Noncommutative coordinate picture of the quantum phase space

Kong, Otto C. W.; Liu, Wei Yin

doi:10.1016/j.cjph.2021.03.014

Cited by 7 publications

(19 citation statements)

References 21 publications

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“…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].…”

Section: Group Theoretically-based Wwgm Framework With Wavefunctions In Coherent State Basismentioning

confidence: 82%

“…Quantum dynamics is completely symplectic, whether described in the Schrödinger picture in terms of real/complex coordinates of the (projective) Hilbert space or the Heisenberg picture as a description in terms of the noncommutative coordinates [12]. The explicit dynamical equation of motion is to be seen as the transformations generated by a physical Hamiltonian characterized by an evolution parameter.…”

Section: The Algebra Of Observables Symmetries and Dynamicsmentioning

confidence: 99%

“…It also gives a direct and intuitive picture of the phase space geometry too. The original position and momentum operators, x µ and p µ , can be seen as noncommutative coordinates of the noncommutative symplectic geometry of the phase space [12]. The contracted versions as x c µ and p c µ are the classical phase space coordinates with no noncommutativity left.…”

Section: Contraction To Classical Theory In Briefmentioning

confidence: 99%

“…As mentioned above, at the quantum level, a G s = G s (p µ , x µ ) operator is a Hamiltonian function of the phase space coordinates p and x , and the corresponding −1 2iG s is the Hamiltonian vector field. It is, of course, well known since Dirac that what has now been identified as a quantum Poisson bracket 1 2i [•, •] [12,13] (and see references therein) reduces exactly to a classical Poisson bracket, which works in our formulation, explicitly shown in Ref. [1]; i.e.,…”

Section: Contraction To Classical Theory In Briefmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

2020

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We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.

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