Schrödinger symmetry: a historical review

TL;DR

The paper traces the origins and development of non-relativistic conformal symmetry, notably Schrödinger symmetry, from Jacobi’s conserved quantities to Lie’s conformal transformations, Eisenhart’s Bargmann geometry, and Bargmann’s central extension. It assembles a coherent picture in which the Schrödinger group ${\rm Sch}(d)$ and the conformal Galilean algebra ${\rm CGA}(d)$ arise via contractions, reductions, and geometric embeddings, with the mass operator $M$ enforcing Bargmann super-selection rules. Local scale-invariance generalizations (Schrödinger–Virasoro, meta-conformal, etc.) extend these ideas to $z\neq 1,2$, yielding covariance-based predictions for two-point and higher correlators, including ageing phenomena. The work highlights the unifying role of central extensions, Bargmann structures, and contraction methods across classical mechanics, quantum theory, and non-equilibrium statistical physics, offering a foundation for future explorations of extended algebras and non-relativistic holographic correspondences.

Abstract

This paper reviews the history of the conformal extension of Galilean symmetry, now called Schrödinger symmetry. In the physics literature, its discovery is commonly attributed to Jackiw, Niederer and Hagen (1972). However, Schrödinger symmetry has a much older ancestry: the associated conserved quantities were known to Jacobi in 1842/43 and its euclidean counterpart was discovered by Sophus Lie in 1881 in his studies of the heat equation. A convenient way to study Schrödinger symmetry is provided by a non-relativistic Kaluza-Klein-type "Bargmann" framework, first proposed by Eisenhart (1929), but then forgotten and re-discovered by Duval {\it et al.} only in 1984. Representations of Schrödinger symmetry differ by the value $z=2$ of the dynamical exponent from the value $z=1$ found in representations of relativistic conformal invariance. For generic values of $z$, whole families of new algebras exist, which for $z=2/\ell$ include the $\ell$-conformal galilean algebras. We also review the non-relativistic limit of conformal algebras and that this limit leads to the $1$-conformal galilean algebra and not to the Schrödinger algebra. The latter can be recovered in the Bargmann framework through reduction. A distinctive feature of Galilean and Schrödinger symmetries are the Bargmann super-selection rules, algebraically related to a central extension. An empirical consequence of this was known as "mass conservation" already to Lavoisier. As an illustration of these concepts, some applications to physical ageing in simple model systems are reviewed.

Figures (5)

• Figure 1: Spatial regions where various un-biased or biased symmetries can be realised. When distances scale isotropically with time as $r\sim \tau^{1/2}$, the space-time dynamical symmetry is the Schrödinger algebra. If a bias occurs and distances scale in the preferred direction as $r_{\|}\sim \tau$ while $r_{\perp}\sim \tau^{1/2}$ in the transverse direction, meta-Schrödinger invariance is realised. But if $r_{\|}\sim r_{\perp}\sim\tau$, meta-conformal symmetry may be realised, under certain conditions.
• Figure 2: Left: Root diagram of the complex Lie algebra $B_2$, with the generators $H,P,M,D,B,K$ of table \ref{['tab:gen']} and the four additional ones $V_{\pm},W,N$. Right: the three minimal standard parabolic sub-algebras $\widetilde{\mathfrak{sch}}(1)=\mathfrak{sch}(1)\oplus\mathbb{C}N$, $\widetilde{\mathfrak{age}}(1)=\mathfrak{age}(1)\oplus\mathbb{C}N$ and $\widetilde{\hbox{\sc cga}}(1)=\hbox{\sc cga}(1)\oplus\mathbb{C}N$.
• Figure 3: (a) Scaling function $f(u)$ of the covariant two-point correlator ${C}(t,r)=t^{-2x_1}f(r/t)$, over against the scaling variable $u=r/t$, for the ortho-conformal, meta-conformal and conformal Galilean algebras, in $(1+1)D$, from eq. (\ref{['gl:correlateurs']}). The inset further underlines the different behaviour for $u\ll 1$ and $u\gg 1$. (b) Comparison with the scaling function obtained from Schrödinger-invariance, clearly distinct from both ortho-conformal and conformal Galilean invariance.
• Figure 4: Schematic free energy before a quench (left panel) and after a quench to either $T=T_c$ or $T<T_c$ (right panel). The state of the system is symbolised by the small ball.
• Figure 5: Illustration of the characteristic data collapse of physical ageing. Panel (a) shows a typical behaviour of a single-time correlator for different times $t_3>t_2>t_1$, while (b) shows the collapse onto a single curve when distances $r=|\boldsymbol{r}|$ are measured in units of the dynamical length scale $L(t)$. Panel (c) similarly illustrates the two-time autocorrelator in dependence of $\tau=t-s$, for different waiting times $s_1<s_2<s_3$ and panel (d) shows that these data collapse when replotted as a function of $y=t/s$. The log-log plot in the inset shows the asymptotic power-law form $f_C(y)\sim y^{-\lambda/z}$.