Hopf-Algebra Description of Noncommutative-Space–time Symmetries

Abstract

In the study of certain noncommutative versions of Minkowski space–time a lot remains to be understood for a satisfactory characterization of their symmetries. Adopting as our case study the κ-Minkowski noncommutative space–time, on which a large literature is already available, we propose a line of analysis of noncommutative-space–time symmetries that relies on the introduction of a Weyl map (connecting a given function in the noncommutative Minkowski with a corresponding function in commutative Minkowski). We provide new elements in favor of the expectation that the commutative-space–time notion of Lie-algebra symmetries must be replaced, in the noncommutative-space–time context, by the one of Hopf-algebra symmetries. While previous studies appeared to establish a rather large ambiguity in the description of the Hopf-algebra symmetries of κ-Minkowski, the approach here adopted reduces the ambiguity to the description of the translation generators, and our results, independently of this ambiguity, are sufficient to clarify that some recent studies which argued for an operational indistinguishability between theories with and without a length-scale relativistic invariant, implicitly assumed that the underlying space–time would be classical. Moreover, while usually one describes theories in κ-Minkowski directly at the level of equations of motion, we explore the nature of Hopf-algebra symmetry transformations on an action.

CANONICAL AND LIE-ALGEBRAIC TWIST DEFORMATIONS OF Κ-Poincaré AND CONTRACTIONS TO Κ-Galilei ALGEBRAS

“…Such a ⋆-product should be consistent with the action of deformed symmetry generators satisfying suitably deformed Leibnitz (coproduct) rules. In the case of κ-deformation the ⋆ κ -multiplication looks as follows (see [31], [32] and references therein)…”

Κ-Deformed STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

“…In our discussion we shall use the quantum κ-deformed Poincare symmetries formulated in modified bicrossproduct basis with classical Lorentz subalgebra, and the κ-deformed mass-shell invariant under the change P 0 → −P 0 [14]. Such a choice is obtained if in standard bicross-product basis [4], [5] we change P i → e P 0 2κ P i , i.e.…”

“…We also observe (see (11)) that S(p i ) = −p i . Using the expansion of noncommutative free field into the κ-deformed plane waves (14) we introduce the creation/annihilation operators satisfying the κ-deformed field oscillators algebra. The novel feature of our construction is the new multiplication rule of creation/anihilation operators which requires putting them off-shell.…”

“…We describe the κ-deformed scalar free field on noncommutative Minkowski space (13) by decomposition into field quanta creation and annihilation operators, with the use of κ-deformed Fourier transform (see (14))…”

Blackbody radiation inκ-Minkowski spacetime

“…Therefore, one has N i ⊲ (x 2 0 − x 2 + 3ix 0 /κ) = 0 and x 2 0 − x 2 + 3ix 0 /κ is a Lorentz-invariant. Likewise, the κ-deformed Poincaré algebra is given for symmetric ordering given in section III B [21]: and its Hopf-algebraic structure, From this algebra one may find the duality relation in coordinate space through the pairing (A4) and duality Eqs. (A5,A8,A9).…”