As I understand it, one way to describe a wave function is as a probability density distribution in phase space. The equations of motion for the wave function would describe how that density distribution evolves over time. Choice of the coordinate system in phase space should not affect the underlying behavior of the the system. If time were included as a coordinate in phase space, evolution of the system would correspond to intersection of the phase space distribution with a hyperplane perpendicular to the time dimension and moving along the time coordinate. This seems analogous to the evolution of a system as described in General Relativity.

So: If time is included in the definition of phase space, would the equations of motion in effect specify a curved geometry in phase space, where the quantity invariant under changes of coordinate system is local curvature as in General Relativity?

  • $\begingroup$ Are you particularly focused on the quantum mechanical issue here? One could talk about a coordinate-free description of the evolution of a probability distribution on classical phase space, for example. Once that is understood, one could turn their attention to the quantum case and its additional subtleties. $\endgroup$ – J. Murray Apr 3 at 17:21
  • $\begingroup$ @J. Murray, I guess it makes good sense to start with classical phase space. $\endgroup$ – S. McGrew Apr 3 at 17:57

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