Conformal Invariance in Classical Field Theory

TL;DR

This paper explores the geometric structure of conformal field theories, demonstrating that conformal invariance can be achieved without Chern-Simons type Lagrangians, by extending the first-order Lagrangian formalism.

Contribution

It introduces a geometric generalization of the first-order Lagrangian formalism to establish strict conformal invariance in classical field theories.

Findings

Global conformal transformations are Noetherian symmetries.

The action functional can be made strictly invariant under conformal transformations.

No Chern-Simons type Lagrangian exists for conformally invariant theories.

Abstract

A geometric generalization of first-order Lagrangian formalism is used to analyse a conformal field theory for an arbitrary primary field. We require that global conformal transformations are Noetherian symmetries and we prove that the action functional can be taken strictly invariant with respect to these transformations. In other words, there does not exists a "Chern-Simons" type Lagrangian for a conformally invariant Lagrangian theory.