The Classical and Quantum Mechanics of Systems with Constraints

In this paper, we discuss the classical and quantum mechanics of finite dimensional mechanical systems subject to constraints. We review Dirac’s classical formalism of dealing with such problems and motivate the definition of objects such as singular and non-singular action principles, firstand second-class constraints, and the Dirac bracket. We show how systems with first-class constraints can be considered to be systems with gauge freedom. A consistent quantization scheme using Dirac brackets is described for classical systems with only second class constraints. Two different quantization schemes for systems with first-class constraints are presented: Dirac and canonical quantization. Systems invariant under reparameterizations of the time coordinate are considered and we show that they are gauge systems with first-class constraints. We conclude by studying an example of a reparameterization invariant system: a test particle in general relativity.