Axioms for Euclidean Green's functions

Abstract

We establish necessary and sufficient conditions for Euclidean Green's functions to define a unique Wightman field theory. Contents1 Introduction 83 2. Test Functions and Distributions 85 3. The Axioms, Main Theorems 87 4. TheoremE->R 90 4.1. Construction of the Wightman Distributions 90 4.

The unitary conformal field theory behind 2D Asymptotic Safety

“…At least on (nondynamical) flat spacetimes the criterion of Hilbert space positivity, alongside with the spectral condition can be translated from the Lorentzian to the Euclidean setting where it reappears as the requirement of reflection-, or Osterwalder-Schrader, positivity [4][5][6][7].…”

“…4 To summarize, for the evaluation of lim ε→0 1 ε S ε it is sufficient to consider the ε-dependence of √ gR, while the integration can be seen as an integration over d 2 x. This prescription can be considered our definition for taking the ε-limit in a well behaved way.…”

Introductory lectures to large- QCD phenomenology and lattice results

“….φ(x n )|0 . Under certain well-defined mathematical conditions [99], it is possible to carry out an analytical continuation to Euclidean spacetime, where the physical information is contained in a set of symmetric Schwinger functions. The symmetry of Schwinger functions is related to the fact that, in a certain sense, Euclidean quantum fields "can be treated as" classical variables.…”

Quantum Field Theory: Where We Are

“…The main modification is the transition form the indefinite Lorentz metric of Minkowski spacetime to a Euclidean metric; the return to the physical Lorentzian metric is expected to be manageable via the so-called Osterwalder-Schrader reconstruction [2] (see Sect. 2).…”

“…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.…”