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 classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

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 of ...

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 In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the ear ...

, were proposed by

Arthur Wightman Arthur Strong Wightman (March 30, 1922 – January 13, 2013) was an American mathematical physicist. He was one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms. With his rigorous treatm ...

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 In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...

from

Lorentz signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative an ...

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