Elementary model of constraint quantization with an anomaly

Quantum gravity is made more difficult in part by its constraint structure. The constraints are classically first-class; however, upon quantization they become partially second-class. To study such behavior, we focus on a simple problem with finitely many degrees of freedom and demonstrate how the projection operator formalism for dealing with quantum constraints is well suited to this type of example.

Figure 1

Figure 1 Exact $L^2$ Eigenvalues of Row 4 and Row 5, Table , as functions of $\beta$

Figure 2

Figure 2 Exact $L^2$ Eigenvalue of Row 6, Table , as a function of $\beta$