# An Alternative to the Lagrangian and Hamiltonian Formulations of Relativistic Field Theories Based on the Energy-Momentum Tensor

## TL;DR

This paper introduces a novel, covariant framework for relativistic field theories based on a Noether-enhanced Legendre transformation to the energy-momentum tensor, facilitating coupling, higher derivatives, and potential gravity theories.

## Contribution

It develops an alternative to Lagrangian and Hamiltonian formulations using energy-momentum tensors, maintaining Lorentz covariance and enabling new applications in gravity and quantization.

## Key findings

- Framework maintains manifest Lorentz covariance.
- Applicable to higher derivative and dissipative systems.
- Offers a covariant canonical quantization scheme for Yang-Mills theories.

## Abstract

A Noether-enhanced Legendre transformation from Lagrange densities to energy-momentum tensors is developed into an alternative framework for formulating classical field equations. This approach offers direct access to the Hamiltonian while keeping manifest Lorentz covariance in the formulation of relativistic field theories. The field equations are obtained by imposing a vanishing divergence of the energy-momentum tensor (in a suitably structured form). The proposed framework is ideally suited for coupling subsystems because their interaction can be expressed as an exchange of energy and momentum. Even higher derivative theories and dissipative systems can be treated. A most promising application is the formulation of alternative theories of gravity. The proposed framework is illustrated for Yang-Mills theories, for which it offers a covariant canonical quantization scheme.