On the Schr{ö}dinger-Newton equation and its symmetries: a geometric view

Abstract

The \SN (SN) equation is recast on purely geometrical grounds, namely in terms of Bargmann structures over $(\d+1)$-dimensional Newton-Cartan (NC) spacetimes. Its maximal group of invariance, which we call the SN group, is determined as the group of conformal Bargmann automorphisms that preserve the coupled Schrödinger and NC gravitational field equations. Canonical unitary representations of the SN group are worked out, helping us recover, in particular, a very specific occurrence of dilations with dynamical exponent $z=(\d+2)/3$.

LaTeX 29 pages; minor corrections

References in corpus (5)

• Non-relativistic conformal symmetries and Newton-Cartan structures
• Hidden Symmetries of Dynamics in Classical and Quantum Physics
• Conformal Killing Tensors and covariant Hamiltonian Dynamics
• Dark Energy and the Schwarzian Derivative
• Self-gravitating darkon fluid with anisotropic scaling

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