By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

Yang-Baxter deformations, AdS/CFT, and twist-noncommutative gauge theory

In this paper, we study the deformed statistics and oscillator algebras of quantum fields defined in $\ensuremath{\kappa}$-Minkowski spacetime. The twisted flip operator obtained from the twist associated with the star product requires an enlargement of the Poincar\'e algebra to include the dilatation generators. Here we propose a novel notion of a fully covariant flip operator and show that to the first order in the deformation parameter it can be expressed completely in terms of the Poincar\'e generators alone. The $R$ matrices corresponding to the twisted and the covariant flip operators are compared up to first order in the deformation parameter and they are shown to be different. We also construct the deformed algebra of the creation and annihilation operators that arise in the mode expansion of a scalar field in $\ensuremath{\kappa}$-Minkowski spacetime. We obtain a large class of such new deformed algebras which, for certain choice of realizations, reduce to results known in the literature.

Deformed oscillator algebras and QFT in κ-Minkowski spacetime

We introduce a new {kappa}-star product describing the multiplication of quantized {kappa}-deformed free fields. The {kappa} deformation of local free quantum fields originates from two sources: noncommutativity of space-time and the {kappa} deformation of field oscillators algebra; we relate these two deformations. We demonstrate that for a suitable choice of {kappa}-deformed field oscillators algebra, the {kappa}-deformed version of the microcausality condition is satisfied, and it leads to the deformation of the Pauli-Jordan commutation function defined by the {kappa}-deformed mass shell. We show by constructing the {kappa}-deformed Fock space that the use of the {kappa}-deformed oscillator algebra permits one to preserve the bosonic statistics of n-particle states. The proposed star product is extended to the product of n fields, which for n=4 defines the interaction vertex in perturbative description of the noncommutative quantum {lambda}{phi}{sup 4} field theory. It appears that the classical four-momentum conservation law is satisfied at the interaction vertices.

Towards quantum noncommutative κ -deformed field theory

Structural and electronic properties of SnO2

We consider the issue of statistics for identical particles or fields in {kappa}-deformed spaces, where the system admits a symmetry group G. We obtain the twisted flip operator compatible with the action of the symmetry group, which is relevant for describing particle statistics in the presence of the noncommutativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the ordering prescription.

Twisted statistics in κ-Minkowski spacetime

Noncommutative Spaces and Poincaré Symmetry

We present Lie-algebraic deformations of Minkowski space with undeformed Poincare algebra. These deformations interpolate between Snyder and kappa-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and kappa-Minkowski in Maggiore-type realizations are discussed. Our construction leads to a new class of deformed special relativity theories.

Kappa-deformed Snyder spacetime

We present Lie-algebraic deformations of Minkowski space with undeformed Poincar\'e algebra. These deformations interpolate between Snyder and $\ensuremath{\kappa}$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that, although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are isomorphic at the level of Hopf-algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $\ensuremath{\kappa}$-deformed Snyder to Snyder space is constructed. The deformed Leibniz rule, the Hopf structure, and the star product are found. Special cases, particularly Snyder and $\ensuremath{\kappa}$-Minkowski in Maggiore-type realizations, are discussed. The same generalized Hopf-algebraic structures are considered as well in the case of an arbitrary allowable kind of realization, and results are given perturbatively up to second order in deformation parameters.

Daniel Meljanac

Papers

Twisted statistics in κ-Minkowski spacetime

Deformed oscillator algebras and QFT in κ-Minkowski spacetime

Noncommutative Spaces and Poincaré Symmetry

Kappa-deformed Snyder spacetime

Kappa Snyder deformations of Minkowski spacetime, realizations, and Hopf algebra