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Register a new userFuzzy Classical Dynamics as a Paradigm for Emerging Lorentz Geometries
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We show that the classical equations of motion for a particle on three dimensional fuzzy space and on the fuzzy sphere are underpinned by a natural Lorentz geometry. From this geometric perspective, the equations of motion generally correspond to forced geodesic motion, but for an appropriate choice of noncommutative dynamics, the force is purely noncommutative in origin and the underpinning Lorentz geometry some standard space-time with, in general, non-commutatuve corrections to the metric. For these choices of the noncommutative dynamics the commutative limit therefore corresponds to geodesic motion on this standard space-time. We identify these Lorentz geometries to be a Minkowski metric on $mathbb{R}^4$ and $mathbb{R} times S ^2$ in the cases of a free particle on three dimensional fuzzy space ($mathbb{R}^3_star$) and the fuzzy sphere ($S^2_star$), respectively. We also demonstrate the equivalence of the on-shell dynamics of $S^2_star$ and a relativistic charged particle on the commutative sphere coupled to the background magnetic field of a Dirac monopole.
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We derive the path integral action for a particle moving in three dimensional fuzzy space. From this we extract the classical equations of motion. These equations have rather surprising and unconventional features: They predict a cut-off in energy, a generally spatial dependent limiting speed, orbital precession remarkably similar to the general relativistic result, flat velocity curves below a length scale determined by the limiting velocity and included mass, displaced planar motion and the existence of two dynamical branches of which only one reduces to Newtonian dynamics in the commutative limit. These features place strong constraints on the non-commutative parameter and coordinate algebra to avoid conflict with observation and may provide a stringent observational test for this scenario of non-commutativity.
Classical point-particle relativistic lagrangians are constructed that generate the momentum-velocity and dispersion relations for quantum wave packets in Lorentz-violating effective field theory.
We generalize the coset procedure of homogeneous spacetimes in (pseudo-)Riemannian geometry to non-Lorentzian geometries. These are manifolds endowed with nowhere vanishing invertible vielbeins that transform under local non-Lorentzian tangent space transformations. In particular, we focus on nonrelativistic symmetry algebras that give rise to (torsional) Newton-Cartan geometries, for which we demonstrate how the Newton-Cartan metric complex is determined by degenerate co- and contravariant symmetric bilinear forms on the coset. In specific cases, we also show the connection of the resulting nonrelativistic coset spacetimes to pseudo-Riemannian cosets via Inonu-Wigner contraction of relativistic algebras as well as null reduction. Our construction is of use for example when considering limits of the AdS/CFT correspondence in which nonrelativistic spacetimes appear as gravitational backgrounds for nonrelativistic string or gravity theories.
A method is presented for deducing classical point-particle Lagrange functions corresponding to a class of quartic dispersion relations. Applying this to particles violating Lorentz symmetry in the minimal Standard-Model Extension leads to a variety of novel lagrangians in flat spacetime. Morphisms in these classical systems are studied that echo invariance under field redefinitions in the quantized theory. The Lagrange functions found offer new possibilities for understanding Lorentz-breaking effects by exploring parallels with Finsler-like geometries.
The current paper is dedicated to determining perturbative expansions for Lagrangians describing classical, relativistic, pointlike particles subject to Lorentz violation parameterized by the nonminimal Standard-Model Extension (SME). An iterative technique recently developed and applied to a Lorentz-violating scalar field theory is now adopted to treat the spin-degenerate SME fermion sector. Lagrangians are obtained at third order in Lorentz violation for the operators $hat{a}_{mu}$, $hat{c}_{mu}$, $hat{e}$, $hat{f}$, and $hat{m}$ for arbitrary mass dimension. The results demonstrate the impact of nonzero spin on classical particle propagation. They will be useful for phenomenological studies of modified gravity and could provide useful insights into explicit Lorentz violation in curved spacetimes.
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