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 algebras
, 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.
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 <math>t</math> with imaginary time <math>tau = -sqrtt</math>; the factors of <math>sqrt</math>... Read More