q-Deformed relativistic one-particle states

On the deformability of Heisenberg algebras

“…Moreover it has been shown that it is hardly possible to find a hermiticity perserving comultiplication on the generators of both the q-difference calculus and of the q-differential calculus. For these reasons one has to think carefully if quantum planes, although they have interesting features [20,21], provide a reasonable base space for quantum field theories.…”

“…Proof: If we assume that the proposition is true, i.e. relations (21) hold for all α ∈ I, the corollary is a direct consequence of these relations. Since we require that for q → 1 the derivatives D should turn into the continuous derivatives it must hold for all α maybe up to some normalization that D ≡ ∂ mod h. Together with the linearity of the calculus and the formal hermiticity of the q-derivatives this requires that in the realization of D α some linear combination of the u k(α) α and (u k(α) α ) −1 must appear.…”

“…Now we can apply proposition 4.1 and equation (21). The statement there is that the quantities u i , u −1 i , or any power of them are defined by by q-difference relations on commutative spaces.…”

Conformal S12 enveloping algebras

“…In this case this is usually done with demanding that a is real and Downloaded by [Otto-von-Guericke-Universitaet Magdeburg] at 03:18 21 October 2014 CONFORMAL sl, ENVELOPING ALGEBRAS see for example [22]. However, then it is clear from the discussions on finite dimensional representations of Habc(s12) in the generic case that for a # &1 one has a discrete family of simples which can hardly be viewed as points in a deformed 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].…”