Scalar field theory in Snyder space-time: alternatives

Abstract

We construct two types of scalar field theory on Snyder space-time. The first one is based on the natural momenta addition inherent to the coset momentum space. This construction uncovers a non-associative deformation of the Poincaré symmetries. The second one considers Snyder space-time as a subspace of a larger non-commutative space. We discuss different possibilities to restrict the extra-dimensional scalar field theory to a theory living only on Sndyer space-time and present the consequences of these restrictions on the Poincaré symmetries. We show moreover how the non-associative approach and the Doplicher-Fredenhagen-Roberts space can be seen as specific approximations of the extra-dimensional theory. These results are obtained for the 3d Euclidian Snyder space-time constructed from $\SO(3,1)/\SO(3)$, but our results extend to any dimension and signature.

24 pages.

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• Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
• Snyder dynamics in a Schwarzschild spacetime
• Classical dynamics on curved Snyder space
• Worldline Formalism in Snyder Spaces
• Nonassociative Snyder phi4 Quantum Field Theory
• Associative realizations of the extended Snyder model
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
• Heisenberg doubles for Snyder type models
• Planck-scale-deformed relativistic symmetries and diffeomorphisms in momentum space
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• UV/IR Mixing in Nonassociative Snyder phi^4 Theory
• Snyder Momentum Space in Relative Locality
• Noncommutative fluid dynamics in the Snyder space-time
• Noncommutative Yang model and its generalizations
• Associative realizations of $\kappa$-deformed extended Snyder model
• Non-commutative Fourier transform for the Lorentz group via the Duflo map
• Tensor models and 3-ary algebras
• Relative-locality phenomenology on Snyder spacetime
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Generalized relativistic kinematics in Poincaré-invariant models
• Twist for Snyder space
• Doubly special relativity as a non-local quantum field theory
• Lifshitz field theories, Snyder noncomutative space-time and momentum dependent metric
• Relative-locality geometry for the Snyder model
• Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries
• Plane waves in noncommutative fluids
• Translation in momentum space and minimal length
• Realizations of the Extended Snyder Model
• $\hslash$-perturbative solutions of quantum Snyder and Yang models with parameters describing spontaneous symmetry breaking
• Snyder Like Modified Gravity in Newton's Spacetime
• Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory