Scalar Field Theory on Non-commutative Snyder Space-Time

Abstract

We construct a scalar field theory on the Snyder non-commutative space-time. The symmetry underlying the Snyder geometry is deformed at the co-algebraic level only, while its Poincaré algebra is undeformed. The Lorentz sector is undeformed at both algebraic and co-algebraic level, but the co-product for momenta (defining the star-product) is non-co-associative. The Snyder-deformed Poincaré group is described by a non-co-associative Hopf algebra. The definition of the interacting theory in terms of a non-associative star-product is thus questionable. We avoid the non-associativity by the use of a space-time picture based on the concept of realization of a non-commutative geometry. The two main results we obtain are: (i) the generic (namely for any realization) construction of the co-algebraic sector underlying the Snyder geometry and (ii) the definition of a non-ambiguous self interacting scalar field theory on this space-time. The first order correction terms of the corresponding Lagrangian are explicitly computed. The possibility to derive Noether charges for the Snyder space-time is also discussed.

10 pages; v2: introduction rewritten, co-algebraic analysis improved, references added; to appear in PRD

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Cited by in corpus (63)

• 30 years in: Quo vadis generalized uncertainty principle?
• Relativistic deformed kinematics from momentum space geometry
• Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance
• Noncommutative Spaces and Poincaré Symmetry
• Anatomy of a deformed symmetry: field quantization on curved momentum space
• Snyder space revisited
• Kappa Snyder deformations of Minkowski spacetime, realizations and Hopf algebra
• Particle Dynamics on Snyder space
• Kappa-Minkowski spacetime, Kappa-Poincaré Hopf algebra and realizations
• Deformed relativistic kinematics on curved spacetime -- a geometric approach
• Differential structure on kappa-Minkowski space, and kappa-Poincare algebra
• The Hidden Quantum Groups Symmetry of Super-renormalizable Gravity
• Twists, realizations and Hopf algebroid structure of kappa-deformed phase space
• Snyder dynamics in a Schwarzschild spacetime
• Classical dynamics on curved Snyder space
• Quantum field theory in generalised Snyder spaces
• Worldline Formalism in Snyder Spaces
• Nonassociative Snyder phi4 Quantum Field Theory
• Asymptotic freedom for $\lambda {\varphi }_{\star }^4$ QFT in Snyder-de Sitter space
• Associative realizations of the extended Snyder model
• Effect of Snyder-de Sitter model on the Black hole thermodynamics in the context of rainbow gravity
• The Kepler problem in the Snyder space
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• Heisenberg doubles for Snyder type models
• Geometry of multi-particle systems with a relativistic deformed kinematics and the relative locality principle
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• UV/IR Mixing in Nonassociative Snyder phi^4 Theory
• Unification of $\kappa$-Minkowski and extended Snyder spaces
• Vector-like deformations of relativistic quantum phase-space and relativistic kinematics
• Snyder Momentum Space in Relative Locality
• Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case
• Noncommutative fluid dynamics in the Snyder space-time
• Constraining Snyder and GUP models with low-mass stars
• Noncommutative Yang model and its generalizations
• Associative realizations of $\kappa$-deformed extended Snyder model
• Spacetime from locality of interactions in deformations of special relativity: the example of $\kappa$-Poincaré Hopf algebra
• Relative-locality phenomenology on Snyder spacetime
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Twin Peaks: A possible signal in the production of resonances beyond special relativity
• Graphene in curved Snyder space
• Snyder Space-Time: K-Loop and Lie Triple System
• Snyder-de Sitter meets the Grosse-Wulkenhaar model
• Relativistic deformed kinematics: from flat to curved spacetimes
• Space-time thermodynamics in momentum dependent geometries
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• A new perspective on Doubly Special Relativity
• Generalized relativistic kinematics in Poincaré-invariant models
• Twist for Snyder space
• Physical velocity of particles in relativistic curved momentum space
• Doubly special relativity as a non-local quantum field theory
• Lifshitz field theories, Snyder noncomutative space-time and momentum dependent metric
• Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra
• Plane waves in noncommutative fluids
• Non-local quantum field theory from doubly special relativity
• Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries
• Localization of particles in quantum field theory
• Translation in momentum space and minimal length
• Refining Bounds for Snyder and GUP Models through Seismic Wave Analysis
• Einstein crystals in Snyder and Snyder-de Sitter noncommutative backgrounds
• Snyder and their representation with creation and annihilation operators
• Some features of a Schwarzchild Black Hole from a Snyder perspective
• Snyder Like Modified Gravity in Newton's Spacetime
• Realizations of the Extended Snyder Model