The Feynman path integral quantization of constrained systems

The quantization of a classical system can be achieved by the canonical quantization method [1]. If we ignore the ordering problems, it consists in replacing the classical Poisson bracket, by quantum commutators when classically all the states on the phase space are accessible. This is no longer correct in the presence of constraints. An approach due to Dirac [2] is widely used for quantizing the constrained Hamiltonian systems [3-5]. The path integral is another approach used for the quantization of constrained systems. This approach was formulated by Faddeev [6]. Faddeev and Popov [7] handle constraints in the path integral formalism by quantizing singular theories with first-class constraints in the canonical gauge. The generalization of the method to theories with second-class constraints is given by Senjanovic [8]. Fradkin and Vilkovisky [9, 10] rederived both results in a broader context, where they improved Faddeev’s procedure mainly to include covariant constraints; also they extended this procedure to the Grassman variables. When the dynamical system possesses some second-class constraints there exists another method given by Batalin and Fradkin [11]: the BFV-BRST operator quantization method. One enlarges the phase space in such a way that the original second-class constraints become converted into the first-class ones, so that the number of physical degrees of freedom remains unaltered. These quantization schemes have the properties that by using them one can easily control important properties of quantum theory such as unitarity and positive-