A note on BRST quantization of SU(2) Yang-Mills mechanics

The quantization of SU(2) Yang-Mills theory reduced to 0 + 1 space-time dimensions is performed in the BRST framework. We show that in the unitary gauge A0 = 0 the BRST procedure has difficulties which can be solved by introduction of additional singlet ghost variables. In the Lorenz gauge Ȧ0 = 0 one has additional unphysical degrees of freedom, but the BRST quantization is free of the problems in the unitary gauge. ∗ e-mail: [email protected] † e-mail: [email protected] 1 SU(2) Yang-Mills mechanics We consider SU(2) Yang-Mills mechanics obtained by the reduction of SU(2) Yang-Mills field theory in (D + 1)-dimensional space-time to a finite-dimensional quantum system, by taking the dynamical variables to depend on the time co-ordinate t only. The lagrangean of such a theory is LYMQM = 1 2 (F a 0i) 2 − 1 4 (F a ij) , (1) where i, j = (1, ..., D), and F a 0i = Ȧ a i − gǫA0Ai , F a ij = −gǫAiAj . (2) Such a system has been widely studied in the context of non-perturbative aspects of (super) Yang-Mills theories [3, 4], and as a first step in the regularized dynamics of membrane theory [5]-[10]. The lagrangean is invariant under time-dependent gauge transformations with parameters Λ(t), taking the infinitesimal form δA0 = Λ̇ a − gǫA0Λ, δAi = −gǫAiΛ. (3) This invariance allows us to impose a gauge condition leaving the physical dynamics unchanged. The simplest choice is A0 = 0. (4) With this condition the effective lagrangean for the remaining D-dimensional vector potentials Aa becomes 1 Leff = 1 2 Ȧa − V [A], (5) with the potential