2005/11/16 by Giulio Ruffini, Ruffini, G.
Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories #Quantum and Classical Electrodynamics #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.gr-qc/0511088
openalex publication_date 2005/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
I study the canonical formulation and quantization of some simple parametrized systems, including the non-relativistic parametrized particle and the relativistic parametrized particle. Using Dirac's formalism I construct for each case the classical reduced phase space and study the dependence on the gauge fixing used. Two separate features of these systems can make this construction difficult: the actions are not invariant at the boundaries, and the constraints may have disconnected solution spaces. The relativistic particle is affected by both, while the non-relativistic particle displays only by the first. Analyzing the role of canonical transformations in the reduced phase space, I show that a change of gauge fixing is equivalent to a canonical transformation. In the relativistic case, quantization of one branch of the constraint at the time is applied and I analyze the electromagenetic backgrounds in which it is possible to quantize simultaneously both branches and still obtain a covariant unitary quantum theory. To preserve unitarity and space-time covariance, second quantization is needed unless there is no electric field. I motivate a definition of the inner product in all these cases and derive the Klein-Gordon inner product for the relativistic case. I construct phase space path integral representations for amplitudes for the BFV and the Faddeev path integrals, from which the path integrals in coordinate space (Faddeev-Popov and geometric path integrals) are derived.