Dancers on a stage with moving walls: particles in a finite quantum system collide with each other and reflect off the edges, but the walls can move according to a prescribed scenario. Physicists have long been able to describe such dances only with fixed walls or in an infinite hall. Now there is a way to account for both the motion of the walls and the reflections from them.

For open boundaries, boundary S-matrices are introduced to describe particle reflection with reversal of direction. Consistency of the whole picture requires that the bulk S-matrices satisfy the Yang–Baxter equations, and the boundary ones the reflection equations. Solving these equations yields the allowed time dependences of the boundary conditions.

In the Gross–Neveu model with SU(2) symmetry and time-dependent interaction g(t), explicit expressions for the bulk S-matrices are obtained in terms of a function f(t) related to g(t). Boundary conditions compatible with integrability are determined by Cherednik K-matrices with parameters $ξ_L$ and $ξ_R$, which in the static case are renormalization-group invariants, and in the dynamic case are conserved quantities.

This opens the way to exact analysis of nonequilibrium phenomena in finite systems, such as ultracold atoms in traps with controllable walls. The connection between renormalization-group invariants of the static model and dynamic invariants of the time-dependent model confirms a deep analogy between time and cutoff scale. The dance of particles on a stage with moving walls can now be described exactly.