User:Sylvain Ribault/Bootstat2021/Henkel

Schrödinger invariance: physical background, algebra, correlations functions, applications...

Dynamics out of equilibrium: quench system to temperature less than critical. The system orders into clusters whose size behaves as $$L\sim t^{\frac{1}{z}}$$. The exponent is determined by equilibrium state, in many cases $$z=2$$. Stationary states are not scale-invariant.

Measure two-time correlation function, and two-time auto-response. Not time translation invariance. But scaling form of two-time fct: basically reduce to fct of $$t/s$$ ratio of times, not $$t-s$$.

No time translation invariance, dynamical scaling, slow dynamics: the system is ageing (physics), terminology from glassy systems, now we give it a precise meaning.

Another example: growth regime in Edwards-Wilkinson equation leads to correlators with scaling properties. System also obeys three defining properties of ageing.

Can we reproduce these results from dynamical symmetry? Problem if nonzero temperature i.e. if there is noise.

Schrödinger-Virasoro algebra: covariant scalar characterized by scaling dimension and mass. Infinite-dimensional extension not observed in physics. It is the maximal extension. If no central charge, represent generators as differential operators in space and time. Not-semisimplicity allows central extension whose charge is the mass.

Solving symmetry eqn for two-point fct leads to Bargman rule: sum of masses vanishes, leads to fields being complex i.e. with negative mass. More generally the rule is to have as many response fields and non-response fields.

Tested in Glauber-Ising model in 2d and 3d.