Axiomatic quantum field theory
Encyclopedia
Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis
and operator algebra
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 give rigorous mathematical constructions of examples satisfying these 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.
from Lorentz signature to Euclidean signature. (Crudely, one replaces the time variable
with imaginary time 
; the factors of 
change the sign of the time-time components of the metric tensor.) The resulting functions are called Schwinger functions. The Schwinger functions are a list of conditions—analycity, permutation symmetry, Euclidean covariance, and reflection positivity—which a set of functions defined on various powers of Euclidean spacetime must satisfy in order to be the analytic continuation of the set of correlation functions of a QFT satisfying the Wightman axioms.
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
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 give rigorous mathematical constructions of examples satisfying these axioms.
Wightman axioms
The first set of axioms for quantum field theories, known as the Wightman axiomsWightman axioms
In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.The axioms exist in...
. were proposed by Arthur Wightman
Arthur Wightman
Arthur Strong Wightman is an American mathematical physicist. He is one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms....
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 continuedAnalytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
from Lorentz signature to Euclidean signature. (Crudely, one replaces the time variable
with imaginary time ; the factors of change the sign of the time-time components of the metric tensor.) The resulting functions are called Schwinger functions. The Schwinger functions are a list of conditions—analycity, permutation symmetry, Euclidean covariance, and reflection positivity—which a set of functions defined on various powers of Euclidean spacetime must satisfy in order to be the analytic continuation of the set of correlation functions of a QFT satisfying the Wightman axioms.