Quantum Field Theory on Curved Backgrounds -- A Primer

TL;DR

This review introduces the algebraic approach to quantum field theory on curved backgrounds, emphasizing its axiomatic foundation and applicability to free fields, with a focus on cosmological applications.

Contribution

It provides a comprehensive introduction to the algebraic method for quantum fields on curved spacetimes, highlighting its ability to handle multiple inequivalent representations.

Findings

Explains the algebraic framework for free quantum fields on curved backgrounds.

Details the process of selecting algebraic states for Hilbert space interpretation.

Prepares readers for recent applications in cosmology.

Abstract

Goal of this review is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, a suitable algebra of observables is assigned to a physical system, which is meant to encode all algebraic relations among observables, such as commutation relations, while, in the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give to the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.