Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction

TL;DR

This paper introduces the algebraic approach to quantum field theory on curved spacetimes, focusing on quasifree Hadamard states, CCR algebras, and the construction of symmetry-preserving states for a real scalar field.

Contribution

It provides an accessible introduction to the algebraic formalism of QFT in curved spacetime, emphasizing Hadamard quasifree states and their geometric and microlocal properties.

Findings

Existence of symmetry-invariant quasifree states in curved spacetime

Construction of Hadamard states using microlocal analysis

Introduction of Wick polynomials in the algebraic framework

Abstract

Within this chapter (published as [49]) we introduce the overall idea of the algebraic formalism of QFT on a fixed globally hyperbolic spacetime in the framework of unital $*$-algebras. We point out some general features of CCR algebras, such as simplicity and the construction of symmetry-induced homomorphisms. For simplicity, we deal only with a real scalar quantum field. We discuss some known general results in curved spacetime like the existence of quasifree states enjoying symmetries induced from the background, pointing out the relevant original references. We introduce, in particular, the notion of a Hadamard quasifree algebraic quantum state, both in the geometric and microlocal formulation, and the associated notion of Wick polynomials.