Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance

Abstract

We investigate the properties of kappa-Minkowski spacetime by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg-Weyl algebra. The deformed algebra consists of kappa-Poincare algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an adventages of this approach to consistently construct a star product which has a property that under integration sign it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal, but not also for kappa-Minkowski spacetime. This star product also has generalized trace and cyclic properties and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and by requiring it to be hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachionic modes and basicaly of the very same form. The issue of Lorentz invariance of the theory is also discussed.

arXiv comment22 pages, no figures, revtex4, in new version comments regarding translation invariance and few references are added, accepted for publication in Int. J. Mod. Phys. A

References in corpus (60)

• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Lorentz invariance with an invariant energy scale
• Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale
• A Generalized Uncertainty Principle in Quantum Gravity
• Testable scenario for Relativity with minimum-length
• Generalized Lorentz invariance with an invariant energy scale
• Quantum Groups, Gravity, and the Generalized Uncertainty Principle
• Classical and Quantum Mechanics of Free \k Relativistic Systems
• Minimal Length Uncertainty Relation and Ultraviolet Regularisation
• 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory
• Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory
• Doubly Special Relativity theories as different bases of $\kappa$--Poincaré algebra
• Non-commutative space-time of Doubly Special Relativity theories
• Quantum symmetry, the cosmological constant and Planck scale phenomenology
• Coproduct and star product in field theories on Lie-algebra non-commutative space-times
• Deformed Field Theory on kappa-spacetime
• Kappa-Minkowski space-time and the star product realizations
• New realizations of Lie algebra kappa-deformed Euclidean space
• Local D=4 Field Theory on $\kappa$--Deformed Minkowski Space
• 2+1 gravity and Doubly Special Relativity
• Twisted Statistics in kappa-Minkowski Spacetime
• Deformed Oscillator Algebras and QFT in $\kappa$-Minkowski Spacetime
• Fock space, quantum fields and kappa-Poincaré symmetries
• kappa-Minkowski spacetime as the result of Jordanian twist deformation
• Field theory on $\kappa$--Minkowski space revisited: Noether charges and breaking of Lorentz symmetry
• Gauge theories on the kappa-Minkowski spacetime
• Towards Quantum Noncommutative $\kappa$-deformed Field Theory
• Scalar Field Theory on Non-commutative Snyder Space-Time
• From noncommutative kappa-Minkowski to Minkowski space-time
• Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
• Modification of Heisenberg uncertainty relations in non-commutative Snyder space-time geometry
• Classical basis for kappa-Poincare algebra and doubly special relativity theories
• On $\kappa$-Deformation and UV/IR Mixing
• Covariant particle statistics and intertwiners of the kappa-deformed Poincare algebra
• Constraints on the quantum gravity scale from kappa - Minkowski spacetime
• Local Field Theory ON $\kappa$-Minkowski Space, Star Products and Noncommutative Translations
• Generalized kappa-deformed spaces, star-products, and their realizations
• Representations of noncommutative quantum mechanics and symmetries
• Scalar field theory on $\kappa$-Minkowski space-time and Doubly Special Relativity
• Lorentz invariant field theory on kappa-Minkowski space
• kappa-deformed Spacetime From Twist
• Kappa-deformed Snyder spacetime
• $\kappa$-Deformed Statistics and Classical Fourmomentum Addition Law
• Noncommutative Differential Forms on the kappa-deformed Space
• Rainbow statistics
• On kappa-deformation and triangular quasibialgebra structure
• Covariant particle exchange for kappa-deformed theories in 1+1 dimensions
• Unified view of multimode algebras with Fock-like representations
• Bosonic realization of algebras in the Calogero model
• Differential structure on the $\kappa$-Minkowski spacetime from twist
• Differential Structure on kappa-Minkowski Spacetime Realized as Module of Twisted Weyl Algebra
• Deformed Heisenberg algebras, a Fock-space representation and the Calogero model
• Kappa-deformed oscillators, the choice of star product and free kappa-deformed quantum fields
• Star product and interacting fields on $\kappa$-Minkowski space
• Jordanian Quantum Deformations of D=4 Anti-de-Sitter and Poincare Superalgebras
• Calogero model, deformed oscillators and the collapse
• Permutation Invariant Algebras, a Fock Space Realization and the Calogero Model
• Lorentz Covariant k-Minkowski Spacetime
• Jordanian Twist Quantization of D=4 Lorentz and Poincare Algebras and D=3 Contraction Limit
• A bound on Planck-scale modifications of the energy-momentum composition rule from atomic interferometry

Cited by in corpus (42)

• Minimal Length Scale Scenarios for Quantum Gravity
• Quantum Spacetime Phenomenology
• Electrodynamics on $\kappa$-Minkowski space-time
• Speed of particles and a relativity of locality in $\kappa$-Minkowski quantum spacetime
• Generalized Poincare algebras, Hopf algebras and kappa-Minkowski spacetime
• Scalar field propagation in the phi^4 kappa-Minkowski model
• Geodesic equation in $k$-Minkowski spacetime
• Kappa-Minkowski spacetime, Kappa-Poincaré Hopf algebra and realizations
• Noncommutative scalar quasinormal modes and quantization of entropy of a BTZ black hole
• A twisted look on kappa-Minkowski: U(1) gauge theory
• $\kappa$-Poincaré invariant quantum field theories with KMS weight
• K-Poincare-Hopf algebra and Hopf algebroid structure of phase space from twist
• Toward the classification of differential calculi on $\kappa$-Minkowski space and related field theories
• Twists, realizations and Hopf algebroid structure of kappa-deformed phase space
• Effects of Noncommutativity on the Black Hole Entropy
• Differential algebras on kappa-Minkowski space and action of the Lorentz algebra
• Quantum field theory in generalised Snyder spaces
• $\kappa$-Deformed Phase Space, Hopf Algebroid and Twisting
• $\kappa$-Poincaré invariant orientable field theories at 1-loop
• Gauge theories on $\kappa$-Minkowski spaces: Twist and modular operators
• Quantum statistics and noncommutative black holes
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• $\kappa$-Deformations and Extended $\kappa$-Minkowski Spacetimes
• Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras
• $\kappa$-Minkowski star product in any dimension from symplectic realization
• Kappa-deformation of phase space; generalized Poincare algebras and R-matrix
• Universal $\kappa$-Poincaré covariant differential calculus over $\kappa$-Minkowski space
• Differential forms and k-Minkowski spacetime from extended twist
• Spacetime and deformations of special relativistic kinematics
• Light-like $\kappa$-deformations and scalar field theory via Drinfeld twist
• Hermitian realizations of kappa-Minkowski spacetime
• Vacuum energy and the cosmological constant problem in $\kappa$-Poincaré invariant field theories
• Twisted BRST symmetry in gauge theories on $\kappa$-Minkowski
• Quantum particles in non-commutative space-time: an identity crisis
• A short introduction to $\kappa$-deformation
• Predictive description of Planck-scale-induced spacetime fuzziness
• Deformed phase spaces with group valued momenta
• Doubly special relativity as a non-local quantum field theory
• Bicrossproduct construction versus Weyl-Heisenberg algebra
• Some Features of Scattering Problem in a $\kappa$-Deformed Minkowski Spacetime
• Twist deformations leading to kappa-Poincare Hopf algebra and their application to physics
• Gauge theory models on $\kappa$-Minkowski space: Results and prospects