On the Gupta-Bleuler Quantization of the Hamiltonian Systems with Anomalies

Abstract

The Gupta-Bleuler quantization method of QED can be generalized to canonically quantized constrained systems with quantum second-class constraints. Such constraints may originate either from the second-class constraints, already presented in the classical description of the theory, or may have their sources in quantum effects, in which case the theory is called anomalous. In this paper, I present a detailed description of how the Gupta-Bleuler ideas can be implemented in these cases and I argue that there are in principle no inconsistencies in quantum anomalous theories. Having quantized the anomalous theories canonically, I derive the path integral formulation of such theories and show that some new terms are necessarily present in this formulation. As an example, I show how the chiral Schwinger model can be quantized in the original fermionic formulation with no reference to the bosonized version used in the literature so far. * e-mail address: t18@ nikhef.nl

Harmonic BRST quantization of systems with irreducible holomorphic bosonic and fermionic constraints

“…[7] We remark that this splitting can also be used with the operatorial quantization in the case that classical first-class constraints become anomalous in the quantum theory, but our path-integral may need to be modified along the lines of ref. [8]. Operatorial constructions somewhat different from ours have been given by Hasiewicz et al [9] and recently by Marnelius.…”

Hamiltonian analysis of \mathsf {SO}(4,1) -constrained BF theory

“…The problem we are facing now is that for non-zero α the constraints are no longer first class and therefore we need a nonstandard procedure to handle them. One possibility would be Gupta-Bleuler quantization [19], but the required procedure of splitting the constraints into holomorphic and anti-holomorphic parts is technically complex and, presumably, leads to explicit breaking of Lorentz covariance (see [20] for discussion in a similar context.) Another possibility would be to make use of the master constraint program [21], [22], [23], and [20], but this is again technically involved.…”

BRST Cohomologies of Mixed and Second Class Constraints

“…The method presented here is a straightforward generalization of the approach proposed a long time ago by Gupta and Bleuler in the context of quantum electrodynamics [3,4]. There were several attempts [5][6][7][8] to join the BRST formalism with ideas of [3,4]. Unfortunately, they were premature at that time and not very successful.…”