Analyticity and positivity of Green's functions without Lorentz

TL;DR

This paper explores how microcausality and positivity impose analyticity and spectral constraints on Green's functions in Lorentz-violating theories, with implications for electromagnetic properties in media.

Contribution

It establishes analyticity and positivity properties of Green's functions without Lorentz invariance, extending dispersion relations and spectral constraints to such theories.

Findings

Microcausality implies analyticity of Green's functions in a forward light-cone domain.

Positivity of spectral density extends to complex frequencies, belonging to the Herglotz-Nevanlinna class.

Verified properties in examples with broken Lorentz invariance, such as non-zero chemical potential or temperature.

Abstract

We study the properties imposed by microcausality and positivity on the retarded two-point Green's function in a theory with spontaneous breaking of Lorentz invariance. We assume invariance under time and spatial translations, so that the Green's function $G$ depends on $\omega$ and $\vec k$. We discuss that in Fourier space microcausality is equivalent to the analyticity of $G$ when $\Im (\omega,\vec k)$ lies in the forward light-cone, supplemented by bounds on the growth of $G$ as one approaches the boundaries of this domain. Microcausality also implies that the imaginary part of $G$ (its spectral density) cannot have compact support for real $(\omega,\vec k)$. Using analyticity, we write multi-variable dispersion relations and show that the spectral density must satisfy a family of integral constraints. Analogous constraints can be applied to the fluctuations of the system, via the fluctuation-dissipation theorem. A stable physical system, which can only absorb energy from external sources, satisfies $\omega \cdot \Im G(\omega,\vec k) \ge 0$ for real $(\omega,\vec k)$. We show that this positivity property can be extended to the complex domain: $\Im [\omega\, G(\omega,\vec k)] >0$ in the domain of analyticity guaranteed by microcausality. Functions with this property belong to the Herglotz-Nevanlinna class. This allows to prove the analyticity of the permittivities $\epsilon(\omega,k)$ and $\mu^{-1}(\omega,k)$ that appear in Maxwell equations in a medium. We verify the above properties in several examples where Lorentz invariance is broken by a background field, e.g. non-zero chemical potential, or non-zero temperature. We study subtracted dispersion relations when the assumption $G \to 0$ at infinity must be relaxed.