# The Diffeomorphism Field Revisited

## TL;DR

This paper revisits the dynamical action of the diffeomorphism field, identifies and corrects an inconsistency, and proposes a modified, Yang-Mills-like action that maintains diffeomorphism invariance without interpreting the field as a tensor.

## Contribution

It provides a corrected formulation of the diffeomorphism field action, aligning it with Yang-Mills theory and resolving previous interpretational issues.

## Key findings

- Modified action is structurally similar to Yang-Mills action
- Full diffeomorphism invariance is achieved in interactions
- Spatial diffeomorphism invariance of the theory is maintained

## Abstract

We revisit the dynamical action, developed in earlier studies [1-3], of the gravitational analog of Yang-Mills field, called the diffeomorphism field. We show an inconsistency in the construction of this action and solve it by a modification. The modified action becomes structurally similar to the Yang-Mills action. Then we explain the problems that arise from interpreting the diffemorphism field as a tensor. We leave out this interpretation, thereby also covariantization as a method to reach diffeomorphism invariance. By introducing corrections that involve affine connection coefficients we recover full diffeomorphism invariance in interactions of the diffeomorphism field with other fields. We also show that this approach maintains the spatial diffeomorphism invariance of the theory itself.