Quantum realizations of Hilbert–Palatini second-class constraints

Abstract

In a classical theory of gravity, the Barbero–Immirzi parameter (η) appears as a topological coupling constant through the Lagrangian density containing the Hilbert–Palatini term and the Nieh–Yan invariant. In a quantum framework, the topological interpretation of η can be captured through a rescaling of the wavefunctional representing the Hilbert–Palatini theory, as in the case of the QCD vacuum angle. However, such a rescaling cannot be realized for pure gravity within the standard (Dirac) quantization procedure where the second-class constraints of Hilbert–Palatini theory are eliminated beforehand. Here, we present a different treatment of the Hilbert–Palatini second-class constraints in order to set up a general rescaling procedure (a) for gravity with or without matter and (b) for any choice of gauge (e.g. time gauge). The analysis is developed using the Gupta–Bleuler and the coherent state quantization methods.