Twisted statistics inκ-Minkowski spacetime

Abstract

We consider the issue of statistics for identical particles or fields in κ-deformed spaces, where the system admits a symmetry group G. We obtain the twisted flip operator compatible with the action of the symmetry group, which is relevant for describing particle statistics in presence of the noncommutativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the ordering prescription. I. INTRODUCTIONNoncommutative geometry is a plausible candidate for describing physics at the Planck scale, a simple model of which is given by the Moyal plane [1]. The models of noncommutative spacetime that follow from combining general relativity and uncertainty principle can be much more general [2]. An example of this more general class is provided by the κ-deformed space [3,4,5], which is based on a Lie algebra type noncommutativity. Apart from its algebraic aspects [6,7,8,9,10,11], various features of field theories and symmetries on κ-deformed spaces have recently been studied [12,13,14,15,16]. Such a space has also been discussed in the context of doubly special relativity [17,18,19].The issue of particle statistics plays a central role in the quantum description of a many-body system or field theory. This issue has naturally arisen in the context of noncommutative quantum mechanics and field theory [20,21,22,23,24]. In the noncommutative case, the issue of statistics is closely related to the symmetry of the noncommutative spacetime on which the dynamics is being studied. If a symmetry acts on a noncommutative spacetime, its coproduct usually has to be twisted in order to make the symmetry action compatible with the algebraic structure. In the commutative case, particle statistics is superselected, i.e. it is preserved under the action of the symmetry. In the presence of noncommutativity, it is thus natural to demand that the statistics is invariant under the action of the twisted symmetry. This condition leads to a new twisted flip operator, which is compatible with the twisted coproduct of the symmetry group [22,23]. The operators projecting to the symmetric and antisymmetric sectors of the Hilbert space are then constructed from the twisted flip operator. While most of the discussion of statistics in the noncommutative setup has been done in the context of the Moyal plane, some related ideas for κ-deformed spaces have also appeared recently [25,26,27].In this paper we set up a general formalism to describe statistics in κ-deformed spaces. Our formalism presented here is applicable to a system with an arbitrary symmetry group, which may include Poincare, Lorentz, Euclidean * trg@imsc.res.in † kumars.gupta@saha.ac.in ‡ harisp@uohyd.ernet.in § meljanac@irb.hr ¶ dmeljan@irb.hr

Noncommutative duality and fermionic quasinormal modes of the BTZ black hole

“…We point out that the formula (A.2) is a general one [28,68,69], being valid for the star products corresponding to any Lie-algebra type of deformation, of which κ-deformation is one particular example ( but interestingly, the θ-deformation is not). For elucidating the origin of the formula (A.2) and other issues related to the κ-deformation, particularly those related to the "method of realizations" and the correspondence between the star product, differential operator realization, coproduct and the operator ordering prescription one may consult [28,68,69]. When expanded up to first order in a, the star product looks as…”

On κ-deformation and triangular quasibialgebra structure

“…This paper is a continuation of earlier work [6,7], following [8], whose aim is systematically to construct κ-deformed quantum field theory from the following particular perspective. (Other approaches can be found in [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].) We recall the viewpoint on quantum field theory taken by Weinberg in [25], namely that quantum field theory takes the form it does because this is essentially the only way to construct a quantum mechanical theory of point particles with Poincaré symmetry -given only a very limited number of additional physical principles, like cluster decomposition.…”

Deformation Quasi-Hopf Algebras of Non-semisimple Type from Cochain Twists

“…This provides, for n = 3, 4, a proof of the results already anticipated in [4]. The result that U κ (iso (3)) and U κ (iso(4)) are twist equivalent to the corresponding undeformed UEAs should not be confused with other statements that exist in the literature, [21], concerning twists and κ-deformed Minkowski space-time, which involve enlarged algebras that include the dilatation generator.…”