Analyticity and positivity of Green's functions without Lorentz

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 $ω$ and $\vec k$. We discuss that in Fourier space microcausality is equivalent to the analyticity of $G$ when $\Im (ω,\vec k)$ 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 $(ω,\vec k)$. 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 $ω\cdot \Im G(ω,\vec k) \ge 0$ for real $(ω,\vec k)$. We show that this positivity property can be extended to the complex domain: $\Im [ω, G(ω,\vec k)] >0$ 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 $ε(ω,k)$ and $μ^{-1}(ω,k)$ 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.

💡 Research Summary

This paper provides a comprehensive study of the fundamental properties of retarded two-point Green’s functions in theories where Lorentz invariance is spontaneously broken, a common scenario in condensed matter physics, finite temperature/chemical potential systems, and cosmology. The central object is the Fourier-space Green’s function G(ω, k), which encodes the linear response of a system to an external perturbation. The work investigates the profound constraints imposed on G(ω, k) by the basic physical principles of microcausality and stability (positivity).

The first major result establishes the equivalence between microcausality in position space (vanishing of commutators outside the lightcone) and analyticity in momentum space. Under the standard assumption that G is a tempered distribution, it is proven that G(ω, k) is analytic in the complex variables ω and k when the imaginary part vector Im(ω, k) lies inside the forward lightcone. This analyticity serves as the foundation for writing multi-variable dispersion relations, which express the full Green’s function as an integral over its imaginary part (spectral density).

A key insight from these dispersion relations is that analyticity imposes stringent, non-trivial constraints on the spectral density Im G(ω, k) for real ω and k. It is shown that Im G cannot have compact support. Furthermore, assuming G vanishes at infinity, Im G must satisfy an infinite family of integral constraints: ∫ dζ Im G(ζ, k + ξζ) = 0 for all real k and |ξ| < 1. The authors verify these constraints in several explicit examples, including Lorentz-invariant theories, response mediated by a single mode, kinetic theory, and systems at finite chemical potential or finite temperature.

The second major result concerns the extension of positivity from the real axis to the complex domain. A stable, passive physical system, which can only absorb energy, satisfies ω·Im G(ω,k) ≥ 0 for real ω and k. The paper demonstrates that this positivity condition can be extended to the entire domain of analyticity guaranteed by microcausality: Im