Schr"odinger invariance and space-time symmetries

TL;DR

This paper explores the Schr"odinger invariance and space-time symmetries, classifying related algebras, proposing a new symmetry group for non-equilibrium aging phenomena, and deriving associated Ward identities and correlation functions.

Contribution

It classifies parabolic subalgebras of conformal algebra, introduces a new dynamic symmetry group for aging systems, and derives Ward identities and correlation functions.

Findings

Classification of parabolic subalgebras of conf_3

Proposal of a new symmetry group for aging phenomena

Derivation of Ward identities and correlation functions

Abstract

The free Schr\"odinger equation with mass M can be turned into a non-massive Klein-Gordon equation via Fourier transformation with respect to M. The kinematic symmetry algebra sch_d of the free d-dimensional Schr\"odinger equation with M fixed appears therefore naturally as a parabolic subalgebra of the complexified conformal algebra conf_d+2 in d+2 dimensions. The explicit classification of the parabolic subalgebras of conf_3 yields physically interesting dynamic symmetry algebras. This allows us to propose a new dynamic symmetry group relevant for the description of ageing far from thermal equilibrium, with a dynamical exponent z=2. The Ward identities resulting from the invariance under conf_d+2 and its parabolic subalgebras are derived and the corresponding free-field energy-momentum tensor is constructed. We also derive the scaling form and the causality conditions for the two- and three-point functions and their relationship with response functions in the context of Martin-Siggia-Rose theory.