Study of some Quantum Field Theoretical models based on Dirac Quantization

A Lagrangian is said to be singular when det[ ∂2L∂q˙i∂q˙j] = 0, and that signifies the presence of constraint in the usual phase space [1, 2]. Constraint means velocity independent relation between coordinate and momentum[1, 2, 3, 4]. So all the velocities of the dynamical variables of a theory cannot be determined in terms of momenta and as a result the precise canonical quantization gets threatened when a system contains constraints in its phase space. So quantization of this type of system is interesting in its own right and this type of quantization is known as Dirac scheme of quantization of constraint system. The canonical method of quantization requires the determination of momenta corresponding to the different field variables.As usual, A0 has no canonical conjugate, so there is a primary constraintπ0 = 0. To preserve this constraint in time, it is necessary to have a further constraint, and this preservation may give Gausss law. If the theory is anomaly free then no further constraint arises, and the above two constraints have vanishing Poisson brackets, i.e., are first class.