Quantization of Dynamical Symplectic Reduction
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Communications in
Mathematical Physics
Quantization of Dynamical Symplectic Reduction Martin Bojowald1 , Artur Tsobanjan2 1 Institute for Gravitation and the Cosmos, The Pennsylvania State University, 104 Davey Lab, University
Park, PA 16802, USA. E-mail: [email protected]
2 King’s College, 133 North River Street, Wilkes-Barre, PA 18711, USA. E-mail: [email protected]
Received: 3 January 2020 / Accepted: 6 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: A long-standing problem in quantum gravity and cosmology is the quantization of systems in which time evolution is generated by a constraint that must vanish on solutions. Here, an algebraic formulation of this problem is presented, together with new structures and results, which show that specific conditions need to be satisfied in order for well-defined evolution to be possible. 1. Introduction When time-reparameterization-invariant dynamical systems are cast as Hamiltonian theories on a symplectic manifold one finds that time evolution and time-reparameterization flows are generated by one and the same phase-space function—the Hamiltonian constraint. The straightforward application of the usual methods of symplectic reduction to such dynamically constrained systems has the undesirable side-effect of also removing their dynamics, and needs to be replaced by dynamical syplectic reduction. This paper describes a method of dynamical reduction for the quantized versions of such systems, where non-commutativity leads to a host of additional complications. However, since dynamically constrained systems are rarely studied outside of canonical approaches to quantum gravity we dedicate most of this introductory section to the review of their classical (i.e. non-quantum) treatment. Our main results and the structure of the rest of this manuscript are outlined at the end of the introduction. Given a symplectic manifold (M, ) and C ∈ C ∞ (M), the symplectic reduction [1] M/C of M by C is the orbit space of M ⊃ MC : C = 0 with respect to the gauge flow FC () = exp( X C ) in MC generated by the Hamiltonian vector field X C of C, dC = (X C , ·). Because L X C C = (X C , X C ) = 0, the flow preserves MC , and the orbit space inherits a unique symplectic form M/C from the presymplectic form i ∗ on MC , where i : MC → M is the inclusion of MC in M. The set of observables of the constrained system, which solve the constraint equation C = 0 and are invariant under the gauge flow, is given by C ∞ (M/C).
M. Bojowald, A. Tsobanjan
In addition to implementing a constraint C = 0 by symplectic reduction, physical systems usually require the definition of a dynamical flow. The canonical way is to select a Hamiltonian function H ∈ C ∞ (M) which generates the dynamical flow FH (t) = exp(t X H ) with the Hamiltonian vector field X H of H . A dynamical flow in the presence of a constraint C = 0 is consistent if it preserves the constraint surface, that is, X H C = (X C , X H ) = −{C, H } = 0 on MC with the Poisson bracket {·, ·}
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