TY - JOUR

T1 - Non-uniqueness of gauge-field potentials

AU - Deser, S.

AU - Wilczek, F.

N1 - Funding Information:
It is well known that the field strength of an Abelian gauge field determines the potential uniquely (in a simply-connected region) up to a gauge transformation. Likewise, for Abelian and non-Abelian gauge fields, vanishing field strength implies that the gauge potential is gauge equivalent to zero. On the other hand, in the non-Abelian case, Wu and Yang \[1\] recently found an explicit example of two inequivalent potentials yielding the same field strength. In this note we will supply further examples of this phenomenon and a necessary condition for its occurrence. The related fact that the cyclic identities do not in general deternrine the potential structure of the field strength is also illustrated. Although the physical significance of these ambiguities is not obvious, we will argue below that it may not be nil. Example I. Let A u be a pure gauge potential, A u = g2-1 aug2, and a a constant. Then the field strength derived from both the potentials aA u and (1 - a)A u is Fur = -a(1 - cO\[Au,Av\]. But, in the generic case, for an SO(3) gauge field the \[Au, Av\] will span 41 all of SO(3) and therefore no gauge transformation leaves Fur invariant. This means that aAv and (1 - a)Au are not generically gauge equivalent. Example 2. A generalization of this example in- e' Supported in part by NSF Grant PHY-76-07299. '~¢~ Supported in part by ERDA contract E(11-1)3072. A.P. Sloan Foundation Fellow. #1 There are six independent commutators; this exceeds the dimension of SO(3). For larger groups, we expect also that the algebra generated by the commutators spans the group.

PY - 1976/12/6

Y1 - 1976/12/6

N2 - Two sets of phenomena peculiar to non-Abelian gauge theories are illustrated. The first concerns the existence of nonsingular potentials which generate the same field strength without being gauge-equivalent; a necessary condition for this ambiguity is obtained. The second exhibits the insufficiency of the cyclic identity in determining field strength-potential relations.

AB - Two sets of phenomena peculiar to non-Abelian gauge theories are illustrated. The first concerns the existence of nonsingular potentials which generate the same field strength without being gauge-equivalent; a necessary condition for this ambiguity is obtained. The second exhibits the insufficiency of the cyclic identity in determining field strength-potential relations.

UR - http://www.scopus.com/inward/record.url?scp=29144506691&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=29144506691&partnerID=8YFLogxK

U2 - 10.1016/0370-2693(76)90250-1

DO - 10.1016/0370-2693(76)90250-1

M3 - Article

AN - SCOPUS:29144506691

VL - 65

SP - 391

EP - 393

JO - Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

JF - Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

SN - 0370-2693

IS - 4

ER -