An exterior-algebraic derivation of the symmetric stress-energy-momentum tensor in flat space-time

TL;DR

This paper derives a symmetric stress-energy-momentum tensor for fields in flat space-time using exterior algebra, providing explicit formulas and a divergence expression that generalizes the Lorentz force, applicable to various field theories.

Contribution

It introduces a novel exterior-algebraic derivation of the symmetric tensor, avoiding canonical methods, with explicit formulas for generic multivector fields and dimensions.

Findings

Derived explicit formulas for the tensor components.

Provided a coordinate-free divergence expression.

Discussed applications to electromagnetism, Proca, Yang-Mills, and conformal invariance.

Abstract

This paper characterizes the symmetric rank-2 stress-energy-momentum tensor associated with fields whose Lagrangian densities are expressed as the dot product of two multivector fields, e. g., scalar or gauge fields, in flat space-time. The tensor is derived by a direct application of exterior-algebraic methods to deal with the invariance of the action to infinitesimal space-time translations; this direct derivation avoids the use of the canonical tensor. Formulas for the tensor components and for the tensor itself are derived for generic values of the multivector grade $s$ and of the number of time and space dimensions, $k$ and $n$, respectively. A simple, coordinate-free, closed-form expression for the interior derivative (divergence) of the symmetric stress-energy-momentum tensor is also obtained: this expression provides a natural generalization of the Lorentz force that appears in the context of the electromagnetic theory. Applications of the formulas derived in this paper to the cases of generalized electromagnetism, Proca action, Yang-Mills fields and conformal invariance are briefly discussed.