Mechanics as a general-relativistic gauge field theory, and Relational Quantization

Abstract

We treat Mechanics as a 1-dimensional general-relativistic gauge field theory, Mechanical Field Theory (MFT), introducing what we call the Mechanical Field Space (MFS) and exploiting its bundle geometry. The diffeomorphism covariance of MFT encodes its relational character, arising - as in all general-relativistic physics - via the conjunction of a hole and a point-coincidence argument. Any putative "boundary problem", meaning the claim that boundaries break diffeomorphism and gauge symmetries, thereby dissolves. It is highlighted that the standard path integral (PI) on the MFS, the exact analogue of the PI used in gauge field theory, is conceptually and technically distinct from the standard PI of Quantum Mechanics. We then use the Dressing Field Method to give a manifestly invariant and relational reformulation of MFT, which reproduces the standard textbook formulation when a clock field is chosen as a (natural) dressing field. The dressed, or basic, PI on the MFS, defining Relational Quantization - i.e. the quantization of invariant relational d.o.f. - is shown to reproduce the standard PI of Quantum Mechanics. This establishes the soundness of Relational Quantization as a general guiding principle: We outline it for general-relativistic gauge field theories.