Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction

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.

Ground states of a Klein-Gordon field with Robin boundary conditions in global anti–de Sitter spacetime

“…The only freedom left is the choice of a quantum state of Hadamard form, a widely accepted condition which entails several relevant physical properties. On the one hand, the quantum fluctuations of all observables are finite, while, on the other hand, it guarantees the existence of a covariant notion of Wick polynomials out of which one can deal with interactions within a perturbation scheme, see e.g., [3,4].…”

Mathematical foundations of quantum mechanics: An advanced short course

“…Here R 3 is the rest space of a given reference frame in the spacetime. Those field operators satisfy commutation relations similar to the ones of X k and P k (e.g., see [26,27,28]). Then the Stone-von Neumann theorem no longer holds.…”

“…Jumping from the finitedimensional case to the infinite-dimensional one corresponds to passing from Quantum Mechanics to Quantum Field Theory (possibly relativistic, and on curved spacetime [28]). The presence of non-equivalent representations of one single physical system shows that a formulation in a fixed Hilbert space is fully inadequate, a least because it insists on a fixed Hilbert space, whereas the physical system is characterized by a more abstract object: An algebra of observables which may be represented in different Hilbert spaces in terms of operators.…”

Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes

“…Nowadays regarded as an indispensable ingredient in the perturbative construction of interacting fields (see e.g. recent reviews [HW3,KM,FV2]), this property accounts for the correct short-distance behaviour of two-point functions. It can be conveniently formulated as a condition on the wave front set of the state's two-point functions [Ra] -a terminology that we explain in the paragraphs below.…”

“…The importance of Hadamard states is primarily due to their pivotal role in renormalization on curved spacetimes [BF2,HW1,HW2,Da], see [FV2,KM,HW3] for recent reviews.…”