Non-commutative Fourier transform for the Lorentz group via the Duflo map

Abstract

We defined a non-commutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the non-commutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map.

References in corpus (39)

• Deformation quantization of Poisson manifolds, I
• Background Independent Quantum Gravity: A Status Report
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Minimal Length Scale Scenarios for Quantum Gravity
• Quantum Spacetime Phenomenology
• The Spin Foam Approach to Quantum Gravity
• The principle of relative locality
• Quantum $\kappa$-Poincare in Any Dimensions
• Group field theory with non-commutative metric variables
• De Sitter space as an arena for Doubly Special Relativity
• Local D=4 Field Theory on $\kappa$--Deformed Minkowski Space
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• The non-commutative Weil algebra
• Diffeomorphisms in group field theories
• Spin foam model for Lorentzian General Relativity
• Waves on Noncommutative Spacetime and Gamma-Ray Bursts
• Field theory on $\kappa$--Minkowski space revisited: Noether charges and breaking of Lorentz symmetry
• Taming nonlocality in theories with Planck-scale-deformed Lorentz symmetry
• Non-commutative flux representation for loop quantum gravity
• From noncommutative kappa-Minkowski to Minkowski space-time
• Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
• Bounds on an energy-dependent and observer-independent speed of light from violations of locality
• Twisting of quantum differentials and the Planck scale Hopf algebra
• Hopf-algebra description of noncommutative-spacetime symmetries
• 4d Deformed Special Relativity from Group Field Theories
• Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups
• Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime
• Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity
• Implementing causality in the spin foam quantum geometry
• Scalar field theory in Snyder space-time: alternatives
• Noncommutative field theory from angular twist
• Poisson structures on the cotangent bundle of a Lie group or a principal bundle and their reductions
• Noncommutative $R^d$ via closed star product
• Quantum Gravity phenomenology: achievements and challenges
• On the $\star$-product quantization and the Duflo map in three dimensions
• Chern-Simons expectation values and quantum horizons from LQG and the Duflo map
• Chern-Simons theory, Stokes' Theorem, and the Duflo map
• Quantum Mechanics on SO(3) via Non-commutative Dual Variables
• Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

Cited by in corpus (8)

• Group field theory condensate cosmology: An appetizer
• Emergent Cosmology from Quantum Gravity in the Lorentzian Barrett-Crane Tensorial Group Field Theory Model
• The Complete Barrett-Crane Model and its Causal Structure
• Mean-field phase transitions in TGFT quantum gravity
• $\kappa$-deformed complex fields and discrete symmetries
• Phase transitions in TGFT: a Landau-Ginzburg analysis of Lorentzian quantum geometric models
• Noncommutative Effective LQC: inclusion of potential term
• Quantum surface holonomies for loop quantum gravity and their application to black hole horizons