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arXiv:1703.04128 · Journal of High Energy Physics Gravitation and Cosmology

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

Chuan Sheng Chew, Otto C. W. Kong, Jason Payne

Published Jan 1, 2019• 12 citations
arXivHTMLPDF

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.

DOI
10.4236/jhepgc.2019.53031

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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.
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 \textit{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.
We present here what we consider a new picture of quantum mechanics with the position and momentum observables as coordinates of the usual quantum phase space of a single particle, which also serves as the model of the physical space. To minimize the mathematics involved, we stick here to the Hilbert space picture of the phase space. We argue that a quantum observable should be seen as taking noncommutative values each of which can equivalently be represented by an infinite number of real numbers. The six noncommutative values of the position and momentum observables hence serve as an alternative system of coordinates for the Hilbert space. The values can, at least in principle, be experimentally determined. This can be seen as a complete resolution of the Einstein–Bohr debate that Einstein would probably be happy with. What we have is a solid noncommutative geometrical picture of the physical space, or spacetime, beyond the Newtonian and Einsteinian framework that is sure to be relevant to Nature directly coupled with the idea that each physical quantity should better be seen as having a value beyond what can be represented by a number. We finish by sketching some implications of the results for the physics and mathematics of quantum spacetime in general.
Abstract Quantum reference frame as in the example of observing a quantum particle from another has been a topic of much recent interest. Quantum spatial translations, quantum rotations, and quantum Lorentz boosts in the sense have been studied to some extent. The article aims at using a consistent formulation of all that to give a full picture of what would be the symmetry of Special Quantum Relativity as special relativity as seen from quantum particles. We first look at a direct formulation of the mathematical quantum analog of any classical one-parameter Lie group of such transformations suggested by our Heisenberg picture thinking of quantum mechanics. A quantum rotation and a quantum Lorentz boost are presented. Analysis of the compositions of two such transformations shows unnatural results with various problems, especially with their commutators. From a physical point of view, we argue for a modified formulation. A quantum reference frame transformation, as the description of firstly the spacetime position of an object, idealized as a particle, as seen from another has to involve a full set of four-coordinate quantum quantities instead of one in a single coordinate only. Unlike the classical case, we illustrate that the analog of a change by the zero classical coordinate values has nontrivial effects in the transformation. A consistent full picture of the symmetry is then obtained. Results that could be interpreted essentially as the quantum analogs of the translation of a single coordinate observable, including angular ones for a rotation or a boost, can be retrieved from that but only as approximations. Under the traditional perspective of relativity symmetry being about spacetime reference frames, our task is then completed. We further look at the quantum version of Heisenberg–Weyl symmetry and perspectives on its relation with the subject matter and argue for the necessity of our background noncommutative geometric perspective for a conceptual consistent story.
The article gives an introductory summary of our recent results from the perspective of the new conceptual notion of noncommutative values of quantum observables, where basic quantum mechanics is seen a theory of particle dynamics that is intrinsically noncommutative.Real numbers as an algebra is really an abstract mathematical model for the description of values of physical quantities.Quantum mechanics illustrates the limitation of that perspective.The noncommutative values preserve the algebraic relations among the observables in their values.Such a value contains the complete information the theory presents for an observable on a given state.All physical quantities, including coordinates of the physical space for describing the location of a particle take definite noncommutative values.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ\hbarℏ-dependent ⋆\star⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.