Thebault, Karim P Y (2010) Symplectic reduction and the problem of time in nonrelativistic mechanics. [Preprint]

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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 fun- damental 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 formal- ization 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 non-relativistic me- chanical theory with a view to both more fully understanding the temporal struc- ture 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 unnec- essary; since the degeneracy involved is benign; and inappropriate; since it leads to a trivial theory. With this anti-reductive position established we will then examine two rival methodologies for consistently representing change and observable func- tions within the original phase space before evaluating the relevant philosophical implications. We will conclude with a preview of the case against symplectic re- duction being applied to canonical general relativity (which will be examined more fully in future work).

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Item Type:	Preprint
Creators:	Creators Email ORCID Thebault, Karim P Y karim.thebault@gmail.com
Keywords:	Problem of Time, Constraints, Symplectic Reduction, Gauge, Quantum Gravity, Reparameterization Invariance
Subjects:	Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances
Depositing User:	Dr Karim Thebault
Date Deposited:	23 Nov 2010 13:51
Last Modified:	19 Dec 2010 14:44
Item ID:	8404
Subjects:	Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances
Date:	23 November 2010
URI:	https://philsci-archive.pitt.edu/id/eprint/8404

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• Symplectic reduction and the problem of time in nonrelativistic mechanics. (deposited 23 Nov 2010 13:51) [Currently Displayed]
  • Symplectic reduction and the problem of time in nonrelativistic mechanics. (deposited 19 Dec 2010 14:44)
    • Symplectic reduction and the problem of time in nonrelativistic mechanics. (deposited 25 May 2011 16:06)

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