Curved-space quantization: Toward a resolution of the Dirac versus reduced quantization question

It is well known that Dirac quantization of gauge theories is not, in general, equivalent to reduced quantization. When both approaches are self-consistent some additional criterion must be found in order to decide which approach is more natural, or correct. Now, in many cases quantization on the physical degrees of freedom is properly curved-space quantization, with a highly nontrivial curvature: neither constant nor Ricci flat. On the other hand, the configuration space of the unreduced gauge theory is often (Ricci) flat, which makes Dirac quantization considerably simpler. We show that the natural ‘‘minimal’’ Dirac quantization scheme, together with certain restrictions we impose on the observables, is sufficient to make the quantum commutator of quadratic observables free of van Hove anomalies. This means the ‘‘minimal’’ Dirac quantization (acting in the physical Hilbert space) is actually a curved-space quantization scheme suitable for the type of curvature mentioned above, at least within a restricted (but still interesting) class of observables. In fact, we demonstrate that this curved-space quantization scheme, unlike ‘‘minimal’’ reduced quantization, has remarkable similarities with other curved-space quantization schemes proposed elsewhere. However, unlike these other schemes, it contains a piece which depends in an essential way on the gauge structure of the unreduced theory, and so could not have been guessed working strictly from within the classical reduced theory.