The Hadamard parametrix on half-Minkowski with Robin boundary conditions: Fundamental solutions and Hadamard states
B. Costeri, C. Dappiaggi, B. A. Juárez-Aubry, R. D. Singh
TL;DR
We study a real Klein-Gordon field on half-Minkowski spacetime ${\mathbb{H}}^d$ with Robin boundary conditions $(\partial_z+\kappa)\Phi|_{z=0}=0$ for $\kappa>0$ and $d\ge 2$. The approach combines a Robin-to-Dirichlet transform (generalizing Bondurant–Fulling) to construct advanced/retarded propagators $G^{\pm}_\kappa$, with a Hadamard parametrix that includes a reflected term in $\sigma_{-}$ to capture boundary effects, and a global/local Hadamard framework with Radzikowski-type wavefront descriptions. It yields Hadamard two-point functions $\omega_{2,\kappa}$ that satisfy both global and local Hadamard conditions, proves a Radzikowski-like equivalence via generalized broken bicharacteristics, and provides a Robin Feynman parametrix for perturbative QFT. These results establish a rigorous foundation for renormalization and perturbation theory of quantum fields on spacetimes with timelike boundaries, incorporating boundary-reflected singularities in a controlled microlocal setting.
Abstract
We address the problem of constructing fundamental solutions and Hadamard states for a Klein-Gordon field in half-Minkowski spacetime with Robin boundary conditions in $d \geq 2$ spacetime dimensions. First, using a generalisation of the Robin-to-Dirichlet map exploited by Bondurant and Fulling [J. Phys. A: Math. Theor. {\bf 38} 7 (2005)] in dimension $2$, we obtain a representation for the advanced and retarded Green operators in terms of a convolution with the kernel of the inverse Robin-to-Dirichlet map. This allows us to prove the uniqueness and support properties of the Green operators. Second, we obtain a local representation for the Hadamard parametrix that provides the correct local definition of Hadamard states in $d \geq 2$ dimensions, capturing `reflected' singularities from the spacetime boundary. We show that our fundamental solutions abide by this local parametrix representation. Finally, we prove the equivalence of our local Hadamard condition and the global Hadamard condition with a wave-front set described in terms of generalized broken bi-characteristics, obtaining a Radzikowski-like theorem in half-Minkowski spacetime.