Generalized quantum dynamics as pre-quantum mechanics
Stephen L. Adler, Millard, A C
We address the issue of when generalized quantum dyanmics, based on a total trace Hamiltonian {\\bf H}, reduces to Heisenberg picture complex quantum mechanics. We begin by showing that when {\\bf H}={\\rm{\\bf Tr}} H_{OP}, with H_{OP} a Weyl ordered operator Hamiltonian, then the generalized quantum dynamics operator equations of motion agree with those obtained from H_{OP} in the Heisenberg picture by using canonical commutation relations. Hence generalized quantum dynamics conicides with complex quantum mechanics in this case, provided the initial values of the operators obey the canonical algebra. The remainder of the paper is devoted to a study of when this initial value condition is satisfied. We first show that for any total trace Hamiltonian which involves no noncommutative constants, there is a conserved operator \\tilde C which is closely related to the canonical algebra. We demonstrate that for a system in equilbrium, the equilbrium ensemble of initial values is governed by the distribution exp(-|\\tilde C|/\\kappa). An argument analogous to that for equipartition of energy, together with the assumption that ensemble averages of products can be replaced by products of ensemble averages, then implies that the ensemble averaged operator initial values obey the standard canonical commutation relations of complex quantum mechanics, with the imaginary unit I identified as -\\tilde C / |\\tilde C|, and with the constant \\kappa playing the role of the reduced Planck constant \\hbar. Thus there is a well--defined sense in which complex quantum mechanics and quantum field theory emerge as a statistical approximation to generalized quantum dynamics.
•Aug 16, 1995•37 citations