Non-relativistic conformal symmetries and Newton-Cartan structures. (English) Zbl 1180.37078

Duval, C.; Horváthy, P. A.

J. Phys. A, Math. Theor. 42, No. 46, Article ID 465206, 32 p. (2009).

Summary: This paper provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational ‘dynamical exponent’, \(z\). The Schrödinger-Virasoro algebra of Henkel et al. corresponds to \(z = 2\). Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger Lie algebra, for which \(z = 2\). For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) of Lukierski, Stichel and Zakrzewski (alias ’\(\mathfrak {alt} \)’ of Henkel), with \(z = 1\). Physical systems realizing these symmetries include, e.g. classical systems of massive and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism.

MSC:

37J15	Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37K05	Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
70G65	Symmetries, Lie group and Lie algebra methods for problems in mechanics
53B15	Other connections

Keywords:

Newton-Cartan space-time; Galilei structure; Schrödinger-Virasoro algebra; Schrödinger-Lie algebra