Continuum Mechanics in the Label Framework: Fluids, Liquid Crystals, Solids and Superfluids by the Symmetry of Their Labels

In the label framework a medium is a field of comoving labels, and its symmetry is a group of label maps. We classify media of three spacelike labels by this group. Maps with unit Jacobian give the perfect fluid of the series; subgroups give a smectic, a columnar phase, the isotropic solid and crystals. We count the elastic moduli of each group and compute stress and sound modes for test media. In these media the zero-frequency modes at a generic wave vector are the local relabellings the group keeps. Symmetry fixes forms and structural numbers, such as 3 moduli for a cubic crystal, but not the values of moduli; tabulated data we checked obey every bound. A superfluid is one timelike label; with its phase on a circle, circulation is quantised in units of h/m, with m the mass of the condensing unit. The results agree with standard continuum theory.