Conformal symmetry of an extended Schrödinger equation and its relativistic origin

Abstract

In this paper two things are done. We first prove that an arbitrary power p of the Schrödinger Lagrangian in arbitrary dimension always enjoys the non-relativistic conformal symmetry. The implementation of this symmetry on the dynamical field involves a phase term as well as a conformal factor that depends on the dimension and on the exponent. This non-relativistic conformal symmetry is shown to have its origin on the conformal isometry of the power p of the Klein-Gordon Lagrangian in one higher dimension which is related to the phase of the complex scalar field.

Non-relativistic holography and singular black hole

“…As long as the field theory dynamics is concerned, one can also perform the Bargmann lift to have relativistic formulation of the conformal dynamics, for examples and general discussions, see [11,12,13,15,16]. Indeed, this is what was used in [8,9] to evaluate the GKP/W-type correlators [2,3] for a non-relativistic CFT from its bulk holographic co-dimension two scalar field theory.…”

“…The Newton-Cartan gravity [10], though formally covariant, is restricted by the Galilean symmetry to have the absolute time, and hence awkward to implement the holography. On the other hand, it is known that the Schrödinger symmetry can be obtained from the Kaluza-Klein (KK) reduction along a null-like direction [11,12,13]. Therefore, we can lift the Newton-Cartan gravity dual theory along a null Killing direction to one-dimensional higher covariant Einstein gravity, and the corresponding geometry is associated with the Bargmann group [14,15,18].…”