Constructing $p,n$-forms from $p$-forms via the Hodge star operator and the exterior derivative

TL;DR

This paper investigates operators combining the Hodge star and exterior derivative on p-forms in n-dimensional spacetimes, constructing higher-order forms and proposing their use in generalized Lagrangians for p-form fields.

Contribution

It introduces a method to generate n-forms with higher derivatives from p-forms and explores their properties and applications in formulating generalized Lagrangians.

Findings

Operators can generate higher-order p-forms with invariant form degree.

Constructed n-forms can be used to define generalized Lagrangians.

Applications include conserved currents and extended field theories.

Abstract

In this paper, we aim to explore the properties and applications on the operators consisting of the Hodge star operator together with the exterior derivative, whose action on an arbitrary $p$-form field in $n$-dimensional spacetimes makes its form degree remain invariant. Such operations are able to generate a variety of $p$-forms with the even-order derivatives of the $p$-form. To do this, we first investigate the properties of the operators, such as the Laplace-de Rham operator, the codifferential and their combinations, as well as the applications of the operators in the construction of conserved currents. On basis of two general p-forms, then we construct a general n-form with higher-order derivatives. Finally, we propose that such an n-form could be applied to define a generalized Lagrangian with respect to a p-form field according to the fact that it incudes the ordinary Lagrangians for the $p$-form and scalar fields as special cases.