Lagrangian quantum theory. IV. Schouten concomitants and the Dirac problem

Abstract

Hamilton's principle of stationary action is formulated in classical mechanics using the Lie algebra of Schouten concomitants of symmetric contravariant tensor fields on the configuration space of the system. Such a formulation is global and coordinate free. It is shown that a directly parallel formulation holds in quantum mechanics so long as all the Poisson brackets involved can be replaced in the quantum version by commutators in a canonical way. An example (where the Hamiltonian possesses a velocity dependent potential) in which this cannot be done is discussed and concluded that in this case the action is stationary only for a subclass of variations, namely those corresponding to Killing vector fields on the configuration manifold.