Covariant particle statistics and intertwiners of the κ-deformed Poincaré algebra

Abstract

To speak about identical particles -bosons or fermions -in quantum field theories with κ-deformed Poincaré symmetry, one must have a κ-covariant notion of particle exchange. This means constructing intertwiners of the relevant representations of κ-Poincaré. We show, in the simple case of spinless particles, that intertwiners exist, and, supported by a perturbative calculation to third order in 1 κ , make a conjecture about the existence and uniqueness of a certain preferred intertwiner defining particle exchange in κ-deformed theories.

Covariant particle exchange for κ-deformed theories in dimensions

“…In [20] we showed this in 3 + 1 dimensions to the first few orders in 1 κ by writing τ : |r µ ⊗ |s µ → |f µ (r, s) ⊗ |g µ (r, s) and solving for the functions f and g explicitly. 5 This has the merit of concreteness, but it obscures the symmetry implicit in τ 2 = id.…”

“…Finding these variables amounts to finding the natural labelling of many-particle states in κ-deformed theories, and is in a sense the central challenge for future work in this direction. Observe that, writing τ for the two-particle intertwiner of [20], one has τ 12 = τ ⊗ 1 when a = 0, while if a = 1 3 then τ 23 = 1 ⊗ τ . But for no value of a is it true that every τ ii+1 acts non-trivially only on the i,i+1st tensor factors -which accords with the results of section 4 above.…”

“…We gave a partial answer in [20], in which we showed that for the case of two scalar particles, and perturbatively to third order in 1 κ , there is a unique covariant map exchanging the particles. Of course, one would like to understand why this should be true exactly, rather than just in perturbation theory, and further to know whether particle exchange can still be defined covariantly for states of more than two particles.…”

On κ-deformation and triangular quasibialgebra structure

“…It is rather the terms linear in Lorentz generators, whose realizations on wavefunctions involve derivatives, that shift the arguments of ψ. And following [6] we expect that…”

“…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].)…”