Canonical quantisation of constrained systems with first class constraints
via Dirac's operator constraint method proceeds by the thory of Rigged Hilbert
spaces, sometimes also called Refined Algebraic Quantisation (RAQ). 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 methods (DID), 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.Comment: 25 page
