Symmetric Dirac Bracket in Classical Mechanics

Abstract

It is known that, in classical systems that have second-class constraints which relate the canonical coordinates and momentum, the ordinary skew-symmetric Poisson bracket must be replaced by the skew-symmetric Dirac bracket. It is also known that in the process of quantization of such systems, the Dirac bracket replaces the Poisson bracket in its correspondence with the quantum commutators. In this paper we obtain the symmetric partner of the Dirac bracket, which is of interest not only to classical mechanics but also in regard to the quantization procedure; i.e., the quantization rules for systems which are restricted by second-class constraints such that the commutation rules involve anticommutators (instead of commutators, as in certain fields with Fermi-Dirac statistics) can be given in terms of this new symmetric bracket. This symmetric bracket is related to the Poisson-Droz-Vincent symmetric bracket.