On the question of gauge ambiguity

It is pointed out that the ambiguity which characterizes gauge conditions of the type 3' S Aa(x)V = PJU C;(x) for nonAbelian gauge theories is also characteristic of the so-called axial-like gauge conditions n.Aa = C;(x), where, here, A; is the nonAbelian gauge potential, B = IJJV g orp 'g -66 n is a four-vector such that ClV PV PV PO vo' p n2 = 0, 1, or-l,andCt 2 are usually Aa-independent functions , CL ofx; g is the Minkowski metric and 6.. is the Kronecker PV 13 delta. (Submitted to Phys. Rev. Letters) "Work supported by the Department of Energy. 2 .As has been emphasized by ~andelstamand Gribov, the specification of -cI the gauge in nonAbelian gauge theory is a delicate matter in the Coulomb gauge and is, in fact, ambiguous in this gauge. 1 It appears to be generally accepted 2,3 that the Mandelstam-Gribov problem is also present in gauges which have gauge conditions aiL Aa' = C;(x) , (1) where Aa 14 is the Yang-Mills field and C;(x) is an Aa-independent function of c1 the space-time coordinate x. However, it does not seem 2,4 to be general knowledge that the ambiguity of Mandelstam and Gribov is also present in gauges of the type (2) where n' is a four-vector such that n2 = 0, 1, or -1, and C;(x) is an Aa14 independent function of x. We should like to clarify this particular point t in this note. Specifically, we use a matrix realization of the Yang-Mills theory so that we introduce A P = A;ta where the hermitian matrices t a carry the adjoined representation of the respective gauge group 8: eta, tb3 =if t C abc (3) where f abc are the real structure constants of & and (talbc = -ifabc . We normalize ta according to (4) t Recently, A. Balachandran et al., Phys. Rev. Lett. 40, 988 (1978),have discussed the ambiguities associated with the U-gauge. tr(tatb) = N bab . (5) Then, 72) reads n*A = C 2 (6) where c2. = c;ta . (7) Now, under a gauge transformation U = exp[-iz*?], A transforms by A +uAPu -' P (aclu)u -1 (8) when we take the Yang-Mills field strength tensor F to be CLV F =aA aYAP + ig[A $ Avl , (9) pv PV t where g is the gauge coupling constant. Thus, the gauge condition (6) becomes nP(UAPU -1 +A g tabu)u3 = c2 (10) or uc2u -' + a (n.au)u -1 = c2 (11) Thus, it has been commonly accepted that, for C2 3 0, for example, the only ambiguity in (6) is an uninteresting Agindependent U(x) which only depends on 4 X I, where 2 are the three coordinates orthogonal to nP. For, if I c2 z 0, then, (11) reads i (n.au)u-' = 0 (12) so that (see footnote (4)) n-au=0 . (13) However, the la'st equation, Eq. (13), for a simple compact Lie group, has other solutions than U(x) = U(:i), since au n.SJ(Z(x)) = 3 . n.aGJ . (14) Hence, Eq. (13) can be satisfied if t The Yang-Mills Lagrangian is GM = z FtiV FPv in our convention.