**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.

## 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 <math>t</math> with imaginary time <math>tau = -sqrtt</math>; the factors of <math>sqrt</math>...

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