Quantum Field Theory in Curved Spacetime

TL;DR

This paper reviews the algebraic approach to quantum field theory in curved spacetime, emphasizing the mathematical foundations, state conditions, and key phenomena like Unruh and Hawking effects, while discussing open issues in the field.

Contribution

It provides a rigorous overview of the algebraic formulation of quantum fields in curved spacetime, including state conditions and key physical effects, highlighting unresolved questions.

Findings

States satisfying the Hadamard condition ensure finite stress-energy tensor

Revisits Unruh and Hawking effects within this framework

Discusses open problems like back-reaction and causality violations

Abstract

We review the mathematically rigorous formulation of the quantum theory of a linear field propagating in a globally hyperbolic spacetime. This formulation is accomplished via the algebraic approach, which, in essence, simultaneously admits all states in all possible (unitarily inequivalent) Hilbert space constructions. The physically nonsingular states are restricted by the requirement that their two-point function satisfy the Hadamard condition, which insures that the ultra-violet behavior of the state be similar to that of the vacuum state in Minkowski spacetime, and that the expected stress-energy tensor in the state be finite. We briefly review the Unruh and Hawking effects from the perspective of the theoretical framework adopted here. A brief discussion also is given of several open issues and questions in quantum field theory in curved spacetime regarding the treatment of ``back-reaction", the validity of some version of the ``averaged null energy condition'', and the formulation and properties of quantum field theory in causality violating spacetimes.