Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

Abstract

We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the Stückelberg-Bogoliubov-Epstein-Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, Köhler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space-times, we construct time-ordered operatorvalued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of timeordered products to all points of space-time. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space-times. Finite renormalizations are deferred to a subsequent paper.As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surface.

Quantum Fields in Curved Spacetime

“…16 Of course, the approximate description leading to this prediction should be valid only when M M P , where M P denotes the Planck mass (∼ 10 −5 gm), but modifications to the evaporation process at this stage (including the possibility of Planck mass remnants) would not significantly alter the discussion below. 17 This result, sometimes called "time-slice property", continues to hold for the enlarged algebra W defined in the next section and also for the algebra of interacting fields B I [26,55].…”

“…In the analysis of quantum field theory on the gravitational collapse spacetime shown in fig. 5-where the black hole remains present forever-the field observables in an open neighborhood of any Cauchy surface comprise the entire algebra 17 , A . Thus, a state that is pure at any initial "time" (i.e., a neighborhood of an initial Cauchy surface, say, prior to the collapse) automatically will remain pure at any final "time" (i.e., a neighborhood of a final Cauchy surface, say, after the black hole has formed).…”

“…The key breakthrough in this regard was made in [68,69]where the equivalence between (17) and (18) was proven-thereby enabling the methods of micolocal analysis to be applied to the renormalization problem, as first done in [16,17].…”

Action Ward Identity and the Stückelberg-Petermann Renormalization Group

“…On the other side, the path integral approach is intrinsically nonlocal; therefore difficulties arise for the treatment of infrared problems and of finite temperatures, and theories on curved back grounds or with external fields cannot properly be formulated. In algebraic quantum field theory, concepts for dealing with these problems have been developed [13,2,4,14,5,17]. We therefore started a program for a perturbative construction of the net of local algebras of observables [4,7].…”

“…In algebraic quantum field theory, concepts for dealing with these problems have been developed [13,2,4,14,5,17]. We therefore started a program for a perturbative construction of the net of local algebras of observables [4,7]. There already exists a local formulation of renormalization due to Epstein and Glaser [11] and based on older ideas of Stückelberg and Bogoliubov [1].…”