Wave equations onq-Minkowski space

Abstract

We give a systematic account of the exterior algebra of forms on q-Minkowski space, introducing the q-exterior derivative, q-Hodge star operator, q-coderivative, q-Laplace-Beltrami operator and the q-Lie-derivative. With these tools at hand, we then give a detailed exposition of the q-d'Alembert and q-Maxwell equation. For both equations we present a q-momentum-indexed family of plane wave solutions. We also discuss the gauge freedom of the q-Maxwell field and give a q-spinor analysis of the q-field strength tensor.

Conservation laws for linear equations on braided linear spaces

“…It includes equations with constant coefficients such as, for example, a scalar, spinor, and vector wave equation on q-Minkowski space. 12,13 For constant coefficients the condition ͑23͒ can be replaced with stronger one:…”

“…For q-Minkowski space investigated equations include the scalar, spinor and vector wave equations. 12,13 In the last section we show how to obtain the conserved currents for a scalar wave equation on q-Minkowski space.…”

Twisted $h$ -spacetimes and invariant equations

“…h case: Relativistic invariant equations have been already discussed for the q-deformation [37,38,39,17,40] and recently the case of the deformed Klein-Gordon and Dirac equations on a deformed homogeneous space (Minkowski) has been considered in [15]. We shall follow here a method similar to that followed in [32].…”

q-Difference intertwining operators and q-conformal invariant equations

“…Then we define the q-difference operators by (cf. (83) Note that our free q-Maxwell equations, obtained from (144) for n = 0, and qJo = 0, are different from the free q-Maxwell equations of [47,48]. (This is natural since they use different q-Minkowski spacetime from [40,41,42].)…”

“…(This is natural since they use different q-Minkowski spacetime from [40,41,42].) The advantages of our equations are: (1) they have simple indexless form; (2) we have a whole hierarchy of equations; (3) we have the full equations, and not only their free counterparts; (4) our equations are q-conformal invariant, not only q-Lorentz [48], or q-Poincar6 [47], invariant. (In fact, it is not clear whether the q-Lorentz algebras of [40,41,42,49] or the q-Poincar6 algebra of [50] are extendable to q-conformal algebras (often easy q = 1 things fail for q ~ 1).…”