Abstract
We consider new Abelian twists of Poincare algebra describing nonsymmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare–Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e. do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.
Highlights
Due to quantum gravity as well as quantized strings effects at Planck distances the notion of classical space-time can not be maintained
The noncommutative structures linked with quantum-deformed dynamical theories appeared recently in two-fold way: i.) as quantum generalization of Lie-algebraic symmetries, described by noncommutative Hopf algebras [9],[10], ii.) as deformed quantum phase spaces, with modified deformed canonical Heisenberg relations, described in the formalism of noncommutative geometry by a generalization of Hopf algebras, called Hopf algebroids [11,12,13]
We add that Hopf algebroid structures of quantum phase spaces with Lie-algebraic space-time sector were already discussed, mostly either without considering the twist quantizations [14] or with twisted Hopf algebroids considered not explicitly but as a part of general mathematical framework [15],[16]; see [17]
Summary
Due to quantum gravity (see [2,3,4,5,6]) as well as quantized strings effects (see e.g. [7,8]) at Planck distances the notion of classical space-time can not be maintained. The relations (1)–(2) can be extended further by the formula pμ £ xρ = −iημρ In such a way we obtain consistent classical action of Poincare algebra P on the Minkowski space M, describing the cross product P #M.4. With the action in (6) provided by the formulae (1), (3) In such a way one can express the noncommutative xμ in terms of classical phase space coordinates (xμ, pμ) and generators Mμν. If relation (7) is valid, the Hopf-algebraic Poincare algebra twist F as well as noncommutative Minkowski space coordinates xμ can be expressed in terms of phase space variables i.e. the Hopfalgebraic formulae are realized in terms of canonical Heisenberg algebra, which provides a classical example of Hopf algebroid. As well for other Lie-algebraic quantum deformations of Poincare algebra, not necessarily described by twist quantization (see [28,14])
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