Analyticity and positivity of Green's functions without Lorentz

TL;DR

This work builds a rigorous framework linking microcausality and positivity to the analyticity of retarded Green's functions in theories with spontaneously broken Lorentz invariance. By treating $G(\omega,\vec{k})$ as a function of multiple complex variables, the authors derive multi-variable dispersion relations (Leontovich-type) and establish integral constraints on the spectral density and fluctuations via the fluctuation-dissipation theorem. They prove that $\omega\,G(\omega,\vec{k})$ belongs to the Herglotz–Nevanlinna class in the forward tube, enabling analytic control of electromagnetic response functions $\epsilon(\omega,k)$ and $\mu^{-1}(\omega,k)$ and extending causality constraints to complex domains. The paper provides concrete examples at finite chemical potential and temperature, discusses subtracted dispersion relations, and generalizes the formalism to broken rotational symmetry, while connecting to spectral representations and S-matrix ideas in non-Lorentz-invariant settings. These results offer a robust foundation for constraining effective field theories and transport in systems where Lorentz symmetry is not realized, with potential applications to condensed matter, cosmology, and beyond.

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 $ω$ 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.

Figures (5)

• Figure 1: The region of analyticity guaranteed by microcausality $D$ in red and the poly upper-half plane (PUHP) in the $(z_1,z_2)$ space in yellow. Plot in the $(\omega_I,k_I)$ plane (left) and in the $(z_{1I},z_{2I})$ plane (right). For $\alpha>1$ the region of analyticity $D$ always includes the $\rm{PUHP}$.
• Figure 2: Integration contour used in \ref{['2dcauchy']}.
• Figure 3: The three cases of compact support of ${\rm Im}\,G(\omega, \vec{k})$ discussed in the text.
• Figure 4: Comparison between constant-$\omega_I$ sections in $\tilde{d}=3$. Two pyramids inside the ${\rm FLC}$ in $\tilde{d}=3$. Varying $\alpha$ changes the area of their constant-$\omega_I$ sections. Setting $\omega_I=3/2$, the green and yellow sections are obtained by choosing $\alpha=2$ (green) or $\alpha=11/10$ (yellow).
• Figure 5: Comparison between constant-$\omega_I$ sections in $\tilde{d}=4$. The constant-$\omega_I$ sections of the hyperpyramids are regular tetrahedrons inscribed in $S^2_{\omega_I/\alpha}$. Setting $\omega_I=2$, the green and yellow tetrahedrons are obtained by the choices $\alpha=2$ and $\alpha=4/3$ respectively.