Algebraic and geometric aspects of generalized quantum dynamics.

We briefly discuss some algebraic and geometric aspects of the generalized Poisson bracket and non–commutative phase space for generalized quantum dynamics, which are analogous to properties of the classical Poisson bracket and ordinary symplectic structure. \pacs{} Typeset using REVTEX 1 Recently, one of us (SLA) has proposed a generalization of Heisenberg picture quantum mechanics, termed generalized quantum dynamics, which gives a Hamiltonian dynamics for general non–commutative degrees of freedom. [1,2] The formalism permits the direct derivation of equations of motion for field operators, without first proceeding through the intermediate step of “quantizing” a classical theory. In a complex Hilbert space, generalized quantum dynamics gives results compatible with standard canonical quantization. It is also applicable to the construction of quantum field theories in quaternionic Hilbert spaces, where canonical methods fail, basically because the matrix elements of operators are themselves elements of the non–commutative quaternion algebra. It is hoped that the methods of generalized quantum dynamics will facilitate answering the question of whether quantum field theories in quaternionic Hilbert space are relevant to the unification of the standard model forces with gravitation at energies above the GUT scale. As applied to quantum theory, generalized quantum dynamics is formulated by defining a Hilbert space VH (based either on complex number or quaternionic scalars) which is the direct sum of a bosonic space V + H and a fermionic space V − H . Next, following Witten [3], one defines an operator (−1) with eigenvalue +1 for states in V + H and −1 for states in V − H . Finally, one needs a trace operation TrO for a general operator O, defined by Tr O = ReTr (−1)O = Re ∑