Motivated by a recent investigation into the notion of a quantum space underlying ordinary quantum mechanics, we reformulate here the WWGM formalism with canonical coherent states and wavefunctions as expansion coefficients in terms of this basis as the starting point. It turns out that this provides us with a transparent and coherent story of simple quantum dynamics where both states (pure and mixed, making use of the Tomita representation), as wavefunctions in Hilbert spaces, and observables arise from a single space/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 (free) 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.

In physics, experiments ultimately inform us about what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the configuration space of a free particle (or the center of mass of a closed system of particles). This configuration space (as well as phase space) can be constructed as a representation space for the relativity symmetry. From the corresponding quantum symmetry, we illustrate the construction of a quantum configuration space, similar to that of quantum phase space, and recover the classical picture as an approximation through a contraction of the (relativity) symmetry and its representations. The quantum Hilbert space reduces into a sum of one-dimensional representations for the observable algebra, with the only admissible states given by coherent states and position eigenstates for the phase and configuration space pictures, respectively. This analysis, founded firmly on known physics, provides a quantum picture of physical space beyond that of a finite-dimensional manifold and provides a crucial first link for any theoretical model of quantum space-time at levels beyond simple quantum mechanics. It also suggests looking at quantum physics from a different perspective.

In this article, we utilize the insights gleaned from our recent formulation of space(-time), as well as dynamical picture of quantum mechanics and its classical approximation, from the relativity symmetry perspective in order to push further into the realm of the proposed fundamental relativity symmetry SO(2, 4) of our quantum relativity project. We explicitly trace how the diverse actors in this story change through various contraction limits, paying careful attention to the relevant physical units, in order to place all known relativity theories -quantum and classical -within a single framework. More specifically, we explore both of the possible contractions of SO(2, 4) and its coset spaces in order to determine how best to recover the lower-level theories. These include both new models and all familiar theories, as well as quantum and classical dynamics with and without Einsteinian special relativity. Along the way, we also find connections with covariant quantum mechanics. The emphasis of this article rests on the ability of this language to not only encompass all known physical theories, but to also provide a path for extensions. It will serve as the basic background for more detailed formulations of the dynamical theories at each level, as well as the exact connections amongst them.This article is a study within our group's Quantum Relativity Project, the key idea of which is to formulate pictures of quantum spacetime and the related dynamics from a (relativity) symmetry perspective. We expect the models of quantum spacetime to be of a noncommutative nature, seeing them as intrinsically quantum; hence, they may not be fully described by any real number geometric picture of finite dimension. The latter, as classical/commutative geometry, is of course applicable to modeling classical spacetime, as in the Newtonian, Minkowskian, as well as the dynamical Einstein general-relativistic spacetime. What we should bear in mind is that all of these are only theoretical models of the notion of spacetime, and as such are only as good as the corresponding model of physical dynamics. The pursuit of better models of fundamental physics should go hand-in-hand with the pursuit of better models of spacetime. Real number geometry may not maintain its role as a successful, not to mention convenient, tool in this endeavor. The point of view underlying our project highlights the difficulty one encounters in appreciating traditional quantum mechanics. Quantum mechanics indeed sounds strange, or even counter-intuitive, when thinking about it as a theory of mechanics on classical spacetime. What we hope to convince people of, however, is that it is no less intuitive than classical mechanics when one thinks about it with the proper model of a quantum (physical) space in hand [1, 2] (a conceptual discussion has been presented in an article in Chinese [3]). A quantum particle, for example, always has a definite position within the quantum model of physical space, though it notably cannot be modeled or represented by a finite number o...

In this pedagogical article, we explore a powerful language for describing the notion of spacetime and particle dynamics intrinsic to a given fundamental physical theory, focusing on special relativity and its Newtonian limit. The starting point of the formulation is the representations of the relativity symmetries. Moreover, that seriously furnishes—via the notion of symmetry contractions—a natural way in which one can understand how the Newtonian theory arises as an approximation to Einstein’s theory. We begin with the Poincaré symmetry underlying special relativity and the nature of Minkowski spacetime as a coset representation space of the algebra and the group. Then, we proceed to the parallel for the phase space of a spin zero particle, in relation to which we present the full scheme for its dynamics under the Hamiltonian formulation, illustrating that as essentially the symmetry feature of the phase space geometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with its space-time, phase space, and dynamics under the appropriate relativity symmetry contraction is presented. While all notions involved are well established, the systematic presentation of that story as one coherent picture fills a gap in the literature on the subject matter.

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