Quantum Field Theory on Curved Backgrounds -- A Primer

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.

36 pages, invited review, typos corrected, references added

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Cited by in corpus (39)

• Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction
• QFT on curved spacetimes: axiomatic framework and examples
• Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions
• Ground states of a Klein-Gordon field with Robin boundary conditions in global anti-de Sitter spacetime
• Quantum fields during black hole formation: How good an approximation is the Unruh state?
• The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States
• Algebraic quantum field theory on spacetimes with timelike boundary
• On the algebraic quantization of a massive scalar field in anti-de-Sitter spacetime
• Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states
• Ground state for a massive scalar field in BTZ spacetime with Robin boundary conditions
• Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary
• Inflation without quantum gravity
• Linear Yang-Mills theory as a homotopy AQFT
• Locally covariant chiral fermions and anomalies
• Covariant canonical formulations of classical field theories
• Quantization of the linearised Einstein-Klein-Gordon system on arbitrary backgrounds and the special case of perturbations in Inflation
• The Fermionic Signature Operator in De Sitter Spacetime
• The algebra of Wick polynomials of a scalar field on a Riemannian manifold
• The Lambda CDM-model in quantum field theory on curved spacetime and Dark Radiation
• Constructing Hadamard states via an extended Møller operator
• A new class of Fermionic Projectors: Møller operators and mass oscillation properties
• Algebraic field theory operads and linear quantization
• Second quantized quantum field theory based on invariance properties of locally conformally flat space-times
• Linear hyperbolic PDEs with non-commutative time
• Probing cosmological singularities with quantum fields: Open and closed FLRW universes
• Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation
• Local Thermal Equilibrium States in Relativistic Quantum Field Theory
• The Casimir effect from the point of view of algebraic quantum field theory
• Curvature fluctuations on asymptotically de Sitter spacetimes via the semiclassical Einstein's equations
• Lorentzian bordisms in algebraic quantum field theory
• A numerical approach to particle creation in accelerating toy models
• A Canonical Complex Structure and the Bosonic Signature Operator for Scalar Fields in Globally Hyperbolic Spacetimes
• Spontaneous particle creation by oscillating compact stars
• Effective action for a quantum scalar field in warped spaces
• Locality in Abelian gauge theories over globally hyperbolic spacetimes
• Quantum field theory on curved spacetime and the standard cosmological model
• On the uniqueness of invariant states
• A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications
• Production of Dirac particle in a deformed Minkowsky space-time