C Duval and P A Horváthy 2009 J. Phys. A: Math. Theor. 42 465206 doi:10.1088/1751-8113/42/46/465206
Non-relativistic conformal symmetries and Newton–Cartan structures
C Duval1,3 and P A Horváthy2
Show affiliationsThis 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 '
' 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.
- PACS
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04.20.Gz Spacetime topology, causal structure, spinor structure
02.20.Sv Lie algebras of Lie groups
02.30.Jr Partial differential equations
04.40.Nr Einstein-Maxwell spacetimes, spacetimes with fluids, radiation or classical fields
- MSC
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37K30 Relations with infinite-dimensional Lie algebras and other algebraic structures
35Q55 NLS-like (nonlinear Schrödinger) equations (See also 37K10)
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (See also 53D20)
83C75 Space-time singularities, cosmic censorship, etc.
22E60 Lie algebras of Lie groups (For the algebraic theory of Lie algebras, see 17Bxx)
- Subjects
- Dates
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Issue 46 (20 November 2009)
Received 8 July 2009, in final form 26 September 2009
Published 22 October 2009
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Non-relativistic conformal symmetries and Newton–Cartan structures
C Duval and P A Horváthy 2009 J. Phys. A: Math. Theor. 42 465206
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