  The analytic continuation to an imaginary velocity of the canonical partition
function of a thermal system expressed in a moving frame has a natural
implementation in the Euclidean path-integral formulation in terms of shifted
boundary conditions. The Poincare' invariance underlying a relativistic theory
implies a dependence of the free-energy on the compact length L_0 and the shift
xi only through the combination beta=L_0(1+xi^2)^(1/2). This in turn implies
that the energy and the momentum distributions of the thermal theory are
related, a fact which is encoded in a set of Ward identities among the
correlators of the energy-momentum tensor. The latter have interesting
applications in lattice field theory: they offer novel ways to compute
thermodynamic potentials, and a set of identities to renormalize
non-perturbatively the energy-momentum tensor. At fixed bare parameters the
shifted boundary conditions also provide a simple method to vary the
temperature in much smaller steps than with the standard procedure.
