  The analytic continuation to an imaginary velocity $i\xi$ 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. Writing the Boltzmann factor as
$\exp[-L_0(H-i\xi.P)]$, the Poincare invariance underlying a relativistic
theory implies a dependence of the free-energy on $L_0$ and the shift $\xi$
only through the combination $\beta= L_0 \sqrt{1+\xi^2}$. This in turn implies
a set of Ward identities, some of which were previously derived by us, among
the correlators of the energy-momentum tensor. In the infinite-volume limit
they lead to relations among the cumulants of the total energy distribution and
those of the momentum, i.e. they connect the energy and the momentum
distributions in the canonical ensemble. In finite volume the Poincare symmetry
translates into exact relations among partition functions and correlation
functions defined with different sets of (generalized) periodic boundary
conditions. They have interesting applications in lattice field theory. In
particular, they offer Ward identities to renormalize non-perturbatively the
energy-momentum tensor and novel ways to compute thermodynamic potentials. At
fixed bare parameters they also provide a simple method to vary the temperature
in much smaller steps than with the standard procedure.
