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\(\kappa \)-Minkowski spacetime and the star product realizations. (English) Zbl 1189.81115
Summary: We investigate a Lie algebra-type \(\kappa \)-deformed Minkowski spacetime with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of \(\kappa \)-Minkowski space. The coproduct and the star product corresponding to each of them are found. An explicit connection between realizations and orderings is established and the relation between the coproduct and the star product, provided through an exponential map, is proved. Utilizing the properties of the natural realization, we construct a scalar field theory on \(\kappa \)-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space. This result is universal and does not depend on the realizations, i.e. the orderings, used.

MSC:
81R60 Noncommutative geometry in quantum theory
53D55 Deformation quantization, star products
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