Implications of Poincaré symmetry for thermal field theories in finite-volume

Abstract

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.