q-deformed relativistic wave equations

Abstract

Based on the representation theory of the q-deformed Lorentz and Poincaré symmeties q-deformed relativistic wave equation are constructed. The most important cases of the Dirac-, Proca-, Rarita-Schwinger-and Maxwell-equations are treated explicitly. The q-deformed wave operators look structurally like the undeformed ones but they consist of the generators of a non-commutative Minkowski space. The existence of the q-deformed wave equations together with previous results on the representation theory of the q-deformed Poincaré symmetry solve the q-deformed relativistic one particle problem.

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.…”

Conservation Laws for Linear Equations on Quantum Minkowski Spaces

“…Lately a number of equations of motion on quantum spaces were studied in the lierature. They include the Klein-Gordon and Dirac equations and their solutions investigated by Podleś [11] as well as equations considered on q-Minkowski space from [16,17,18,19,20,21,22] built within the framework of braided differential calculus [12,13,14,23]. In addition some quantum models on non-commutative spaces, in which deformation of commutation relations is motivated by Heisenberg principle and classical gravity, were studied by Doplicher et al in [24,25].…”

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).…”