On the geometry of quantum constrained systems

Geometric methods have proved useful in the Hamiltonian description of classical constrained systems. We develop the corresponding geometry for quantum constrained systems, using the formulation of quantum mechanics in which the space of rays $\text{Ps}$ is a K"ahler manifold and unitary transformations are its canonical transformations. The Dirac conditions ${\widehat{C}}_i\mathrm{\Psi }=0$ define the zero set of a single function $\text{Cq}=\text{ev}\text{Mop}$, the expectation value of the master constraint operator. Our central identity is $\text{Cq}=\mathrm{\mid }\mu {\mathrm{\mid }}^2+\frac{1}{2}\mathrm{\parallel }{X}_{\mu }{\mathrm{\parallel }}^2$, where $\mu =(\text{ev}{\widehat{C}}_i)$ is the moment map of the gauge action on $\text{Ps}$ and $X_{\mu }$ its fundamental vector fields: the quantum condition imposes the classical-type condition $\mu =0$ \emph{and} the collapse of the gauge orbits to points. This explains why no second reduction is needed. The physical state space is a totally geodesic K"ahler submanifold. $\text{Cq}$ is a Morse–Bott function which gradient flow is heat-kernel-smeared group averaging, converges to the nearest-point projection onto physical states, and its fibres are invariant under the complexified gauge group. Physical observables are characterised to all orders by the double-commutator condition, which is equivalent to $[\widehat{O},{\widehat{P}}_0]=0$, with ${\widehat{P}}_0$ the projector onto physical states. When zero lies in the continuous spectrum, the gradient flow converges to Dirac states in the sense of Grundling and Hurst, and its large-time asymptotics reproduce the rigging map of refined algebraic quantization. Finally, pulled back along coherent states, $\text{Cq}$ equals the classical master constraint plus an $O(\mathrm{\hslash })$ term given by the squared speed of the classical gauge orbits. Together these results connect the Dirac procedure, the master constraint programme, refined algebraic quantization and symplectic reduction within one geometric picture.