Axiomatic quantum field theory

The first set of axioms for quantum field theories, known as the Wightman axioms, were proposed by Arthur Wightman in the early 1950s. These axioms attempt to describe QFTs on flat Minkowski spacetime by regarding quantum fields as operator-valued distributions acting on a Hilbert space. In practice, one often uses the Wightman reconstruction theorem, which guarantees that the operator-valued distributions and the Hilbert space can be recovered from the collection of correlation functions.

The correlation functions of a QFT satisfying the Wightman axioms often can be analytically continued from Lorentz signature to Euclidean signature. (Crudely, one replaces the time variable ${\displaystyle \;t\;}$ with imaginary time ${\displaystyle \;\tau =-{\sqrt {-1\,}}\,t~;}$ the factors of ${\displaystyle \;{\sqrt {-1\,}}\;}$ change the sign of the time-time components of the metric tensor.) The resulting functions are called Schwinger functions. For the Schwinger functions there is a list of conditions — analyticity, permutation symmetry, Euclidean covariance, and reflection positivity — which a set of functions defined on various powers of Euclidean space-time must satisfy in order to be the analytic continuation of the set of correlation functions of a QFT satisfying the Wightman axioms.

The Haag–Kastler axioms axiomatize QFT in terms of nets of algebras.

These axioms (see e.g.[1]) are used in the conformal bootstrap approach to conformal field theory in ${\displaystyle \mathbb {R} ^{d}}$. They are also referred to as Euclidean bootstrap axioms.