A Quantum Space behind Simple Quantum Mechanics

Abstract

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.

Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit

“…The dynamical theory is naturally a Hamiltonian theory from the symmetry of the phase space as symplectic geometry. The dynamics is better described on the algebra of observables as essentially the matching representation of the group C * -algebra [1,14,21]. Moreover, all those fit in well with the idea of the position and momentum operators as noncommutative coordinates of the phase space [12,13,21].…”

“…Our idea of having the pseudo-unitary metric on the space of states comes mostly from the intuition on the need to take the pseudo-unitary Minkowski metric seriously as a quantum notion, for the noncommutative position and momentum operators as coordinates for the space [21]. The quantum phase space, exactly the projective Hilbert space for the 'non-relativistic' theory, has been shown to serve as the quantum model of the physical space [1,14]. Well known as an infinite dimensional symplectic manifold, a noncommutative geometric picture of it has been presented [12] with the position and momentum operators as coordinates.…”

“…Each group element can be identified with a point in the H R (1, 3)/SO(1, 3) coset space [14,17]. X (ς) andP (ς) are operators on the abstract representation space H ς spanned by the p…”

“…The matching representation of the group C * -algebra [6,7], further extended to a proper class of distributions, gives the observable algebra as functions, and distributions, of position and momentum operators,X i = x i andP i = p i , as given by the Weyl-Wigner-Groenewold-Moyal(WWGM) formulation [8][9][10][11]. The operators α(p i , x i ) = α(p i , x i ) act as differential operators on the wavefunctions on coherent state basis φ(p i , x i ) by the Moyal star-product α φ; α β = (α β) .X i andP i can be seen as operator coordinates of the quantum phase space [12,13], which has been argued to serve as a proper quantum model for the physical space [1,14]. We naturally seek a Lorentz covariant version of that with a c → ∞ contraction of the symmetry taking the Lorentz boosts to that of the Galilean ones [15].…”

Special Relativity and Its Newtonian Limit from a Group Theoretical Perspective

“…There is, indeed, an additional question motivating this note, namely, how the symmetry perspective can be used to understand better quantum mechanics and its classical limit. The relativity symmetry contraction picture can be seen as a way to understand the classical phase space as an approximation to the quantum phase space [3], and even suggests a notion of a quantum model for the physical space [12]. In a broader scope, relativity symmetry deformations was much pursued as a probe to possible dynamical theories at the more fundamental levels [13][14][15][16][17].…”