Hadamard states from null infinity

TL;DR

This paper presents a method to construct Hadamard states for free field theories on spacetimes with null boundaries, focusing on massless scalar fields and linearized gravity in asymptotically flat backgrounds.

Contribution

It introduces a scheme for constructing Hadamard states using null infinity, applicable to conformally coupled scalar fields and linearized gravity.

Findings

Constructed Hadamard states from null infinity for specific fields.

Applicable to globally hyperbolic, asymptotically flat spacetimes.

Provides a framework for physically acceptable quantum states.

Abstract

Free field theories on a four dimensional, globally hyperbolic spacetime, whose dynamics is ruled by a Green hyperbolic partial differential operator, can be quantized following the algebraic approach. It consists of a two-step procedure: In the first part one identifies the observables of the underlying physical system collecting them in a *-algebra which encodes their relational and structural properties. In the second step one must identify a quantum state, that is a positive, normalized linear functional on the *-algebra out of which one recovers the interpretation proper of quantum mechanical theories via the so-called Gelfand-Naimark-Segal theorem. In between the plethora of possible states, only few of them are considered physically acceptable and they are all characterized by the so-called Hadamard condition, a constraint on the singular structure of the associated two-point function. Goal of this paper is to outline a construction scheme for these states which can be applied whenever the underlying background possesses a null (conformal) boundary. We discuss in particular the examples of a real, massless conformally coupled scalar field and of linearized gravity on a globally hyperbolic and asymptotically flat spacetime.