The uniqueness of the energy momentum tensor in non-Abelian gauge field theories

The uniqueness of the energy momentum tensor in non-Abelian gauge field theories is established under minimal hypothesis.


I. INTRODUCTION
In the general theory of relativity, the interaction of the gravitational field (characterized by a metric tensor gij ) and a source-free gauge field (characterized by a curvature form It is easy to see that with these definitions, the following identity holds: Since the Einstein tensor given by the left-hand side of (1) is divergence-free, the same must be true for Tg, the right-hand side of ( 1 ). This is the case because of the identity and Eq. (2). For any Tij in the right-hand side it must be true that T ijlli = 0, at least when (2) holds. In other words, it must be true that The uniqueness of the energy momentum tensor was established recently 1 under the restrictive hypothesis Tijlli = C af3 Hf3j r Fairjli' Clearly (7) is weaker and it is mandatory because of ( 1 ) and (2). In this paper we will prove that To ij is essentially the only solution to the following problem: to find all gauge invariant symmetric tensors Tij = Tij(ghk;F~k) such that (7) holds. Our result generalizes Ref. 3.
We want to point out that, due to the condition (7), one cannot generate energy momentum tensors by adding terms to the action.

Then (2) reads
because of the gauge invariance of T ii and its tensorial character. Then Tiilli = 0 written out in full in the above mentioned coordinate system is ( T,'223 Let us choose, for arbitrary but fixed ghk, F~k' the derivatives F~,h such that (8) holds. Taking account of (8) and (5), it is clear that theF~lh appearing in (10) are arbitrary and independent. Then we deduce Taking i = 1,2,3,4 in (11)  where, for the sake of simplicity, we have used the notation for a fixed p.

III. THE UNIQUENESS OF THE ENERGY MOMENTUM TENSOR
Let us denote, for fixed a, p, and y, a 3 Tij Tijhkrslm = _ _ _ _ _ _ aF~k aFt;. aFfm We will prove that all these derivatives are zero. From (12)-(18) it is clear that it is enough to consider the cases ijhk = 1213, 1214, 1223, 1224, 1323, 1314. In case (a) using (16), (15), and 1 TZhkl for i#j and h #k, we have In case (b) we have Finally, in case (c) it is where we have also used the equality of the cross derivatives. We conclude that  iJ,h,k,r,s,l,m, (27) and so T ij is a polynomial in Fij of degree not greater than two. Consequently The tensorial concomitants of glm were recently found 4 • 5 for any valence of the tensor. Taking account of the fact that we are dealing with all coodinate systems, and not merely with those belonging to an oriented atlas, then it follows that where d a /3' A., and aa/3 are real numbers and aa/3 = a/3a' d a /3 = d/3a' Then Assuming FaijllJ = 0, it follows that TijllJ = 0, and so, using the identity (5) It is easy to see that if S tk is the term within brackets in (31) then, because of (12)-(18), we have From (32) It follows easily from the gauge invariance of T ij that the aa/3 are Ad G invariant.
In summary, we have proved the following. Theorem: If Tij = Tij(ghk: F~k) is a gauge invariant tensor whose divergence vanishes when the divergence of Faij is zero, and if Tij = Tji, then Tij = Toij + ,1.gij, where Tg is the usual energy momentum tensor.