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Aspects of hot Galilean field theory

Kristan Jensen

Abstract

We reconsider general aspects of Galilean-invariant thermal field theory. Using the proposal of our companion paper, we recast non-relativistic hydrodynamics in a manifestly covariant way and couple it to a background spacetime. We examine the concomitant consequences for the thermal partition functions of Galilean theories on a time-independent, but weakly curved background. We work out both the hydrodynamics and partition functions in detail for the example of parity-violating normal fluids in two dimensions to first order in the gradient expansion, finding results that differ from those previously reported in the literature. As for relativistic field theories, the equality-type constraints imposed by the existence of an entropy current appear to be in one-to-one correspondence with those arising from the existence of a hydrostatic partition function. Along the way, we obtain a number of useful results about non-relativistic hydrodynamics, including a manifestly boost-invariant presentation thereof, simplified Ward identities, the systematics of redefinitions of the fluid variables, and the positivity of entropy production.

Aspects of hot Galilean field theory

Abstract

We reconsider general aspects of Galilean-invariant thermal field theory. Using the proposal of our companion paper, we recast non-relativistic hydrodynamics in a manifestly covariant way and couple it to a background spacetime. We examine the concomitant consequences for the thermal partition functions of Galilean theories on a time-independent, but weakly curved background. We work out both the hydrodynamics and partition functions in detail for the example of parity-violating normal fluids in two dimensions to first order in the gradient expansion, finding results that differ from those previously reported in the literature. As for relativistic field theories, the equality-type constraints imposed by the existence of an entropy current appear to be in one-to-one correspondence with those arising from the existence of a hydrostatic partition function. Along the way, we obtain a number of useful results about non-relativistic hydrodynamics, including a manifestly boost-invariant presentation thereof, simplified Ward identities, the systematics of redefinitions of the fluid variables, and the positivity of entropy production.

Paper Structure

This paper contains 36 sections, 284 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction and summary
  2. Summary of results for fluid mechanics
  3. Summary of results for thermal partition functions
  4. The plan
  5. A primer to Galilean-invariant field theory in curved spacetime
  6. Newton-Cartan geometry and Milne boosts
  7. Symmetries and Ward identities
  8. A Milne covariant derivative, and a boost-invariant energy current
  9. Including a magnetic moment
  10. Basics of life at $T > 0$
  11. Euclidean field theory and thermal circles
  12. Hydrodynamic variables
  13. Constitutive relations and the gradient expansion
  14. Positivity of entropy production
  15. Reformulating non-relativistic hydrodynamics
  16. ...and 21 more sections

Figures (1)

  • Figure 1: A depiction of the Euclidean manifolds from which we construct thermal field theory. There is a thermal circle at each point on the spatial slice, whose size and orientation vary smoothly as one moves around in space. The spacetime is invariant under uniform translation along the circle. The total structure is that of a (thermal) circle bundle.