On the Second Quantization of Gauge Systems

Abstract

Within the stochastic quantization scheme the second quantization of constrained Hamiltonian systems is exemplified for the case of the relativistic point particle. In addition, the related quantization of the abelian Chern-Simons field theory is discussed. 389 One of the most appealing features of stochastic quantization lies in the quantiza­ tion of gauge field theories where, remarkably, no gauge fixing terms are needed.l) As a consequence stochastic quantization preserves the full gauge symmetry throughout and constitutes a manifestly gauge covariant quantization scheme. In the first part of my contribution I would like to summarize some generaliza­ tions2>.a> of the stochastic quantization scheme to study the second quantization of classical Hamiltonian systems with constraints. 4 >- 6 > A key ingredient for this approach is the use of the ERST formalism 7 >- 9 > which constitutes a powerful and elegant tool for dealing with constrained systems. I discuss the method for the simple case of the interacting relativistic point particle; it will be seen upon second quantiza­ tion that the standard perturbation expansion of a self interacting scalar field is obtained. In the second part I would like to point out some similarities of the above method to the stochastic quantization of the Chern-Simons field theory. 10 >-Iz> Here I restrict myself to the abelian case only. A classical system with phase space variables (x, p) is constrained or possesses a gauge invariance, if the Legendre transformation between the velocities and the momenta is singular. As a consequence not all the velocities can be expressed by the momenta and some of the Euler-Lagrange equations reduce to constraints Gi(x, p)=O, i=l, ···, m.