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Physica Scripta

Displacement formulae for noncommutative operators

Bailey C. Hsu, Jean-François S. Van Huele

Published Dec 1, 2009• 2 citations
Publisher

Abstract

We present analytical expressions for displacements that involve noncommutative operators. We generalize the standard displacement formula D a f(x)=f(x-a) by considering successively the displacement effect of momentum operators in noncommutative quantum mechanics [p i ,p j ]=iθ ij , d-dimensional radial momentum operators p r (d)=- iℏ(∂ r +(d−1)/2) and spin-dependent operators. The noncommutativity results in three characteristic effects: the appearance of phase factors, space-modulated displacements and staggered spin-dependent displacements.

DOI
10.1088/0031-8949/80/06/065011

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We present analytical expressions for displacements that involve noncommutative operators. We generalize the standard displacement formula D a f(x)=f(x-a) by considering successively the displacement effect of momentum operators in noncommutative quantum mechanics [p i ,p j ]=iθ ij , d-dimensional radial momentum operators p r (d)=- iℏ(∂ r +(d−1)/2) and spin-dependent operators. The noncommutativity results in three characteristic effects: the appearance of phase factors, space-modulated displacements and staggered spin-dependent displacements.
We present analytical expressions for displacements that involve noncommutative operators. We generalize the standard displacement formula Da f(x) = f(x a) by considering successively the displacement effect of momentum operators in noncommutative quantum mechanics [pi, pj] = i ij, d-dimensional radial momentum operators p (d) r = i¯(@r +(d 1)/2) and spin-dependent operators. The noncommutativity results in three characteristic effects: the appearance of phase factors, space-modulated displacements and staggered spin-dependent displacements.
Abstract In quantum mechanics, the operator representing the displacement of a system in position or momentum is always accompanied by a path-dependent phase factor. In particular, two non-parallel displacements in phase space do not compose together in a simple way, and the order of these displacements leads to different displacement composition phase factors. These phase factors are often attributed to the nonzero commutator between quantum position and momentum operators, but such a mathematical explanation might be unsatisfactory to students who are after more physical insight. We present a couple of simple demonstrations, using classical wave mechanics and classical particle mechanics, that provide some physical intuition for the phase associated with displacement operators.
Some consequences of promoting the object of noncommutativity theta(ij) to an operator in Hilbert space are explored. Its canonical conjugate momentum is also introduced. Consequently, a consistent algebra involving the enlarged set of canonical operators is obtained, which permits us to construct theories that are dynamically invariant under the action of the rotation group. In this framework it is also possible to give dynamics to the noncommutativity operator sector, resulting in new features.
In this letter, firstly, the Schrddoto\\ddot{o}ddotodinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space-space and space-momentum as well as momentum-momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.