Schrödinger Symmetry: A Historical Review

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 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.