Electromagnetism in terms of its two degrees of freedom

ANNALS OF PHYSICS 105, 407-419 (1977)

Electromagnetism in Terms of

Its Two Degrees of Freedom

R. GAMBINI* AND S. HOJMAN

International Centre for Theoretical Physics, Trieste, Italy

Received October 6, 1976

Electromagnetism is described in terms of two scalar gauge-invariant variables. The role of gauge freedom in the quantization process when A@(x) is written in momentum space appears as the indetermination of a two-dimensional space-like plane orthogonal to Ihe (null) momentum k@ plus a c-number arbitrary function. Duality rotations can naturally be described as the freedom to choose two orthonormal vectors in the two-dimensional space-like plane. The action written in terms of the two dynamical degrees of freedom is explicitly invariant under finite duality rotations and the associated Noether current is gauge invariant. Finally, we establish the equivalence for the Poisson bracket relations (PBR’s) (based on equal time and equal null-time PBR for the two degrees of freedom) between A@(X) and AW’) for any gauge.

1. INTRODUCTION

The usual canonical quantization procedure does not apply directly to gauge- invariant theories. This is due to the fact that the constraint equations present in such theories are not always consistent with the canonical Poisson bracket relations (PBR). There are, however, different ways of solving the problems posed by the gauge theories, such as the Gupta-Bleuler method [1, 21 for quantum electrodynamics, or Dirac’s procedure [3] to deal with constrained systems. Of course, a very direct approach consists in solving, when possible, the constraints and quantizing only the two degrees of freedom as is usually done, for instance, in the Coulomb gauge in electromagnetism. Nevertheless, this kind of quantization is almost always done in a given gauge.

The quantization of linear gauge theories can also be achieved without fixing a gauge. For instance, two different solutions to this problem are presented in Refs. [4, 51. There is a difference in the approaches of ([A, see Ref. [4]) and ([B], see Ref. [.5]), namely the PBR for the dynamical degrees of freedom are defined for “simultaneity” hypersurfaces which are light-like and space-like, respectively.

One of the main goals of this work is to prove the equivalence of the PBR’s [AiL(x), /i~~(.u’)] obtained from both procedures for any gauge and arbitrary .Y and s’.

* On leave of absence from Departamento de Fisica, Universidad Simbn Bolivar, Apartado Postal 80659, Caracas 108. Venezuela.

407 Copyright c 1977 by Academic Press, Inc. All rights of reproduction in any form reserved. ISSN 0003-4916

408 GAMBINI AND HOJMAN

The relationship between the two approaches appears naturally when considering the Fourier transform of the variables used in [A]. In this context, as shown in [B], gauge freedom appears as the partial indeterminancy of two space-like vectors of an orthonormal null tetrad and one arbitrary c-number function.

We find here the connection between gauge transformations and the corresponding changes in the tetrad. The gauge freedom does not exhaust the indeterminancy. In fact, once a gauge is chosen, there is still one degree of freedom left to choose the two space- like vectors (a rotation in that spacelike plane) that corresponds to duality rotations. The electromagnetic action, when written in terms of the two (scalar, gauge-invariant) dynamical variables, is explicitly invariant underjinite duality rotations. Furthermore, the Noether current associated with this invariance is explicitly gauge invariant.

In Section 2 we review some of the results obtained in [A] and [B]. In Section 3 we establish the connection between the two approaches. In Section 4 we study duality rotations starting from the electromagnetic action written in terms of the two degrees of freedom only. In Section 5 we obtain the PBR for Aw(x) and A&(x’) (using the null-plane formalism) for any gauge, and we prove they are equivalent to those obtained in [B] by imposing the usual equal-time PBR for the two dynamical degrees of freedom. Finally, in the Appendix, the equal-null-time (x r = .r+‘) PBR for the two degrees of freedom are established using Dirac’s formalism.

2. REVIEW

(a) The Electromagnetic Field in Null Coordinates

Let us briefly review the results obtained in Ref. [4]. The infinite momentum frame [6] is the reference frame obtained by choosing new

space-time coordinates (xi, x-, x2, 9) related to the usual coordinates (go, ?I, 5?, 9)

by

x+ = 20 + 21 $0 - 21

--p-T X-=2112’ $3 z $2 * 9 x3 Z p3 0.1)

thus the x+ and X- coordinates lie on light planes. Let us consider four orthonormal vectors l$, labeled by 01 = 0, I, 2, 3 such that

n h3) iL&&) = %oo) = 77 = diag(-1, I, 1, I), (2.2)

where g,, is the Minkowski space-time metric with the same signature as +jeio . Intro- ducing the null vectors

ELECTROMAGNETISM: TWO APPROACHES 409

we obtain another tetrad that satisfies

(2.4)

The completeness relations

+j ‘““‘Eg)B;,,3) = gU”, (2.5)

hold for both tetrads. It is now straightforward to relate the ordinary temporal components of any tensor with its null components.

In order to show a formulation free of constraints in terms of the two physical variables of the electromagnetic field, it is convenient to start from a first-order action written in terms of the ten independent variables {A, , P”J

dI = {AG,yFuv + $FuVFUY) d2X dx- dx’ (2.6)

The attention is now shifted to the evolution along x+ = var, which is a null-line. Therefore, constraints and dynamical equations are not the same ones that appear in the usual treatment. The constraints correspond to the field equations which do not contain a+ derivatives, i.e.,

F+,- - F,-,, = 0,

Fat, = Aboa - Aa,t, , (2.7) Fa- = A-,, - A,,-,

where {a, b E (2,3)}. The constraints may be solved by introducing the two-dimensional decomposition of the two spatial components A, of the electromagnetic potential, given by

A, = E,~A~,~ $ AL,, = A=, + ALP, , (2.8)

where AT and AL are scalars under transformations which leave the null-plane decomposition invariant and F,,~ is the Levi-Civita symbol. In fact, we obtain

A- = AL,- - (-,)-1’2 A +-,- ,

Fat, = A:,a - A&, (2.9)

Fa’,- = -(-4)-l” A+-,-,, - A:,-,

where A+- 3 (-LQ-‘/~ I;;-; d = ax28x2 + i?2ax3.


Introducing the values of A-, F,- , Fob into Eq. (2.6), the action may be written in terms of the two physical variables AT and A+- only

dl = (a+&* 2-A/ + 2+A+ 2a4,- - +A$-,, - +A$ d”x, d- dx+. (2.10)

This structure is typical of the null-plane formalism, a sum of terms 2+q(“)2-q(“) minus a positive quantity which plays the role of the Hamiltonian.

Independent variations of AaT, A~,+ lead to massless Klein-Gordon equations which are gauge independent,

with

OAT=0 a q A,- = 0 (2.11)

0 = 28+8- - A.

The four-potential Au may be written in terms of the two physical quantities AT, A+- and of the gauge quantity AL,

A, = A,* + AL,, ,

A, = -(-d)-1/2A+-,- + AL,-, (2.12)

A+ = (-d)+A+-,+ + AL,+.

The potential Au identically satisfies Maxwell equations for any choice of the gauge quantity AL.

(b) The Electromagnetic Field in Momentum Space

Now, we shall briefly describe one way of achieving the isolation of the two degrees of freedom of the electromagnetic potential for any gauge, in the realm of the 3 + 1 treatment of conventional electromagnetism (as done in [B]).

We begin by considering the Fourier transform Au(k) of the electromagnetic potential Au(x) defined by

-wd = &p 1 eikxA”(k) d4k, (2.13)

where A*“(k) = A”(-k) because A@(x) is real. One can now define four scalars P)(k) such that

V”(k) = e$)(k) Y@(k) (2.14)

for any vector V”(k), where the four vectors e&,(k) are defined by

4,) = k”/l k I, 1 k [ = (I k2 I)“’

(r Y guve(de(fd) = m3) = rl tar& = diag(sign k‘-‘, - sign k2, 1, 1) for k2 # 0


and

el;,, = k”, !J Y .ww=x3) = 17kd3) = rl

(al3

(with y(,,) defined by (2.4))

for k” : 0.

The completeness relation

is satisfied for any k<. From Maxwell equations

(2.15)

(2.16)

it follows that

@A(m) - k’dk(,,A’R’ = 0 (2.17)

(i) A(“)(k) is arbitrary (for any k),

(ii) A(l) = 0 (for any ku),

(iii) Ac2) = Ac3) = 0 for k2 # 0, A(2) and A(3) are arbitrary for k” = 0.

Statement (iii) is a consequence of the fact that At2’ and Ac3’ satisfy Klein-Gordon equations for zero mass (in momentum space). A(?) and Ac3) are the dynamical degrees of freedom of the electromagnetic potential while Ato) (completely arbitrary) and A(l) r= 0 play no dynamical role whatsoever. In fact, the electromagnetic potential can be dejned in terms of two scalar massless fields A(“)(k) satisfying massless Klein- Gordon equations and an arbitrary scalar (gauge) field A(O)(k) as

A”(x) = &)@ 1 ko,,k, $S 4dk)M (a) (Wei’:” + A (a) (+pkr]

+ (27i)3i3 J d4k elia, ,4”‘(k) e i1m , (2.18a)

with A(“)(k) = i3(k2) A(“)(k)

and

A’““-‘(k) = B(kO) A’“‘(k),

e%(k) = --eyJ--k).

A’“““(k) = -e(k”) /f’“‘(k)

(2.18b)

Clearly, no constraints appear in this approach. Gauge freedom appears connected to the arbitrariness of A(O)(k) and the freedom left in the definition of the tetrad.

The PBR’s

[A(+l(k), ~‘b”“(k’)] = 17’“Ws(k _ k’) 2’ko~‘)“” (2.19)


are defined in agreement with the fact that the Ata) ‘s are massless scalar fields. Further- more, the PBR [A(O), anything] = 0 = [Au), anything] were defined. For AU(k) = c&(k) A(*)(k) one obtains

[A”‘-‘(k), ,+)(k’)] = rl(ab)e;l,)e~b)~(k - k’) 2(k”y’)1’z

and for AU(x)

for any gauge, where qu” = 7cab) eya, eyb, . From Eq. (2.21) the PBR’s [AU, FeB] and [FUV, F”B] can be derived. From [AU(x), A”(x’)] it can be proven that the constraint div E = 0 has vanishing PBR’s with any quantity without the need of fixing a gauge.

3. RELATIONSHIP BETWEEN GAUGE FREEDOM AND NULL TETRADS

In order to prove the equivalence of the PBR’s between A”(x) and A”(x’) based on light-like and space-like approaches for any gauge we need to establish the relationship between them. Furthermore, the role played by the gauge variable in the PBR’s has to be clarified.

For the viewpoint developed in [A] we need, in addition to the PBR’s for AT(x) and A+-(x), the functional dependence of AL(x) in terms of the basic quantities AT and A+- . This dependence is not determined by the theory, and it allows an arbitrary choice of a functional f such that

AL = f(A+- , AT, x”). (3.1)

This exhausts the gauge freedom and determines completely the PBR for Au(x). Here we shall only consider a more restrictive class of gauges, which ensures the linearity of the equations for the vector potential Au and which allows us to reobtain the family of gauges considered in [B], namely

AL = DTAT + D+-A+- + 4(x“), (3.2)

where D+- , DT are integrodifferential scalar operators with no explicit xu-dependence and 4 is a c-number arbitrary function, i.e., it has vanishing PBR with the dynamical quantities. We shall hereafter call this kind of restriction “linear gauges” because it is equivalent to imposing a linear gauge condition on the components of Au. The usual gauges belong to this family. For instance, the null gauge [63 A- = 0 gives AL,- = (--LI)-~/~A+-,- , i.e.,

AL = (-d)-1/2A+- + 4(x+, xa). (3.3)

We shall now link the results obtained with the null-plane approach with what has


been done in [B]. It is convenient to rewrite the action (2.10) in the following way (up to divergences)

and define

(II = (ii+(-L!l)li” AT P-(-A)‘,‘” AT -:- ?+A,- %L4+

- +((-A)“’ A ‘,,,” - &4:_,n;. d2x dx- dx+

,4’2’(4 = (-d)lP A’,

A’3’(x) = A+- .

(3.4)

(3.5)

In terms of these new variables (3.4) takes the form of a sum of two actions for real scalar massless fields,

dI = {~+~‘q-~‘2 - $(~‘2’,,)2 + a~‘~‘ii-A(~) - &4’3’,n)2> d2x dx- d.v+. (3.6)

The Fourier transforms of At2)(x), At3)(x) are given by

1 ei%@)(k) 8(k2) d4k,

(3.7)

s eiksffc3)(k) 6(k2) d”k,

ensuring that A’@(x) satisfy massless Klein-Gordon equations. d*ca)(k) = -Ata) because AC@(x) is real. From (3.2) the gauge variable AL may be written as

AY.4 = & J h,,(k) eikzAt2)(k) 6(k2) d”k

- ___ (2& . 1

D(,)(k) eik”Af3)(k) 6(k2) d4k (3.8)

where

Da,(k) = DW (k,k”)l/” ’

Ddk) = D+_(k). (3.9)


From Eqs. (2.12), (3.7), and (3.8) the components of AU(x) are

1 * + (2n)3;2 1

k+D(,)(k) eikzA(al(k) S(k2) d4k

I - -i- Qq”i” i

kgii,aA’O’(k) &k,

1 - ~,,bkt, A0 '= (&,.)3'" !

___- ei"<",@'(k) &(kZ) d4k (kek")lP

t L j- kJh(k) (27T)3/3 &1=/i(b)(k) 6(k2) rt4k

1 * + &-r)3/2

__ k,ei”zA(o)(k) d4k. !

(3.10)

Comparing (3.10) with (2.18a) and (2.18b) we can identify1 the tetrad vectors $,(k) in the following way:

e(,),(k) = JL ,

(3.11)

and with

-k- '(3dk) = (kc;+2 ' (k,ke)l/2 ' t (40) + &3,(k) k, 3

edk) = 6, JW + W2t2) + @id e(,),(k) + &2$?2),(k) + 43)%,(k>

they already constitute a null-orthonormal tetrad in the sense

where 7taS) is given by (2.4) and ;(a)~ = e(a)p for Dc2) = 0 and Dt3) = 0. It is worthwhile to note that A(O)(k) so defined is an arbitrary (gauge) c-number in

agreement with [B] (for nonlinear gauges A(O) is no longer a c-number). The gauge freedom appears also connected with the choice of the operators D(Z) and Db) . The invariance of the action (3.6) under rotations in the (A (2), Af3)) plane implies that the tetrad is defined up to a rotation in the (e(,) , ec3)) plane. As we shall prove, this invari-

1 These vectors coincide with those defined in [B] in all the Lorentz reference frames with the same null-plane decomposition. Imposing that they transform as vectors under any Lorentz transformations one ensures that the ACar’s transform as scalars.


ante turns out to be equivalent to the electromagnetic field invariance under duality rotations.

From Eqs. (2.14) and (3.11) it can be seen that AU(k) is always contained in the null- hyperplane defined by k”, a$, , and k;L3, because Au(k) has a vanishing component along e’;“,,(k) for any gauge. The three-dimensional metric in this hyperplane is singular. In spite of the fact that this plane is gauge invariant, the fourth vector, r&, , needed to complete the “orthonormal” tetrad is not, as can clearly be seen from Eqs. (3.11).

We may now summarize the relationship between the freedom left in the tetrad vectors and gauge invariance. There are thirteen equations to define the sixteen components of the four vectors of the tetrad. It was already shown in [B] that two of the three degrees of freedom left are related to gauge freedom (through their appearance in the definition of the PBR’s for AU(~) and AV(.\-‘)). The third degree of freedom left is related to duality rotations (see Section 4).

Two different kinds of gauge freedom should be distinguished, one related to the value chosen for a combination of AU(s) and the other related to those quantities which are chosen to behave as c-numbers (in the classical theory c-numbers are defined as quantities having vanishing PBR’s with anything). The two degrees of freedom left in the definition of ~(~1 in the tetrad are precisely related to this last kind of gauge freedom.

4. DUALITY ROTATIONS

The action (3.6) is invariant under rotations given by

A’*)’ = A(“) cos 0 + ,4(3) sin 0,

At3)’ = -A(z) sin B + At3) cos 8. (4.1)

We shall immediately see that they correspond to duality rotations [7]. In fact, the electromagnetic fields E and B can be written in terms of the scalar gauge-invariant quantities At2) and A@) only, as follows,

El = (-,y A(3), E,, = (-A)-“’ A(3),In - e,,(-0)“’ A(2),b,, ,

B, = (-d)l’” ,&), B,, = Enb(-d)-1”2 A(3),0b + (-4)-1!2 A(“),,, , (4.2)

and they transform under (4.1) according to

Ei = Ei cos 0 - Bi sin 8,

Bi = E, sin 0 + Bj cos f3, (4.3)

which is a duality rotation. It is important to remark that the Lagrangian density, that can be written in terms

of AU only, is invariant under the transformation (4.1) for any 0. This is in clear


contrast with the usual treatments where, to prove invariance, either 8 is restricted to be an infinitesimal or one is forced to introduce an extra potential.

The conserved current associated with the invariance under duality rotations is

p(x) = A’3’(x) 8JL4(2)(X) - A’2’(x) 8~A’3’(X). (4.4)

Note thati” is explicitly gauge invariant. We shall now prove that the conserved charge obtained from (4.4) coincides with

that given in Ref. [8]. In fact, from their Eq. (2.6),

Taking (4.2) into account, we obtain

Q = 3 f d3.y [A'"'(--)@ + a&-l A (3),o f A (2),1(d + ala,) k3),10

A(3) .O (d + a a)-'A'2' 11 ,oo - (2 w 311 (4.6)

and, by using the equations of motion for Af2)(x) and Af3)(x), we finally obtain

Q = 1 d3x (A(2)A(3),o - A(3h(2),0) = j-j”(x) d3x, (4.7)

where j0 is given by (4.4).

5. EQUIVALENCE BETWEEN THE POISSON BRACKET RELATIONS IN THE NULL-PLANE AND USUAL FORMALISM IN ANY GAUGE

We shall begin this section by deriving the PBR’s for A”(x) in any gauge in the null- plane formalism. To do this, we start from the PBR’s for the scalar gauge-invariant dynamical variables written in [A],

[A’“‘(x), A(yX’)],+=z., = * E(X- - r’) 6,(x - x’). (5.1)

These relations were obtained from the action (3.6) taking into account that the .4ta)‘s are scalar fields. The same PBR’s are obtained by using Dirac’s formalism [3], as is shown in detail in the Appendix.

Considering Eqs. (3.7),

A’“‘(x) = (2;):,2 s &k=/+)(k) S(k2) d4k, (5.2)


with J*(a)(k) = --A(@-k) and integrating over dk-, we obtain

Using (5.1) we prove that

[A(“)(k), .@*(k’)] = %i~+~~) j k- / S3’(k - k’), (5.4)

which allows us to find the PBR’s between Am and A”(Y) for any gauge and arbitrary xu and .xu’. From (3.10) and (3.11) AU(S) is written as

- 1

and using the PBR’s (5.4),

[A++, x-, x), AY(.U ', SC', x')]

(2k)3 5

% dk_ i d’k (e ikb.I.‘)

o 2/k-j. - epik(‘-“)) ~‘“*)e~,,,(k) e&,(k). (5.6)

It is convenient to obtain the four-dimensional expression of the PBR’s (5.6) to be able to compare with the results of [B] obtained from the usual PBR’s for the physical quantities for equal time, so = 9”. This form is

[A”(x), A”(Y)] == 2(;?r)” L d”k S(k2) c(kJ(e i ik(S-2’) - e@‘-“)) qcab)e’i,)(k) e:,,(k).

(5.7) It is well known that, for k2 = 0, c(k-) = c(ko), and using this fact in Eq. (5.7) one

finally obtains (2.21) taking into account that e(,)(-k) = --e(,)(k). This establishes the equivalence of the PBR’s between Au(x) and A”(x’) for any linear gauge obtained from the usual and null-plane formalisms. (It is clear from Eq. (3.1) how to proceed to compute the PBR’s when nonlinear gauges are considered.)

APPENDIX

In this appendix we shall show that the PBR (5.1) for the two degrees of freedom may be obtained from Dirac’s constraint formalism [3].

Let us start by writing the Lagrangian density coming from the action (3.6),

22 = a+A(a) a-.4(b) 8(&) - &P),cA(b)J I?+&) .

The canonical momenta are

(A-1)

(A.21


therefore, these momenta are not functionals of the “velocities” ii+A(“). In fact, they are constrained by

&)(xq = 17(,,)(XU) - aL4(,)(Xq a 0. (A.3)

Following Dirac, these are second-class constraints, as we shall prove. Let us take the canonical PBR given by

{.a, )(.Y’, 0 x-, x), P(x+ 2 x-y>; = - p)q y- ~ a - A-‘) 62(x - x’). (A.4)

The PBR between two constraints are

C(o&i y) = {17(,)(x’-, s) - L4(,)(XT, .x), D(,)(*Y-, y) - b_A(,)(x+, y))

= 26,,&- 6(x- - J’-) 83(x - y). (A.9

Consequently, they are second-class constraints because the PBR’s between them are different from zero.

We may now proceed to construct from the PBR (A.4) the Dirac bracket defined by

where C&(z, w) is the inverse of C(a~)(-y, Y), that is

s d3z C&(,,(x-, z) C(bc)(z. y) = 6&3(x - y).

It is easy to see that the inverse of the Cc&x, y) given in Eq. (A.5) is

&cd) C&)(2, w) = 7 E(W - z-) P(w - 2)

(A.7)

C.4.8)

and the Dirac brackets turn out to be

W,,,(x+, -4, A&i-, .a* = -&Ib) a36 - v> + * J a,- s3yx - z)

x E(W- - z-) S2(w - z) P(w - J-) d3z d3w

=-Ij- ( b) * x- __ 3’-) a2(x - y). (A.9)

If all Poisson brackets are now replaced by Dirac brackets, the second-class constraint may be set strongly equal to zero. In our formalism, the first-class con-


strain& have already been solved in order to obtain the action (3.6), therefore, the Hamiltonian is

with the brackets

ff = s d‘% dx- &da)e,db)‘r8(&) , (A.lO)

or

{F-&)(x+, X),A(b)(X’, y): * = - 2 *(ab) 6(s- - y-) P(x - y)

hub) (A(,)(x+, x), A(,)(X t, y): * = - 7 4.r - 1-j WC - Y). (A.11)

ACKNOWLEDGMENTS

The authors wish to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

REFERENCES

1. S. GIJPTA, Proc. Roy. Sot. (London) 63 (1950), 681. 2. K. BLEULER, Helu. Phys. Acta 23 (1950), 567. 3. P. A. M. DIRAC, “Lectures on Quantum Mechanics,” Belfer Graduate School of Science, Yeshiva

University, New York, 1964. 4. C. ARAGONE AND R. GAMBINI, Nuovo Cimetlto B 18 (1973), 311. This paper is referred to as [A]. 5a. S. HOJMAN, Ann. Physics 103 (19JJ), 74, referred to as [B]; b. S. HOJMAN, in preparation. 6. J. B. KOGUT AND D. E. SOPER, Phys. Rev. D 1 (1970), 2901. 7. C. W. MISNER AND J. A. WHEELER, Ann. Physics 2 (1957), 525. 8. S. DESER AND C. TEITELBOIM, Phys. Rev. D 13 (1976), 1592.

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Electromagnetism in terms of its two degrees of freedom