Analyticity and positivity of Green's functions without Lorentz

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.

64 pages, 5 figures