Abstract
With approaching quantum/noncommutative models for the deep microscopic spacetime in mind, and inspired by our recent picture of the (projective) Hilbert space as the model of physical space behind basic quantum mechanics, we reformulate here the WWGM formalism starting from the canonical coherent states and taking wavefunctions as expansion coefficients in terms of this basis. This provides us with a transparent and coherent story of simple quantum dynamics where both the wavefunctions for the pure states and operators acting on them arise from the single space/algebra, which exactly includes the WWGM observable algebra. Altogether, putting the emphasis on building our theory out of the underlying relativity symmetry -- the centrally extended Galilean symmetry in the case at hand -- allows one to naturally derive both a kinematical and a dynamical description of a quantum particle, which moreover recovers the corresponding classical picture (understood in terms of the Koopman-von Neumann formalism) in the appropriate (relativity symmetry contraction) limit. Our formulation here is the most natural framework directly connecting all of the relevant mathematical notions and we hope it may help a general physicist better visualize and appreciate the noncommutative-geometric perspective behind quantum physics. It also helps to inspire and illustrate our perspective on looking at quantum mechanics and quantum physics in general in direct connection to the notion of quantum (deformed) relativity symmetries and the corresponding quantum/noncommutative models of spacetime as various levels of approximations all the way down to the Newtonian.
Highlights
Introduction to the Quantum Relativity PerspectiveSome years before the turn of the century, the idea that physical spacetime should be modeled, at least at the deep microscopic scale, by some form of noncommutative geometry [1] started to get more and more appreciation from physicists
There is no notion of configuration space and the phase space can only be obtained from the so-called projective representation of G(3), which really means a unitary representation of a bigger group, the U(1) central extension G (3)
Our key perspective is that the latter should be taken as the relativity symmetry behind quantum mechanics and the phase space taken as the quantum model for the physical space
Summary
Some years before the turn of the century, the idea that physical spacetime should be modeled, at least at the deep microscopic scale, by some form of noncommutative geometry [1] started to get more and more appreciation from physicists. We expect looking at the WWGM formalism through the Weyl-Wigner transform starting instead from the Hilbert space of wavefunctions over the coherent state basis will provide a interesting picture of quantum mechanics which would be suitable for the analysis of the classical limit. Taken as representation spaces of the relativity symmetry, the limit of the two picture under the contraction can be traced to give what are essentially the phase and configuration space of the classical Newtonian picture The latter is more directly given, naively, in the language of Hilbert spaces; the Koopman-von Neumann formulation—a part of the story to be described explicitly here.
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