Axiomatic quantum field theory is a mathematical discipline which aims to describe

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

in terms of rigorous axioms. It is strongly associated with

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

and

operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study o ...

s, but has also been studied in recent years from a more geometric and functorial perspective. There are two main challenges in this discipline. First, one must propose a set of axioms which describe the general properties of any mathematical object that deserves to be called a "quantum field theory". Then, one gives rigorous mathematical constructions of examples satisfying these axioms.

Analytic approaches

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\,t~;

the factors of

\;\sqrt\;

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 The conformal bootstrap is a non-perturbative mathematical method to constrain and solve Conformal field theory, conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of ...

approach to

conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...

\mathbb^d

. They are also referred to as Euclidean bootstrap axioms.

References