Wightman axioms 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.
Osterwalder–Schrader axioms 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 \;t\; with imaginary time \;\tau = -\sqrt{-1\,}\,t~; the factors of \;\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.
Haag–Kastler axioms The
Haag–Kastler axioms axiomatize QFT in terms of nets of algebras.
Euclidean CFT axioms These axioms (see e.g.) are used in the
conformal bootstrap approach to
conformal field theory in \mathbb{R}^d. They are also referred to as
Euclidean bootstrap axioms. ==See also==