Non-Relativistic BMS algebra

We construct two possible candidates for the non-relativistic ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra in 4 space-time dimensions by contracting the original relativistic ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra. The ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra is infinite-dimensional, and it contains the generators of the Poincar'e algebra, together with the so-called super-translations. Similarly, the proposed ${\mathfrak{n}\mathfrak{r}\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebras can be regarded as two infinite-dimensional extensions of the Bargmann algebra. We also study a canonical realisation of one these algebras in terms of the Fourier modes of a free Schr"odinger field, mimicking the canonical realisation of the relativistic ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra using a free Klein-Gordon field.