Canonical quantization of the Dirac field in a finite volume

The $1+1$ dimensional Dirac field in a finite volume is quantized canonically by using two different methods in this paper. First, we take the boundary conditions (BCs) as primary Dirac constraints, and prove that those BCs as well as the intrinsic constraints are entangled and they form the second class constraints. The quantization is performed canonically by using Dirac’s procedure. And then, we study this model in the reduced phase space. It shows that the Poisson brackets in the reduced phase space are equivalent to the Dirac brackets in the original phase space.