On the geometry of quantum constrained systems

The use of geometric methods has proved useful in the hamiltonian description of classical constrained systems. In this note we provide the first steps toward the description of the geometry of quantum constrained systems. We make use of the geometric formulation of quantum theory in which unitary transformations (including time evolution) can be seen, just as in the classical case, as finite canonical transformations on the quantum state space. We compare from this perspective the classical and quantum formalisms and argue that there is an important difference between them, that suggests that the condition on observables to become physical is through the double commutator with the square of the constraint operator. This provides a bridge between the standard Dirac procedure --through its geometric implementation-- and the Master Constraint program.