Lagrangian quantum theory. IV. Schouten concomitants and the Dirac problem
- 1 February 1975
- journal article
- Published by IOP Publishing in Journal of Physics A: General Physics
- Vol. 8 (2) , 179-185
- https://doi.org/10.1088/0305-4470/8/2/007
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.Keywords
This publication has 6 references indexed in Scilit:
- Lagrangian quantum theory. III. Coordinate-free formulationJournal of Physics A: Mathematical, Nuclear and General, 1974
- Lagrangian quantum theory (II). Several degrees of freedomNuclear Physics B, 1973
- On Killing tensors and constants of motionJournal of Mathematical Physics, 1973
- Lagrangian quantum theoryNuclear Physics B, 1973
- Velocity-Dependent Potentials in the Heisenberg PicturePhysical Review B, 1969
- Symmetric tensors of the second order whose first covariant derivatives are zeroTransactions of the American Mathematical Society, 1923