Schr"odinger invariance and space-time symmetries

TL;DR

The paper addresses Schrödinger invariance and space-time symmetries in non-relativistic systems, clarifying how mass can be treated as a dynamical variable to convert the Schrödinger group’s projective action into a true representation via a Fourier dual field. It establishes a precise inclusion $\mathfrak{sch}_d \subset (\mathfrak{conf}_{d+2})_{\mathbb{C}}$, analyzes parabolic subalgebras that govern ageing dynamics, and develops Ward identities with an explicitly improved energy-momentum tensor. It then derives causal two- and three-point functions consistent with both conformal and Schrödinger covariance and connects these results to Martin-Siggia-Rose theory for response functions. The work provides a representation-theoretic foundation for non-equilibrium ageing phenomena and points toward infinite-dimensional extensions and applications to disordered and ageing systems.

Abstract

The free Schrödinger 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 Schrödinger 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.

Figures (3)

• Figure 1: Roots of $B_2$ and their relation with the generators of the Schrödinger algebra $\mathfrak{sch}_1$. The double circle in the centre denotes the Cartan subalgebra $\mathfrak{h}$.
• Figure 2: Integration contours in (\ref{['B:eq:I3']}) if $t_{23}>0$, indicated as oriented full lines. The thick black lines indicate the fixed cuts on the negative real axis and the grey lines indicate the moving cuts which occur for $u-u'+{\rm i} v$ real negative. On the left the contour is shown when one first integrates over $u$, with $\alpha =\frac{r_{13}^2}{2t_{13}}$ and $\beta=\frac{r_{23}^2}{2t_{23}}-v$. On the right one integrates first over $u'$, with $a=\frac{r_{23}^2}{2t_{23}}$ and $b=\frac{r_{13}^2}{2t_{13}}+v$.
• Figure 3: Integration contour in (\ref{['B:eq:I3']}) if $t_{23}<0$. The notation is the same as in figure \ref{['Abb2']}.