Planar Carrollean dynamics, and the Carroll quantum equation

Abstract

We expand on the known result that the Carroll algebra in $2+1$ dimensions admits two non-trivial central extensions by computing the associated Lie group, which we call extended Carroll group. The symplectic geometry associated to this group is then computed to describe the motion of planar Carroll elementary particles, in the free case, when coupled to an electromagnetic field, and to a gravitational field. We compare to the motions of Carroll particles in $3+1$ dimensions in the same conditions, and also give the dynamics of Carroll particles with spin. In an electromagnetic background, the planar Carroll dynamics differ from the known Carroll ones due to 2 new Casimir invariants, and turn out to be non-trivial. The coupling to a gravitational field leaves the dynamics trivial, however. Finally, we obtain the quantum equation obeyed by Carroll wave functions \textit{via} geometric quantization.

24 pages, 1 figure; added a section about gravitational coupling; to be published in Journal of Geometry and Physics

References in corpus (6)

• Conformal Carroll groups and BMS symmetry
• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• Flat Holography: Aspects of the dual field theory
• Carroll contractions of Lorentz-invariant theories
• Conformal Carroll groups
• Dynamics of Carroll Particles

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• Carroll Expansion of General Relativity
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• Universal Corner Symmetry and the Orbit Method for Gravity
• Carrollian Motion in Magnetized Black Hole Horizons
• Quantization of Carrollian conformal scalar theories
• Carrollian limit of quadratic gravity
• Memory Effect and Carroll Symmetry: 50 years later
• Uniqueness of Galilean and Carrollian limits of gravitational theories and application to higher derivative gravity
• Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei
• MultiCarroll dynamics
• Fracton and Non-Lorentzian Particle Duality: Gauge Field Couplings and Geometric Implications
• Carroll black holes in (A)dS and their higher-derivative modifications
• Conformally related vacuum gravitational waves, and their symmetries
• Post-Carrollian Mechanics, Ideal Gas, and Gravity
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• Asymptotic symmetries in Carrollian theories of gravity with a negative cosmological constant
• Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs