3. — The field equations.

We come now to the field equations for a gauge theory which will gen­eralize Maxwell's equations. Written in terms of the covariant derivative V^, and in units where the velocity of light is 1, the equations can be written in terms of commutators

 

 

In (3.2) we sum over /i and in Minkowski space the term corresponding to the time component has a minus sign (whereas the Euclidean analogue has all positive signs).

These two equations involving the potential have a rather different char­acter, in that the first is an identity (the Bianchi identity of differential geometry) and only the second, the Yang-Mills equation, imposes a condi­tion on the potential. For G = 17(1) these equations written in terms of the field Fpv are Maxwell's equations in vacuo. The first equation is then just the integrability condition on JV which asserts that (at least locally) we can introduce a potential Au so that

 

 

For non-abelian G it is not possible to write these equations in terms of Fpv alone because the covariant derivatives explicitly involve the po­tential An. This emphasizes once more the primary role of the potential as opposed to the field.

The Yang-Mills equation (3.2) is derived from a Lagrangian C by inte­grating over II1 a Lagrange density which is an invariantly defined qua­dratic expression in the curvature. For G = U(n) or SU(n) one puts (up to constant factor)

 

 

where Ft" is obtained in the usual way from Fuv, raising indices by the standard metric tensor of Minkowski or Euclidean space and we sum over all fi, v. Equations (3.2) are the corresponding Euler Lagrange equations. The minus sign was inserted in (3.3) so that in the Euclidean case we geta positive Lagrangian, the point being that —Trace {AB) is positive definite on the Lie algebra of U(n). For other Lie groups one can either use an embed­ding in U(n) or more intrinsically one replaces Trace (AB) by the Killing form which is the standard invariant bilinear form on L(G): the two methods give the same answer up to a positive scalar multiple.

In the Euclidean case the Lagrangian can be viewed as the natural L2-norm of the curvature, that is the integral over II* of the sums of the squares of the absolute values of all its components in a standard ortho- normal base. More invariantly one can rewrite this as follows. First we recall that a skew tensor corresponds to an exterior differential 2-form

 

 

The dual 2-form *oc defined relative say to the standard Euclidean metric is given by replacing a12 by a34 etc. with an appropriate minus sign if the indices involve an odd permutation. It satisfies *2= where A denotes the exterior multiplication and gives here an exterior 4-form, i.e. a volume form which can therefore be integrated. Passing now to the curvature F which is a 2-form with values in the Lie algebra L(GJ:) we define *F in the same way, switching spatial indices and leaving untouched the Lie algebra variables. Then we put and this is the invariant way of writing the Lagrangian.

Equation (3.1) asserts that the covariant derivative of F, skew-sym­metrized, is zero or symbolically

 

Using the duality operator * we see that (3.2) can then be written

 

 

 

This way of writing the Yang-Mills Lagrangian and equations exhibits

clearly its in variance and covariance properties. First of all the equations clearly make sense on a curved (Eiemannian) 4-space, since the *-operator uses only the infinitesimal duality. Secondly the *-operator on 2-forms in 4-space is conformally invariant in the sense that two metrics ds2 and Q(x)ds2 give the same *. Thus the Yang-Mills equation (and the Yang-Mills Lagrangian) depend only on the conformal structure of 4-space. This im­portant property of Maxwell's theory is therefore preserved in the non- abelian case.

The apparent symmetry or duality between (3.5) and (3.6) is delusory as we have explained, although in Maxwell theory it reflects the duality between electricity and magnetism and attempts have been made to understand the non-abelian analogue. This is a deep question and the proper understanding of this duality is likely to be found only at the quan­tum level [22] [40]. However at the classical level we note an elementary consequence of (3.5) and (3.6), namely that (3.6) follows from the identity (3.5) if the field F satisfies one of the equations

 

 

Thus we have here first-order non-linear equations for the potential which imply the second-order Yang-Mills equations. These equations have a par­ticularly simple significance in the Euclidean case as we shall see in the next section. Note that the definition of * involves an orientation of Rl (an order­ing of the coordinates xr, ..., x}) and that (3.7) and (3.8) switch when we reverse the orientation. Thus there is no essential mathematical difference between the two cases.