Analyticity and positivity of Green’s functions without Lorentz

A bstract 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 $$ \overrightarrow{k} $$ k → . We discuss that in Fourier space microcausality is equivalent to the analyticity of G when Im( ω , $$ \overrightarrow{k} $$ 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 , $$ \overrightarrow{k} $$ 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 ·Im G ( ω , $$ \overrightarrow{k} $$ k → ) ≥ 0 for real ( ω , $$ \overrightarrow{k} $$ k → ). We show that this positivity property can be extended to the complex domain: Im[ ω G ( ω , $$ \overrightarrow{k} $$ 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 → 0 at infinity must be relaxed.