Monopole and Berry phase in momentum space in noncommutative quantum mechanics

Abstract

To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter θ to a field operator, which one proves to be only momentum dependent. We find consequently that this field must be obligatorily a dual Dirac monopole in momentum space. Recent experiments in the context of the anomalous Hall effect provide evidence for a monopole in the crystal momentum space. We suggest a connection between the noncommutative field and the Berry curvature in momentum space which is at the origine of the anomalous Hall effect. . Noncommutative gauge theories were also found as being naturally related to string and M-theory [4].In this framework an antisymmetric θ µν parameter usually taken to be constant [5,11] is introduced in the commutation relation of the coordinates in the space time manifold [x µ , x υ ] = iθ µν . This relation leads to the violation of the Lorentz symmetry, a possibility which is intensively studied theoretically and experimentally [12]. Applications of noncommutative theories were also found in condensed matter physics, for instance in the Quantum Hall effect [13? ] and the non-commutative Landau problem [15,16,17] i.e., a quantum particle in the noncommutative plane, coupled to a constant magnetic field with a constant selected θ parameter as usual.In this letter, we generalize the quantum mechanics in non-commutative geometry by promoting the θ parameter with a new field obeying its own field equations. Note that some authors, (for example [18]) introduced a position dependent θ field using a Kontsevich product [19] in the study of gauge theory. Contrary to these approaches we find that the θ field must be momentum dependent.The physical motivations of our work are twofold: (i) For a constant θ field, we show that a quantum particle in a harmonic potential has a behavior similar to a particle in a constant magnetic field θ in standard quantum mechanics, since a paramagnetic term appears in the Hamiltonian. Moreover the particle in the presence of the θ field acquires an effective dual mass in the same way that an electron moving in a periodic potential in solid state physics. Thus it is legitimate to interpret this field like a field having properties of the vacuum.In this context it is natural to extend the theory to a non-constant field. This proposal is strongly enforced by the lack of rotation generators in noncommutative space with a constant θ parameter, i.e. the angular momentum does not satisfy the usual angular momentum algebra. We then show that this θ field is only momentum dependent and that the requirement of the angular momentum algebra, that is the existence of an angular momentum, necessarily imposes a dual Dirac monopole in momentum space field configuration. Thereafter we will intensely use the concept of duality between the quantities defined in momentum space compared with those defined in the position space.(ii) The second motivation comes for recent theoretical works [20] concerning the anomal...

Noncommutative mechanics and exotic Galilean symmetry

“…A remarkable example for its phenomenological implications is provided by the monopole field in momentum space κ = θ p | p| 3 , which is indeed the only possibility consistent with the spherical symmetry and the canonical relations {r i , p j } = δ ij [29]. Its expression appears to be consistent, at least qualitatively, with the data reported in ( [5]) and in Spin Hall Effects [6], [30].…”

Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

“…(25,28) and the MM in momentum space has been of much theoretical interest. [3,4,10,31]. From a more general point of view, physics in noncommutative space-time coordinates has been of great theoretical interest, rather in high-energy physics community, in particular, in the context of string and M theories.…”

Geometrical spinoptics and the optical Hall effect

“…Various approaches, including a full-fledged computation of refraction and reflection of arbitrarily polarized Gaussian electromagnetic wave packets [7], have been put forward. Of particular interest are recent extensions of geometrical optics, within a Maxwellian context, using a certain "Berry connection" whose curvature (in momentum space) yields a modification of the Fermat equations of motion for polarized light beams [29,4,5,8,31]; see also [2,3,21]. A quasi-classical formula for the above-mentioned transverse shift of polarized light beams has also been proposed by Onoda, Murakami, and Nagaosa [31], together with an experimental set up using photonic crystals in order to reveal the OHE for reflected and refracted light beams.…”