Conserved symmetries in noncommutative quantum mechanics

Abstract

We consider a problem of the consistent deformation of physical system introducing a new features, but preserving its fundamental properties. In particular, we study how to implement the noncommutativity of space-time without violation of the rotational symmetry in quantum mechanics or the Lorentz symmetry in field theory. Since the canonical (Moyal) noncommutativity breaks the above symmetries one should work with more general case of coordinate-dependent noncommutative spaces, when the commutator between coordinates is a function of these coordinates. First we describe in general lines how to construct the quantum mechanics on coordinate-dependent noncommutative spaces. Then we consider the particular examples: the Hydrogen atom on rotationally invariant noncommutative space and the Dirac equation on covariant noncommutative space-time.

Energy-dependent noncommutative quantum mechanics

“…For free particle, namely V ( x, y) = 0, m * = m, B h = B e (E), and K h = k e (E). The Hamiltonian (55) gives a unified description of the quantum evolution for both the free particle and for the harmonic oscillator in the energy-dependent noncommutative geometry. By taking k = 0 we reobtain immediately the case of the free particle.…”

“…An interesting feature of this model is the dependence of nonlocality on the energy of the system, so that the increase of the energy leads to the increase in nonlocality. The physical properties of systems with dynamic noncommutativity were considered in [51][52][53][54][55][56][57][58]. A quantum mechanical system on a noncommutative space for which the structure constant is explicitly timedependent was investigated in [59], in a two-dimensional space with nonvanishing commutators for the coordinates X, Y and momenta P x , P y given by…”

“…An interesting feature of this model is the dependence of nonlocality on the energy of the system, so that the increase of the energy leads to the increase in nonlocality. The physical properties of systems with dynamic noncommutativity were considered in [51][52][53][54][55][56][57][58].…”

Effect of coordinate noncommutativity on the mass of a particle in a uniform field and the equivalence principle

“…It is worth noting that in the case of canonical version of noncommutative space (1)-(3) one faces the problem of rotational symmetry breaking [4,24]. Therefore, different classes 1 of noncommutative algebras were considered to preserve the rotational symmetry (see, for instance, [25,26] and references therein).…”

Minimal length, area, and volume in a space with noncommutativity of coordinates

“…У цiй статтi ми розглядаємо мiнiмальну довжину, площу та об'єм у некомутативному просторi (1)-(3). Зауважимо, що у тривимiрному просторi з некомутативнiстю координат канонiчного типу виникає проблема порушення сферичної симетрiї (див., для прикладу, [5,24,25]). У нашiй статтi [26] побудовано некомутативну алґебру, яка еквiвалентна алґебрi канонiчного типу та є сферично-симетричною.…”