A new “null-plane” quantum Poincaré algebra

Abstract

A new quantum deformation, which we call null-plane, of the (3+1) Poincaré algebra is obtained. The algebraic properties of the classical null-plane description are generalized to this quantum deformation. In particular, the classical isotopy subalgebra of the null-plane is deformed into a Hopf subalgebra, and deformed spin operators having classical commutation rules can be defined. Quantum Hamiltonian, mass and position operators are studied, and the null-plane evolution is expressed in terms of a deformed Schrödinger equation.

Unified description for $$\kappa $$ κ -deformations of orthogonal groups

“…We shall specify later the expression for C τ for this particular case. At the moment one can observe that the inverse to (9), the formula…”

“…The r-matrix mentioned above corresponds to this case. 1 Another option is the so-called light-like (null-plane) deformation corresponding to null-vectors, which was firstly considered in [9] (then also in [10,11]) with quantum-deformed direction on the light cone (x + = x 0 + x 3 ) and with the corresponding symmetry the so-called 'null-plane quantum Poincaré Lie algebra'. It was inspired by the central problem of quantum relativistic systems in the Hamiltonian formulation, which has been studied for the null-plane evolution.…”

“…Particularly, the explicit formulas expressing classical basis in terms of bicrossproduct one have been obtained therein. Similarly, the null-plane deformation has been originally obtained and investigated in the basis inherited from the so-called deformation embedding method [9]. The classical Lie algebra basis in this context has not been explored yet.…”

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

“…Quantum deformations of Lie algebras and Lie groups [2][3][4][5][6][7][8] have been broadly applied in the construction of deformed symmetries of spacetimes [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], especially for the Poincaré and Galilei cases, for which the deformation parameter is known to play the role of a fundamental scale. Among all these quantum kinematical algebras the well known -Poincaré algebra [9,13,14,16,18] has been frequently considered.…”

Quantum deformations of space-time symmetries with fundamental mass parameter

“…Recently there were also considered the κ-deformations along one of the space axes, for example x 3 (see [25]; this is so called tachyonic κ-deformation with O(2, 1) classical subalgebra). Other interesting κ-deformation is the null-plane quantum Poincaré algebra [26] with the "quantized" light cone coordinate x + = x 0 + x 3 and classical E(2) subalgebra. The generalized κ-deformations of D = 4 Poincaré symmetries were recently proposed in [27] and describe the κ-deformation in any direction y 0 = R 0µ x µ in Minkowski space.…”