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Quantum systems that are weakly influenced by an uncontrolable external environment do not evolve unitarily. The Schrödinger picture of states evolving in Hilbert space must be abandoned along with traditional semiclassical (WKB) approximations. However, these methods can be adapted to describe the evolution of a density operator by doubling the corresponding classical phase space, because projection operators, dyadic operators and unitary operators all correspond to particular sorts of Lagrangian surfaces in double phase space.
Unitary evolution then results from the direct generalization of WKB theory. This relies on the classical motion of the corresponding Lagrangian surface, driven by a restricted double phase space Hamiltonian. When this constraint is lifted, decoherence, diffusion and dissipation are naturally included in a broad picture of nonunitary evolution. The new terms which modify the amplitude of the WKB-solution are derived from the Lindblad Equation that governs quantum Markovian sytems. |
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