Legendre transformations and Clairaut-type equations

TL;DR

This paper explores the mathematical structure of Legendre transformations in quantum field theory, revealing their Clairaut-type form and presenting new solutions relevant to theories with composite fields.

Contribution

It demonstrates that Legendre transformations in quantum field theory are of Clairaut-type and introduces novel solutions for equations involving composite fields.

Findings

Legendre transformations have Clairaut-type form in quantum field theory.

Loop corrections preserve the Clairaut-type structure with composite fields.

A new solution to the functional Clairaut-type equation is derived.

Abstract

It is noted that the Legendre transformations in the standard formulation of quantum field theory have the form of functional Clairaut-type equations. It is shown that in presence of composite fields the Clairaut-type form holds after loop corrections are taken into account. A new solution to the functional Clairaut-type equation appearing in field theories with composite fields is found.