On the relation between rigging inner product and master constraint direct integral decomposition
Abstract
Canonical quantization of constrained systems with firstclass constraints via Dirac's operator constraint method proceeds by the theory of Rigged Hilbert spaces, sometimes also called refined algebraic quantization. This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the master constraint method has been proposed which replaces the individual constraints by a weighted sum of absolute squares of the constraints. Now the direct integral decomposition (DID) methods, which are closely related to Rigged Hilbert spaces, become available and have been successfully tested in various situations. It is relatively straightforward to relate the rigging inner product to the path integral that one obtains via reduced phase space methods. However, for the master constraint, this is not at all obvious. In this paper we find sufficient conditions under which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at least locally in phase space. Then the master constraint DID for those Abelian constraints can be directly related to the rigging map and therefore has a path integral formulation.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 September 2010
 DOI:
 10.1063/1.3486359
 arXiv:
 arXiv:0911.3431
 Bibcode:
 2010JMP....51i2501H
 Keywords:

 Dirac equation;
 Hilbert spaces;
 integral equations;
 Lie algebras;
 master equation;
 quantisation (quantum theory);
 03.65.Aa;
 03.65.Fd;
 03.65.Db;
 02.30.Rz;
 02.20.Sv;
 Algebraic methods;
 Functional analytical methods;
 Integral equations;
 Lie algebras of Lie groups;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 25 pages