A quantization quandary on the canonical road to quantum gravity

Abstract

Karim Thebault Canonical quantization techniques are generally considered to provide one of the most rigorous methodologies for passing from a classical to a quantum description of reality. For classical Hamiltonian systems with constraints a number of such techniques are available (i.e. gauge fixing, Dirac constraint quantization, BRST quantization and geometric quantization) but all are arguably equivalent to the quantization of an underlying reduced phase space that parameterizes the "true degrees of freedom" and displays a symplectic geometric structure. The philosophical coherence of making any ontological investment in such a space for the case of canonical general relativity will be questioned here. Further to this, the particular example of Dirac quantization will be critically examined. Under the Dirac scheme the classical constraint functions are interpreted as quantum constraint operators restricting the allowed state vectors. For canonical general relativity this leads to the Wheeler-de Witt equation and the infamous problem of time but, prima facie, seems to rely on our interpretation of the classical Poisson bracket algebra of constraints as the phase space realization of the theory's local symmetries (i.e. the group of space-time diffeomorphisms). As with the construction of an interpretively viable symplectic reduced phase space, this straight forward connection between constraints and local symmetry will be questioned for the case of GR. These issues cast doubt on the basis behind the derivation of the so-called wave function of the universe and give us some grounds for re-examining the entire canonical quantum gravity program as currently constituted.