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Generalized quantum dynamics

We propose a generalization of Heisenberg picture quantum mechanics in which a Lagrangian and Hamiltonian dynamics is formulated directly for dynamical systems on a manifold with non--commuting coordinates, which act as operators on an underlying Hilbert space. This is accomplished by defining the Lagrangian and Hamiltonian as the real part of a graded total trace over the underlying Hilbert space, permitting a consistent definition of the first variational derivative with respect to a general operator--valued coordinate. The Hamiltonian form of the equations is expressed in terms of a generalized bracket operation, which is conjectured to obey a Jacobi identity. The formalism permits the natural implementation of gauge invariance under operator--valued gauge transformations. When an operator Hamiltonian exists as well as a total trace Hamiltonian, as is generally the case in complex quantum mechanics, one can make an operator gauge transformation from the Heisenberg to the Schrödinger picture. When applied to complex quantum mechanical systems with one bosonic or fermionic degree of freedom, the formalism gives the usual operator equations of motion, with the canonical commutation relations emerging as constraints associated with the operator gauge invariance. More generally, our methods permit the formulation of quaternionic quantum field theories with operator--valued gauge invariance, in which we conjecture that the operator constraints act as a generalization of the usual canonical commutators.