Observables and Dynamics Quantum to Classical from a Relativity Symmetry and Noncommutative-Geometric Perspective

Abstract

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.

Analysis on Complete Set of Fock States with Explicit Wavefunctions for the Covariant Harmonic Oscillator Problem

“…None of all that requires the observables to be Hermitian. In fact, in the noncommutative geometric picture the geometry is the dual object of the observable algebra, which is basically the representation of the group C * -algebra matching with the quantum theory as the representation of the basic/relativity symmetry of H R (3) [24]. We plan on going with the studies of a pseudounitary Lorentz covariant quantum theory along this line, with the latter generalized to the H R (1, 3) symmetry, results and lessons from which would help us fully understand the physics of the covariant harmonic oscillator solutions given here.…”

The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including all Known Relativity Symmetries, Classical and Quantum

“…Whatever nonzero value for 1 c we admit, Lorentz symmetry provides one with a correct description of the relevant physics, while the Newtonian model can never be confirmed to be (exactly) correct -merely correct up to some limitation in our measurements. Our proposed fundamental quantum relativity symmetry of SO (2,4) [7] comes from the idea of a fully stabilized symmetry, incorporating all the known fundamental constants G, , and c into the algebra structure, with and the Poincaré symmetry as (part of) a contraction limit. For the sake of convenience, we note first the SO (2,4) algebra is defined by…”

“…The primary goal of the recent articles [1,2] was to present a detailed picture of the feasibility of this whole scheme at the first level, namely that of the coset space representations. We will illustrate here not only that a (1 + 3)D picture of what we have done in Refs.…”

“…We will illustrate here not only that a (1 + 3)D picture of what we have done in Refs. [1,2] can be formulated, but also that the relevant coset space representations (of the H R (1, 3) relativity symmetry) can be incorporated in sequences of representations starting from SO(2, 4), pass through H R (1, 3), as well as including the extended symmetries of H R (3) or G(3), before eventually arriving at those relevant for Newtonian physics. We will, however, only briefly discuss the key notions relating to the full formulation of the associated dynamical theories, leaving such detailed investigations to be reported on in future publications.…”

“…As the Poincaré symmetry does not even admit a nontrivial U(1) central extension, this ten generator framework is certainly not enough. The SO (2,4) symmetry is supposed to possess some manner of invariant length and in-variant momentum, characterizing the noncommutativity among momentum and position observables, respectively, eventually. The X µ generate what are called "momentum boosts," which would be contracted to commuting momentum translations at the lower levels.…”