Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2512.10843 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913097924476928 |
|---|---|
| author | Creminelli, Paolo Longo, Alessandro Salehian, Borna Zahed, Ahmadullah |
| author_facet | Creminelli, Paolo Longo, Alessandro Salehian, Borna Zahed, Ahmadullah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analyticity and positivity of Green's functions without Lorentz Creminelli, Paolo Longo, Alessandro Salehian, Borna Zahed, Ahmadullah High Energy Physics - Theory Statistical Mechanics Complex Variables 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. |
| title | Analyticity and positivity of Green's functions without Lorentz |
| topic | High Energy Physics - Theory Statistical Mechanics Complex Variables |
| url | https://arxiv.org/abs/2512.10843 |