Remarks on the Formulation of Quantum Mechanics on Noncommutative Phase Spaces

Abstract

We consider the probabilistic description of nonrelativistic, spinless one-particle classical mechanics, and immerse the particle in a deformed noncommutative phase space in which position coordinates do not commute among themselves and also with canonically conjugate momenta. With a postulated normalized distribution function in the quantum domain, the square of the Dirac delta density distribution in the classical case is properly realised in noncommutative phase space and it serves as the quantum condition. With only these inputs, we pull out the entire formalisms of noncommutative quantum mechanics in phase space and in Hilbert space, and elegantly establish the link between classical and quantum formalisms and between Hilbert space and phase space formalisms of noncommutative quantum mechanics. Also, we show that the distribution function in this case possesses 'twisted' Galilean symmetry.

25 pages, JHEP3 style; minor changes; Published in JHEP

References in corpus (5)

• Lectures on Noncommutative Geometry
• Twisted Galilean symmetry and the Pauli principle at low energies
• U(1) gauge invariant noncommutative Schrödinger theory and gravity
• Very Basic Noncommutative Geometry
• Noncommutative Field Theory: Nonrelativistic Fermionic Field Coupled to the Chern-Simons Field in 2+1 Dimensions

Cited by in corpus (7)

• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Probing Noncommutativities of Phase Space by Using Persistent Charged Current and Its Asymmetry
• On the Plethora of Representations Arising in Noncommutative Quantum Mechanics and An Explicit Construction of Noncommutative 4-tori
• Constrains of Charge-to-Mass Ratios on Noncommutative Phase Space
• On the Quantization of Length in Noncommutative Spaces
• Deformation of Noncommutative Quantum Mechanics
• Quantization of Length in Spaces with Position-Dependent Noncommutativity