Marcin Daszkiewicz

κ-DEFORMED STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

We consider κ-deformed relativistic symmetries described algebraically by modified Majid–Ruegg bi-cross-product basis and investigate the quantization of field oscillators for the κ-deformed free scalar fields on κ-Minkowski space. By modification of standard multiplication rule, we postulate the κ-deformed algebra of bosonic creation and annihilation operators. Our algebra permits one to define the n-particle states with classical addition law for the four-momentum in a way which is not in contradiction with the nonsymmetric quantum four-momentum co-product. We introduce κ-deformed Fock space generated by our κ-deformed oscillators which satisfy the standard algebraic relations with modified κ-multiplication rule. We show that such a κ-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of κ-deformed algebra of oscillators in field-theoretic noncommutative framework.

κ-deformed oscillators, the choice of star product and free κ-deformed quantum fields

The aim of this paper is to study in D = 4 the general framework providing various κ-deformations of field oscillators and consider the commutator function of the corresponding κ-deformed free fields. In order to obtain free κ-deformed quantum fields (with c-number commutators) we proposed earlier a particular model of a κ-deformed oscillator algebra (Daszkiewicz M, Lukierski J and Woronowicz M 2008 Mod. Phys. Lett. A 23 9 (arXiv:hep-th/0703200)) and the modification of κ-star product (Daszkiewicz M, Lukierski J, Woronowicz M 2008 Phys. Rev. D 77 105007 (arXiv:0708.1561 [hep-th])), implementing in the product of two quantum fields the change of standard κ-deformed mass-shell conditions. We recall here that other different models of κ-deformed oscillators recently introduced in Arzano M and Marciano A (2007 Phys. Rev. D 76 125005 (arXiv:0707.1329 [hep-th])), Young C A S and Zegers R (2008 Nucl. Phys. B 797 537 (arXiv: 0711.2206 [hep-th])), Young C A S and Zegers R (2008 arXiv: 0803.2659 [hep-th]) are defined on a standard κ-deformed mass shell. In this paper, we consider the most general κ-deformed field oscillators, parametrized by a set of arbitrary functions in 3-momentum space. First, we study the fields with the κ-deformed oscillators defined on the standard κ-deformed mass shell, and argue that for any such choice of a κ-deformed field oscillators algebra we do not obtain the free quantum κ-deformed fields with the c-number commutators. Further, we study κ-deformed quantum fields with the modified κ-star product and derive a large class of κ-oscillators defined on a suitably modified κ-deformed mass shell. We obtain a large class of κ-deformed statistics depending on six arbitrary functions which all provide the c-number field commutator functions. This general class of κ-oscillators can be described by the composition of suitably defined κ-multiplications and the κ-deformation of the flip operator.

Newton equation for canonical, Lie-algebraic, and quadratic deformation of classical space

The Newton equation describing particle motion in a constant external field force on canonical, Lie-algebraic, and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of the particle is generated. We also indicate that in the case of spatial coordinates commuting in a Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and position-dependent forces.

CANONICAL AND LIE-ALGEBRAIC TWIST DEFORMATIONS OF GALILEI ALGEBRA

We describe various nonrelativistic contractions of two classes of twisted Poincare algebra: canonical one (θμν-deformation) and the one leading to Lie-algebraic models of noncommutative spacetimes. The cases of contraction-independent and contraction-dependent twist parameters are considered. We obtain five models of noncommutative nonrelativistic spacetimes, in particular, two new Lie-algebraic nonrelativistic deformations of spacetime, respectively, with quantum time/classical space and with quantum space/classical time.

Classical mechanics of many particles defined on canonically deformed nonrelativistic space-time

We provide the classical mechanics of many particles moving in canonically twist-deformed spacetime. In particular, we consider two examples of such noncommutative systems — the set of N particles moving in gravitational field as well as the system of N interacting harmonic oscillators.

Phase spaces of doubly special relativity

Abstract We show that depending on the direction of deformation of κ -Poincare algebra (time-like, space-like, or light-like) the associated phase spaces of single particle in Doubly Special Relativity theories have the energy–momentum spaces of the form of de Sitter, anti-de Sitter, and flat space, respectively.

Non-critical light-cone string

A free string quantum model is constructed directly in the light-cone variables in the range of dimensions 1 < D < 25. The longitudinal degrees of freedom are described by an abstract Verma module with the central charge restricted by the requirement of closure of the non-linear realization of the Poincare algebra. The spin content of the model is analyzed. In particular for D = 4 the explicit formulae for the character generating functions of the open and closed strings are given and the spin spectrum of the first twelve excited levels is calculated. It is shown that for the space-time dimension in the range 1 < D < 25 this light-cone quantum model is equivalent to the critical massive string and to the non-critical Nambu-Goto string.

Quantum Mechanics of Many Particles Defined on Twisted N -enlarged Newton–Hooke Space-times

We provide the quantum mechanics of many particles moving in twisted N-enlarged Newton-Hooke space-time. In particular, we consider the example of such noncommutative system - the set of M particles moving in Coulomb field of external point-like source and interacting each other also by Coulomb potential.

CANONICAL, LIE-ALGEBRAIC AND QUADRATIC TWIST DEFORMATIONS OF GALILEI GROUP

New Galilei quantum groups dual to the Hopf algebras proposed in Ref. 1 are obtained by the nonrelativistic contraction procedures. The corresponding Lie-algebraic and quadratic quantum spacetimes are identified with the translation sectors of considered algebras.

TWIST DEFORMATIONS OF NEWTON–HOOKE HOPF ALGEBRAS

We construct five new quantum Newton–Hooke Hopf algebras with the use of Abelian twist procedure. Further we demonstrate that the corresponding deformed spacetimes with quantum space and classical time are periodic or expanding in time.