Equivalence of the Euclidean and Wightman Field Theories

TL;DR

This paper establishes the equivalence between Euclidean and Wightman field theories by proving that Osterwalder--Schrader axioms and the weak spectral condition are mutually consistent, using a new inversion formula for Laplace transforms.

Contribution

It introduces a new inversion formula for Laplace transforms of tempered distributions and proves the equivalence of Euclidean and Wightman axioms under the weak spectral condition.

Findings

Proved the new inversion formula for Laplace transforms.

Established the equivalence of Osterwalder--Schrader and Wightman axioms.

Demonstrated the consistency of Euclidean Green's functions with Wightman field theory.

Abstract

The new inversion formula for the Laplace transformation of the tempered distributions with supports in the closed positive semiaxis is obtained. The inverse Laplace transform of the tempered distribution is defined by means of the limit of the special distribution constructed from this distribution. The weak spectral condition on the Euclidean Green's functions requires that some of the limits needed for the inversion formula exist for any Euclidean Green's function with even number of variables. We prove that the initial Osterwalder--Schrader axioms (1973) and the weak spectral condition are equivalent with the Wightman axioms.