Abstract
The Schrödinger and Heisenberg evolution operators are represented in phase space T*
n by their Weyl symbols. Their semiclassical approximations are constructed in the short and long time regimes. For both evolution problems, the WKB representation is purely geometrical: the amplitudes are functions of a Poisson bracket and the phase is the symplectic area of a region in T*n bounded by trajectories and chords. A unified approach to the Schrödinger and Heisenberg semiclassical evolutions is developed by introducing an extended phase space χ2 ≡ T*(T*n). In this setting, Maslov's pseudodifferential operator version of WKB analysis applies and represents these two problems via a common higher dimensional Schrödinger evolution, but with different extended Hamiltonians. The evolution of a Lagrangian manifold in χ2, defined by initial data, controls the phase, amplitude and caustic behaviour. The symplectic area phases arise as a solution of a boundary condition problem in χ2. Various applications and examples are considered. Physically important observables generally have symbols that are free from rapidly oscillating phases. The semiclassical Heisenberg evolution in this context has been traditionally realized as an ℏ power series expansion whose leading term is classical transport. The extended Heisenberg Hamiltonian has a reflection symmetry that ensures this behaviour. If the WKB initial phase is zero, it remains so for all time, and semiclassical dynamics reduces to classical flow plus finite order ℏ corrections.