On the geometry of quantum constrained systems

Abstract

The use of geometric methods has proved useful in the Hamiltonian description of classical constrained systems. Here we put forward the first steps towards 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, which 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–Bergmann procedure—through its geometric implementation—and the master constraint program.