A simple and complete Lorentz-covariant gauge condition

Volume 90B, number 3

PHYSICS LETTERS

25 February 1980

A SIMPLE AND COMPLETE LORENTZ-COVARIANT GAUGE CONDITION

C. CRONSTROM 1

Nordita, DK-2100 Copenhagen O, Denmark Received 16 October 1979

We discuss a very simple gauge condition, for abelian or non-abelian gauge theories, which is both Lorentz-covariant and complete. Using this gauge a straightforward solution to the problem of expressing the vector potentialA in terms of the field strength G is obtained.

1. Introduction. In this note a very simple Lorentzcovariant and complete gauge condition is proposed and discussed. The gauge condition in question fixes the vector potential uniquely, apart from global gauge transformations. Furhtermore, using the gauge condition in question, a straightforward solution is obtained to the inversion problem, i.e. the problem of expressing the vector potential A in terms of the field tensor G. We consider a general theory of the Yang-Mills type [ 1 ], with a semi-simple compact gauge group G. A hermitian matrix description is used for the vector potential A, Au(x ) = Aa~(x)Ta ,

(1)

where the Ta'S are hermitian matrix representatives of the generators of the group G. The gauge condition we propose is simply the following,

xUAu(x ) = 0 .

(2)

The gauge condition (2) is deceptively simple, but has not, to the best of my knowledge, been discussed previously in the literature. In fact, the only previously given gauge condition which is complete in non-abelian gauge theories, is the following superaxial gauge condition,

Ao(xO, O, O, O)

= O,

Al(X°,xl,O,O)

=0,

A2(xO,xl,x2,0)

=0,

(3)

A3(xO,xl,x2,x3)=O . To the best of my knowledge, the condition (3) was first given by Coleman [2]. In eq. (2) the point xU = 0 occupies a somewhat special position in that eq. (2) implies that Au(0, 0, 0, 0) = 0 .

(4)

However, instead of eq. (2) we could equally well take (x. - z#)A.(z

+ (x - z ) ) = O ,

(S)

where z is a fixed point in Minkowski space. Taking z = 0 simplifies the notation, and implies no essential losss of generality, whence we restrict our consideration to z = 0 in what follows. Besides the usefulness of any particular gauge condition, there are two questions which arise about any gauge condition [3]. The first question is whether the gauge condition is attainable, i.e. given an arbitrary potential A u not satisfying the gauge-condition, can one find a gauge transformation ~2 such that the gauge-transformed At,,

Au ~ Aua = g2-1Aug 2 _ (i/g)g2-1aug 2 , 1 On leave from Department of Theoretical Physics, University of Helsinki. Supported in part by the Academy of Finland.

(6)

satisfies the gauge condition. The second question concerns the completeness 267

Volume 90B, number 3

PHYSICS LETTERS

(or uniqueness) of the gauge condition, i.e. if one has one potential Au satisfying the gauge condition, can one find a gauge transform of that potential which also satisfies the gauge condition in question. If not, then the gauge condition is complete.

2. Attainability and completeness o f the gauge condition. We assume from the outset that space-time is simply connected and that the vector potentials we consider are regular, i.e. are continuously differentiable up to second order. Let us consider an arbitrary regular vector potential A u, not satisfying eq. (2). Let

B u = co-lAuco - (i/g)co-l(~t, co).

(7)

25 February 1980

co(x) ~ co~(x) = ~ - l ( x ) c o ( x ) ~ l ~ 2 ( O ) ~

.

(13)

Thus, by requiring that the initial value w 0 in eq. (9) transforms as coS = ~-l(o)coO ,

(14)

we remove the global gauge transformation ~2(0) of

B u defined by eq. (7), which otherwise would be induced by a local gauge transformation ~2(x) o f A u. 3. The inversion problem. The field tensor G uv(x ) is defined as follows, Guy(x) : a~Au(x) - a uAv(x) - ig[Au(x), A~(x)l (15)

Imposing the condition (2) on B u leads to the following differential equation for co,

The field-tensor therefore satisfies the Bianchi identity,

xUauco = - i g x U A u ( x ) co .

~7a(A)Go, + V#(A)G,a + ~7.r(A)aa~ = O,

(8)

The differential equation (8) has the solution

where XTa denotes the covariant derivative,

V a ( A ) ( a - aaq~ + ig[A a, ~bl. dt xUAu(tx)?coO ,

co(x) = p exp [-ig L

(9)

o

where coO is an arbitrary initial value for co, and P stands for "time-ordering" in the variable t. Thus, the gauge condition (2) is attainable by means of the gauge transformation co(x) given in eq. (9). We then consider the question of completeness. Assume that there is one vector potential A u satisfying (2) and also a gauge transform A ~ of that potential which also satisfies the condition (2). Using the relation (6) we obtain then a differential equation for ~ , namely

xUOu~ = 0 .

(10)

However, according to a classical theorem (Euler's theorem), eq. (10) implies that g2 is a homogeneous function of zeroth degree,

~ ( x ) = ~2(tx) ,

(I 1)

where t is arbitrary. Since ~ ( x ) is regular, in particular at the origin, we can take t = 0 in eq. (11) and obtain g2(X0 , X 1 , X 2 , X 3) = ~ ( 0 ) = const.

(12)

The gauge condition (2) is thus a complete gauge condition, apart from global (space-time independent) gauge transformations. The function co(x) defined by eq. (9) transforms as follows under a gauge transformation ~2, 268

(16)

(17)

Assume now that the vector potentialA u in eq. (15) satisfies the gauge condition (2). Then we have

xvauv(x ) : xVavAu(x ) + A u ( x ) .

(18)

Replacing x by tx in eq. (18), where t is an arbitrary parameter, we obtain, (d/dt) (tAu(tx)) = tx VGuv(tx ) .

(19)

Integrating eq. (19), we then obtain, 1

Au(x) = f dt txVGuv(tx ) . 0

(20)

The solution (20) to the inversion problem Guv(x ) -+ Au(x ) is very simple compared to the solution given by Halpern [4], who uses a variant of the super-axial gauge (3) in his construction o f A u in terms of Guy. One still has to check that the solution (20) actually solves eq. (15), i.e. that one actually obtains the field tensor Guv(x ) by inserting At, given by eq. (20) in eq. (15). It is not difficult to show that the Bianchiidentities (16), withA u given by eq. (20), are necessary and sufficient conditions for A u to be the solution to eq. (15), in the gauge specified by eq. (2). The Bianchi identities are thus constraint equations of the Guv(x ) in the gauge given by eq. (2).

Volume 90B, number 3

PHYSICS LETTERS

4. Discussion and symmary. The gauge condition we have considered here, eq. (2), is a very simple, Lorentz-covariant gauge condition, which has been shown to give a unique (apart from global gauge transformations) specification o f the vector potential A u in any non-abelian (or abelian) gauge theory. We have restricted our considerations to regular potentials A u (and therefore regular gauge transformations) i.e. twice continuously differentable potentials in a simply connected Minkowski space. The gauge condition considered has been shown to lead to a very simple solution to the inversion problem o f obtaining the vector potential from the field-strength. Quantising gauge theory in the complete covariant gauge given here, is an interesting question which will be dealt with in future communications.

25 February 1980

I am indebted to the staff of NORDITA for hospitality, and to P. Cvitanovi6, H.B. Nielsen and H. R6mer for useful discussions. References [1] R.L. Mills and C.N. Yang, Phys. Rev. 96 (1954) 191. [2] S. Coleman, Erice Lectures 1975, in: New Phenomena in subnuclear physics, Part A, ed. Z. Zichichi; see also: C.W. Bernard, N.H. Christ, A.H. Guth and E.J. Weinberg, Phys. Rev. D16 (1977) 2967. [ 3] R.J. Jackiw, Gauge-specification in a non-abelian gauge theory, talk given at Orbis Scientia (Coral Gables, FL, January 1978), CTP-preprint 697 (February 1978). [4] M.B. Halpern, Phys. Rev. D19 (1979) 517.

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