The equivariant inverse problem and the Maxwell equations

In affirmative, the equivariant inverse problem for Maxwell‐type Euler–Lagrange expressions is solved. This allows the proof of the uniqueness of the Maxwell equations.


I. INTRODUCTION
It is very well known that Maxwell equations can be written in covariant form as! where Fij is a skew-symmetric tensor and *Fij = 1Jij hk F hk . Equation (1.2) is equivalent to the existence of a covector field tPi such that Fij=tPi,J-tPJ,i ' (1.3) The Maxwell equations can be deduced from a variational principle as follows. If L=L(gij;tPi;tPi,J) ' (1.4) then from a variation of tPi we obtain as Euler-Lagrange equations, The left-hand side of ( 1.1) has two properties of covariance: (1) by a transformation of coordinates it changes as a vector; and (2) by a change of gauge, i.e., by a transformation ofthe type tPi -+tPi + f/J,i> where f/J is a scalar, it is invariant. These properties are also possessed by the Lagrangian ( 1.6). However, the last assertion is not mandatory since the Lagrangian does not have, in general, any physical meaning, although the Euler-Lagrange expressions do have a meaning. The main purpose of this article is to prove that the situation already encountered with the Maxwell equations is found always in spaces of dimension 4, i.e., the assumption of the two covariance properties for the Euler-Lagrange expressions implies that the Lagrange is equivalent to (it has the same Euler-Lagrange expressions as) a Lagrangian with the same properties. 2 This solves for the affirmative the equivariant inverse problem 3 for Maxwell-type Euler-Lagrange expressions.
Precisely, we consider a quantity B i(gij;gij,h ;tPi;tPi,J;tPi,Jh) ( 1.7) such that Here we do not assume any covariance property for L! with respect to transformations of coordinates or changes of gauge. We will prove that (1.7) and (1.8) imply the existence of L = L (gij ;tPi ;tPi,J ), which is a gauge-invariant scalar density such that Ei(L) =Bi.

II. THE EQUIVARIANT INVERSE PROBLEM
The condition (iii) in (1. 8) written in full is The coefficients of (2.1) depend only on gij' tPi' and tPi,J' Since B i is gauge invariant, by the replacement theorem 4 we have Since B i is a tensorial density, the same is true for 3) is also a tensorial density. Hence, by (2.2), Hi -! HijhF hJ is a vector density and depends only on gij and Fij . It is known 5 that such vector densities are zero, and so we obtain hklJ . (2.4) where ( at/J a(J Now, (2.24) and (2.27) are the integrability conditions we need to establish the existence of a scalar T = T(t/J,t/!) such that (2.28) In this case, a straightforward computation proves that (2.29) where L =,[gT is a gauge-invariant scalar density. This solves the equivariant inverse problem since det (Fii ) ¥-0 is a dense subset of the space of variables gii ,Fii [and then (2.29) is valid everywhere by a continuity argument].

III. THE MAXWELL EQUATIONS
We have proved in Sec. II that if B I is of the type (1.7) and it satisfies (1.8), then follows that a 2 L I a~ = O. By a continuity argument we deduce that everywhere, L=,[g(Ct/l+A), (3.6) where A and C are real numbers, and so L iili = C ,[gFlJ li ' which makes (3.2) the usual Maxwell equations.

ACKNOWLEDGMENT
We thank the referee for helpful suggestions. 'The tensor gij is the metric tensor of a four-space-time manifold and g = Idet(gij ) I. We use the summation convention and we raise and lower indices with go and ii. We denote 1J Qhk = (..Ji) -'E'Jhk, where E'Jhk are the Levi-Civita permutation symbols. The vertical bar stands for covariant derivative and the comma stands for the usual partial derivative. Here, Ji is the charge-current vector.
2This is not true for all dimensions. If it is 3, then B i = E'Jh Fin is of the form