Thebault, Karim P Y
(2010)
Symplectic reduction and the problem of time in nonrelativistic mechanics.
[Preprint]
Abstract
Symplectic reduction is a formal process through which degeneracy within the mathematical representations of physical systems displaying gauge symmetry can be controlled via the construction of a reduced phase space. Typically such reduced spaces provide us with a formalism for representing both instantaneous states and evolution uniquely and for this reason can be justifiably afforded the status of fundamental dynamical arena  the otiose structure having been eliminated from the original phase space. Essential to the application of symplectic reduction is the precept that the first class constraints (which feature in the Hamiltonian formalization of any gauge theory) are the relevant gauge generators. This prescription becomes highly problematic for reparameterization invariant theories within which the Hamiltonian itself is a constraint; not least because it would seem to render prima facie distinct stages of a history physically identical and observable functions changeless. Here we will consider this problem of time within nonrelativistic mechanical theory with a view to both more fully understanding the temporal structure of these timeless theories and better appreciating the corresponding issues in relativistic mechanics. For the case of nonrelativistic reparameterization invariant theory application of symplectic reduction will be demonstrated to be both unnecessary; since the degeneracy involved is benign; and inappropriate; since it leads to a trivial theory. With this antireductive position established we will then examine two rival methodologies for consistently representing change and observable functions within the original phase space before evaluating the relevant philosophical implications. We will conclude with a preview of the case against symplectic reduction being applied to canonical general relativity (which will be examined more fully in future work).
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