Construction of Hadamard states by characteristic Cauchy problem

TL;DR

This paper develops a method to construct Hadamard states for Klein-Gordon fields in a spacetime region defined by a lightcone, using boundary symplectic spaces and pseudodifferential calculus, ensuring the states satisfy the physical Hadamard condition.

Contribution

It introduces a novel boundary-based approach to construct Hadamard states via characteristic Cauchy problems, extending the toolkit for quantum field theory in curved spacetime.

Findings

Constructed a boundary symplectic space on the lightcone.

Identified conditions for boundary states to induce Hadamard states.

Characterized pure boundary states that lead to pure bulk states.

Abstract

We construct Hadamard states for Klein-Gordon fields in a spacetime $M_{0}$ equal to the interior of the future lightcone $C$ from a base point $p$ in a globally hyperbolic spacetime $(M, g)$. Under some regularity conditions at future infinity of $C$, we identify a boundary symplectic space of functions on $C$, which allows to construct states for Klein-Gordon quantum fields in $M_{0}$ from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on $C$ ensuring that the bulk state it induces in $M_{0}$ satisfies the Hadamard condition. Using pseudodifferential calculus on the cone $C$ we construct a large class of Hadamard boundary states on the boundary with pseudodifferential covariances, and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in $M_{0}$.