Conservation Laws for Linear Equations on Quantum Minkowski Spaces

Abstract

The general, linear equations with constant coefficients on quantum Minkowski spaces are considered and the explicit formulae for their conserved currents are given. The proposed procedure can be simplified for * -invariant equations. The derived method is then applied to Klein-Gordon, Dirac and wave equations on different classes of Minkowski spaces. In the examples also symmetry operators for these equations are obtained. They include quantum deformations of classical symmetry operators as well as an additional operator connected with deformation of the Leibnitz rule in non-commutative differential calculus.

Conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces

“…The following propositions hold for equation (1 -3) and are the extension of results derived in [11,12,13]. …”

“…In the previous papers we discussed the conservation laws for linear equations with constant coefficients for discrete differential calculus [10,11] as well as for a wider category of equations acting on noncommutative spaces namely on quantum Minkowski spaces [12,15] and the braided linear ones [13,14,16]. Now we would like to present our results for an extended class of equations with variable coefficients of the form:…”

Stationarity-conservation laws for certain linear fractional differential equations

“…The derived formula (8) is similar to the multiplicity properties of the transformation operators in the discrete [29] and noncommutative [30,31] differential multidimensional calculi for standard product of functions:…”

“…The Leibniz's rule for the introduced differintegrable operator of positive order ν is similar to the one known from the discrete and noncommutative calculus [29,30,31]:…”

“…As is well-known in classical field theory the conservation laws for linear differential systems can be derived using TakahashiUmezawa method [28]. This procedure has been extended to discrete and noncommutative models [29,30,31]. Our aim is to show that similar procedure can be applied to fractional equations in the convolution algebra of functions in order to obtain the stationarity-conservation laws which are analogs of conservation equations known for models from classical differential calculus.…”