The History and Present Status of Quantum Field Theory in Curved Spacetime

Abstract

Quantum field theory in curved spacetime is a theory wherein matter is treated fully in accord with the principles of quantum field theory, but gravity is treated classically in accord with general relativity. It is not expected to be an exact theory of nature, but it should provide a good approximate description in circumstances where the quantum effects of gravity itself do not play a dominant role. Some of the earliest applications of the theory were to study particle creation effects in an expanding universe. A major impetus to the theory was provided by Hawking's calculation of particle creation by black holes, showing that black holes radiate as perfect black bodies. During the past 30 years, considerable progress has been made in giving a mathematically rigorous formulation of quantum field theory in curved spacetime. Major issues of principle with regard to the formulation of the theory arise from the lack of Poincare symmetry, the absence of a preferred vacuum state, and, in general, the absence of asymptotic regions in which particle states can be defined. By the mid-1980's, it was understood how all of these difficulties could be overcome for free (i.e., non-self-interacting) quantum fields by formulating the theory via the algebraic approach and focusing attention on the local field observables rather than a notion of "particles". However, these ideas, by themselves, were not adequate for the formulation of interacting quantum field theory, even at a perturbative level, since standard renormalization prescriptions in Minkowski spacetime rely heavily on Poincare invariance and the existence of a Poincare invariant vacuum state. However, during the past decade, great progress has been made, mainly due to the importation into the theory of the methods of "microlocal analysis". This article will describe the historical development of the subject and describe some of the recent progress.

States of low energy in homogeneous and inhomogeneous expanding spacetimes

“…This characterization is reminiscent of the spectral condition in Minkowski spacetime. It has also been shown that the expectation value of the energymomentum tensor on these states is coherently renormalizable [8,9]. For the case of non-quasifree states, the singularities of all n-point functions can be determined from the singularities of their two-point function and the canonical commutation relations [10].…”

Vacuum fluctuation of conformally coupled scalar field in FLRW space-times

“…Several derivations of this effect can be found in literature [9,10,11,12,13] and recently the search for "experimental" verification making use of analogue models has been pursued by many investigators (see for example [14,15]). …”

“…the coordinates are x = (t, r), the areal radius is R = a(t) r and the normal metric simply reads 11) namely the trapping horizon is located at r H = 1 a and in term of the areal radius reads 12) where the Hubble parameter is given by H(t) = 1 a da dt . The quantity R H is known as Hubble Sphere, but we may also refer to it as Hubble dynamical horizon in the Hayward's terminology.…”

Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations

“…The ansatz of Born and Oppenheimer 12 is usually derived in the context of molecular Hamiltonians with external magnetic field 27,34 , 4) and project to the component equations of the eigenvalue equation, 5) w.r.t. this basis ( = 1):…”