Phase space properties and the short distance structure in quantum field theory

Abstract

The paper investigates relations between the phase space structure of a quantum field theory ("nuclearity") and the concept of pointlike localized fields. Given a net of local observable algebras, a phase space condition is introduced that allows a very detailed description of the theory's field content. An appendix discusses noninteracting models as examples.

Operator product expansions as a consequence of phase space properties

“…II, we recall the relevant facts and results on the point field structure established in Ref. 9, which lie at the root of our investigation. Section III defines products of pointlike fields and gives a proof of their product expansion.…”

“…A natural phase space condition, similar but not identical to the usual compactness or nuclearity conditions, 12 was proposed in Ref. 9 and shown to allow a precise description of the theory's field content. We will give a brief review of the results established in Ref.…”

“…We make use of the theory's phase space behavior, which has recently been shown to have a strong impact on the field content. 8,9 A phase space condition proposed in Ref. 9 can be taken as a physically natural assumption to ensure a regular short distance behavior.…”

“…8,9 A phase space condition proposed in Ref. 9 can be taken as a physically natural assumption to ensure a regular short distance behavior. On these grounds, we will establish operator product expansions rigorously in the sense of an asymptotic series.…”

Quantum Field Theory: Where We Are

“…Various axiomatic approaches (Wightman QFT, Euclidean QFT, Algebraic QFT) may differ in the technical way the principles are formulated. Several theorems establishing (partial) equivalences among these approaches have been established, such as the Osterwalder-Schrader reconstruction theorem [2] stating the precise prerequisites for the invertibility of the passage from real time QFT to Euclidean QFT ("Wick rotation"), or the possibility to recover local fields from local algebras [3]. In the Wightman formulation, one postulates the existence of fields as operator-valued distributions defined on a common dense domain within a Hilbert space.…”