Abstract
The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a "phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a "quantum Darboux theorem" that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.
Highlights
A fundamental problem in quantum mechanics is to compute correlators hfje−ħiHðtf−tiÞjii; where the states jii and jfi are elements of some Hilbert space H, the operator His a quantum Hamiltonian, and ti and tf are classical times measured in some laboratory
Requiring nontrivial hthoalot n∇omisiefsl)a,tth(aencdorifrenlaetcoersshafrjyP, γqeuxoptðie−nRtiγnAg Þbjiyi only depends on the endpoints of γ and points jii and jfi in the fibers of HZ above these two endpoints: We attach the moniker quantum dynamical system to the data ðHZ; ∇Þ of a Hilbert bundle equipped with a flat connection ∇
We treat the Hamiltonian-Jacobi theory in terms of contact geometry. (Note that a modern treatment of HamiltonJacobi actions and their formal quantization is given in [4].) Thereafter we describe the quantum analog of the Darboux theorem for quantum dynamical systems [5]
Summary
A fundamental problem in quantum mechanics is to compute correlators hfje−ħiHðtf−tiÞjii; where the states jii and jfi are elements of some Hilbert space H, the operator His a quantum Hamiltonian, and ti and tf are classical times measured in some laboratory. It is useful to view the coordinates ti and tf as labels belonging to points in an odd-dimensional phase-spacetime manifold Z that corresponds to all possible classical laboratory measurements of generalized times, positions, and momenta. Requiring nontrivial hthoalot n∇omisiefsl)a,tth(aencdorifrenlaetcoersshafrjyP, γqeuxoptðie−nRtiγnAg Þbjiyi only depends on the endpoints of γ and points jii and jfi in the fibers of HZ above these two endpoints: We attach the moniker quantum dynamical system to the data ðHZ; ∇Þ of a Hilbert bundle equipped with a flat connection ∇. The remainder of the article is devoted to developing a diagrammatic calculus for computing the asymptotics of the gauge transformations appearing in the quantum Darboux theorem of [5], and applying these to the computation of correlators
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