Quantum geodesics in quantum mechanics

Abstract: We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $Ω^1$ depending on the Hamiltonian $H=p^2/2m+V(x)$ and a flat quantum connection with torsion on it such that a quantum formulation of autoparallel curves (or `geodesics') reduces to Schrödinger's equation. The connection is compatible with a natural quantum symplectic structure and associated generalised quantum metric. A remnant of our approach also works on any symplectic manifold where, by extending the calculus, we can encode any hamiltonian flow as `geodesics' for a certain connection with torsion which is moreover compatible with an extended symplectic structure. Thus we formulate ordinary quantum mechanics in a way that more resembles gravity rather than the more well-studied idea of formulating geometry in a more quantum manner. We then apply the same approach to the Klein Gordon equation on Minkowski space with a background electromagnetic field, formulating quantum `geodesics' on the relevant relativistic Heisenberg algebra. Examples include a proper time relativistic free particle wave packet and a hydrogen-like atom. based on a joint work with Shahn Majid.

general relativity and quantum cosmologyHEP - theorymathematical physicsquantum algebra

Audience: researchers in the topic

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Mathematical Physics Seminar

Series comments: Description: Noncommutative geometry, field theory, gravity.