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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

The Heisenberg picture is not privileged for the canonical quantization of dissipative systems

Nivaldo Agostinho Lemos

Published Oct 15, 1981• 14 citations
Publisher

Abstract

An old result of Brittin to the effect that dissipative systems belonging to a certain class lend themselves to canonical quantization in the Heisenberg picture but not in the Schr\"odinger picture is shown to be incorrect. It is shown that canonical quantization is impossible in the presence of nonconservative forces, no matter which of the two pictures one chooses. This restores the equivalence of the Heisenberg and Schr\"odinger pictures.

DOI
10.1103/physrevd.24.2338

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An old result of Brittin to the effect that dissipative systems belonging to a certain class lend themselves to canonical quantization in the Heisenberg picture but not in the Schr"odinger picture is shown to be incorrect. It is shown that canonical quantization is impossible in the presence of nonconservative forces, no matter which of the two pictures one chooses. This restores the equivalence of the Heisenberg and Schr"odinger pictures.
The work of E. Kanai shows that the quantization of dissipative systems by use of a Hamiltonian formalism involving the time explicitly leads to results in disagreement with experience. We discuss the quantization of dissipative systems in a manner such that the time does not enter explicitly. It is shown that in the Heisenberg representation the only kind of dissipative forces consistent with the formalism are those which are functions of the coordinates alone. There exists no Schr\"odinger representation unless the forces are non-dissipative.
The equivalence of Schrödinger and Heisenberg pictures is often thought to break down in nonlinear quantum mechanics due to the absence of a state-independent unitary propagator. We show that for Hamiltonians depending on the state through instantaneous expectation values, the equivalence is preserved. While the unitary intertwiner becomes initial-state dependent, it remains well-defined and ensures identical physical predictions in both pictures. We illustrate this with the analytically solvable examples of a nonlinear spin precession and a mean-field harmonic force, as well as in local field theory with mean-field coupling.