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Noncommutative spaces and Poincaré symmetry. (English) Zbl 1397.81106
Summary: We present a framework which unifies a large class of noncommutative spacetimes that can be described in terms of a deformed Heisenberg algebra. The commutation relations between spacetime coordinates are up to linear order in the coordinates, with structure constants depending on the momenta plus terms depending only on the momenta. The possible implementations of the action of Lorentz transformations on these deformed phase spaces are considered, together with the consistency requirements they introduce. It is found that Lorentz transformations in general act nontrivially on tensor products of momenta. In particular, the Lorentz group element which acts on the left and on the right of a composition of two momenta is different and depends on the momenta involved in the process. We conclude with two representative examples which illustrate the mentioned effect.

MSC:
81R60 Noncommutative geometry in quantum theory
81T75 Noncommutative geometry methods in quantum field theory
81S05 Commutation relations and statistics as related to quantum mechanics (general)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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