Only in the Majorana theory is the probability current given by the electromagneticenergy-momentum density 4-vector, and so only in this theory can we observe such a con-nection. We thank an anonymous referee for pointing this out.

where

j mn= 2 12 w *t

mnw ( 58)

( tmn) ab= gmagn

b+ gnagmb ( 59)

is a particular symmetric tensor satisfying

um j mn= 0 (60)

The current Jm is then given by

Jm = unT mn ( 61)

from which it is clear that the conservation of Jm follows from that of T mn,as stated earlier.

A very peculiar role is played by the current j 0m; as we shall see in thenext section, this is a Lorentz boost term, whose presence is not evident inthe noncovariant formulation of Majorana± Maxwell electromagnetism.Obviously, this `̀ boost current’’ is not a conserved one,

¶ m j 0m= ¶ 0(u . J) 2 (u . ¶ ) J0 ( 62)

and it is interesting to observe, from (60), that it does not contribute to thecurrent Jm.

4. HAMILTONIAN FORM. SPIN AND INTRINSIC BOOST

From the covariant equation of motion (31) one can deduce a usefulLorentz noninvariant Hamiltonian formulation left-multiplying (31) byc 2 1

0 . The equations of motion can then be cast in the form H w = E w with

H= a . p, a= c 2 10 c ( 63)

The explicit form of the matrix c 2 10 is given by

(c 2 10 ) a

b = u0 gab 2 ( ga0ub + uag0

b ) +uaub

u0

+ga0g0

b

u02 ie0a

bc uc ( 64)

and the following relations hold:

u0(c2 10 + c0) = u0(c 0* 2 c0 ) + ( c 2

0+ 1) ( 65)

( c 2 10 ) ² = c 2 1

0 ( 66)

240 Esposito

the latter saying that c 2 10 is Hermitian, as the other c matrices. It is then

easy to see that, in the form (63), the matrices a are not Hermitian, ingeneral.

However, this is just an accident since, using (46) and the m= 0 equa-tion of (45), we can recast the equations of motion in the form H w = E w

with, now,

H= ( a + s ) . p ( 67)

where the matrices

a1= 0

0000

0000

000

2 i

00i0 1 , a

2= 00000

000i

0000

0

2 i00 1 , a

3= 00000

00

2 i0

0i00

0000 1

t1= 0

0100

1000

0000

0000 1 , t

2= 00010

0000

1000

0000 1 , t

3= 00001

0000

0000

1000 1

are involved. Note that

(ai) ab= 2 ie0ia

b ( 68)

are the generalization to four dimensions of the a -matrices in (8) , while

( t i) ab = g0ag i

b + g iag0b ( 69)

coincide with the matrices t0i-defined in (59).It is somewhat interesting to observe that the Hamiltonian in (67) is

explicitly independent of um ( and holds true for arbitrary um) , the equationsof motion H w = E w being themselves explicitly independent on um :

E w 0= p . y ( 70)

Ey= p w 0 2 ip 3 y ( 71)

However, not all the solutions of ( 70) , ( 71) are allowed in the generallyaccepted formulation of electromagnetism (even if, in the literature, theE= 0 solutions of Maxwell equations have been considered as well ) , since


in this form these equations do not contain the orthogonality condition(46). This is a fundamental condition, expressing the antisymmetricbehavior of Fmn , FÄ mn ; so, in this case, together with (70), ( 71) also Eq. ( 46)must be taken into account, and the considered solutions of H w = E w arethose satisfying (46).

Obviously, in the limit (21) we recover the Majorana± MaxwellEqs. ( 43) and (44).

Let us now turn to the matrices ( 68) and (69). These verify the followingcommutation rules:

[ai, a

j]= 2 ieijka

k ( 72)

[ai, t

j]= 2 ieijkt

k ( 73)

[ t i, t j] = 2 ieijkak ( 74)

and we recognize in these the Lorentz algebra.By requiring that the total angular momentum of the photon

J= L+ S be conserved ( i.e., [H, J] = 0), we can then identify the spinmatrices with

S= 2 a ( 75)

In an analogous way, we can further identify the intrinsic part of Lorentzboost matrices with

Ks= i s ( 76)

as we have already anticipated.With the introduction of the spin matrices, we can now rewrite the

equations of motion in the form

S . p

Ew = 2 w +

s . p

Ew ( 77)

where in the l.h.s. we recognize the helicity operator. With a direct calculus,using (47), we see that

sm

n . p

Ew

n=p m

E 2 (p . y ) ( 78)

so that the boost term in the equations of motion is proportional to p . y .If p . y= 0 the boost term vanishes, and then only in this case is the photonin a helicity eigenstate (with l= 6 1 eigenvalues) . But, from (47) , p . y= 0means w 0= 0, and this, from (46), implies that um= (1, 0 ).

242 Esposito

We then conclude that the physical transversality of the photonimplies:

· the photon is in a helicity eigenstate (with l= 6 1 eigenvalues) ;

· a Lorentz boost term in the equations of motion is forbidden.

Conversely:

· if the photon has also longitudinal degrees of freedom, then it is notin a helicity eigenstate ( as happens for the Dirac equation too);

· longitudinal degrees of freedom ( if present ( 3) ) are described by aLorentz boost term in the equations of motion.

5. CONCLUSIONS

In this paper we have given a version of the Majorana formulation ofMaxwell electromagnetism which is explicitly invariant under the Lorentzgroup. The main advantage of the Majorana formulation is that it does notrequire the use of electromagnetic potentials, thus avoiding nonphysicaldegrees of freedom. This has also been achieved here in the explicitLorentz-invariant version with the introduction of `̀auxiliary fields,’’ linear( covariant) combinations of the electric and magnetic fields built using thebasic antisymmetric properties of the electromagnetic field strenghts Fmn

and FÄ mn .As also in the noncovariant version, ( 6, 7) the Majorana formulation of

electromagnetism is strongly based on the fact that there are three spatialand one temporal dimensions, since only in 3+ 1 dimensions is the dualtensor of Fmn a two-index tensor as Fmn .

Here we have shown not only that the Maxwell equations can be writ-ten as a Dirac-like equation for the photon, but also that the interpretativestructure of the Dirac equation can be maintained for the photon as well.In fact, interpreting w * as the adjoint field of w ( see Eq. ( 48) ), the probabilitycurrent for the photon has the same form as in the Dirac case ( cf. Eq. ( 50) ),provided the new c-matrices are defined as in (30).

The `̀ transversality’’ property of the photon (47) ( together with (46) )is contained in the Dirac-like Eq. ( 31) and it is thus not necessarily afurther constraint, as in the noncovariant formulation ( see Eq. ( 7) ).

Finally, the spin matrices, and also the ( intrinsic) boost ones, havebeen derived in Sec. 4 using the Hamiltonian form of the Dirac-like equa-tion (31), and the helicity properties of the photon has been recovered aswell.


ACKNOWLEDGMENTS

This paper has its origin in a fruitful exchange of correspondence withProf. E. Recami. The author is indebted to him. Very interesting commentsby two anonymous referees are also most kindly acknowledged.

REFERENCES

1. S. Esposito, Phys . Lett . A 225 , 203 (1997).2. D. Mugnai, A. Ranfagni, R. Ruggeri, and A. Agresti, Phys. Rev. Lett . 68, 259 (1992).3. M. W. Evans, Found. Phys. 24 , 1519, 1671 (1994) , and references therein.4. M. W. Evans, `̀The derivation of the Majorana form of Maxwell’s equations from the

B-cyclic theorem,’’ submitted for publication. M. W. Evans to E. Recami, private com-munication.

5. B. Gaveau, T. Jacobson, M. Kac, and L. S. Schulman, Phys . Rev. Lett . 53, 419 (1984) .6. E. Majorana, Scientific Papers, unpublished, deposited at the `̀Domus Galileana,’’ Pisa,

quaderno 2, p. 101/1; 3, pp. 11 and 160; 15, p. 16; 17, pp. 83 and 159; see also the firstreference in Ref. 7 and the catalog of Majorana’s scientific unpublished papers published byR. Mignani, M. Baldo, and E. Recami in E. Majorana, Lezioni all’Universita di Napoli(Bibliopolis, Naples, 1987). E. Majorana, Nuovo Cimento 9, 335 (1932) .

7. R. Mignani, E. Recami, and M. Baldo, Lett . Nuovo Cimento 11, 568 (1974). See alsoE. Giannetto, Lett . Nuovo Cimento 44, 140, 145 (1985).

8. L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields, 4th edn. (Pergamon,Oxford, 1975).

9. D. V. Ahluwalia, M. B. Johnson, and T. Goldman, Phys. Lett . B 316 , 102 (1993).

244 Esposito

Covariant Majorana Formulation of Electrodynamics

Documents

Transcript of Covariant Majorana Formulation of Electrodynamics