On the Generality of Refined Algebraic Quantization

TL;DR

This paper analyzes the generality of Refined Algebraic Quantization (RAQ) within the Dirac constraint quantization framework, focusing on systems whose constraints form a Lie algebra with structure constants. It formalizes a precise Dirac procedure by fixing a $*$-algebra of observables ${\cal A}_{obs}$ on a dense subspace ${\Phi}$ of an auxiliary Hilbert space ${\cal H}_{aux}$, and then seeks physical states in the dual ${\Phi^*}$ via a Gel'fand triple ${\Phi}\hookrightarrow {\cal H}_{aux} \hookrightarrow {\Phi^*}$. RAQ is shown to define the physical inner product through a rigging map $\eta: {\Phi} \to {\Phi^*}$, with the inner product $(\eta(\phi_1),\eta(\phi_2))_{phys}=\phi_1[\eta(\phi_2)]=\eta(\phi_2)[\phi_1]$, and the physical space ${\cal H}'_{phys}$ as the closure of $Im(\eta)$, while ensuring $\eta$ intertwines the action of ${\cal A}_{obs}$. Under technical conditions on operator domains, RAQ can reproduce a nontrivial subrepresentation of the observables on ${\cal H}_{phys}$, and with an extra assumption about superselection sectors, can recover the full representation; the work also discusses the role of unbounded operators, the potential shift to group averaging results, and implications for extending RAQ beyond simple Lie algebras, such as in gravitational contexts.

Abstract

The Dirac quantization `procedure' for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular interpretation of the Dirac procedure known as refined algebraic quantization. We find technical conditions under which refined algebraic quantization can reproduce the general implementation of the Dirac scheme for systems whose constraints form a Lie algebra with structure constants. The main result is that, under appropriate conditions, the choice of an inner product on the physical states is equivalent to the choice of a ``rigging map'' in refined algebraic quantization.