Unitarily Equivalent Classes of First Order Differential Operators

TL;DR

This paper investigates the classification of first order differential operators based on a common vector field, demonstrating their isomorphism and relating non-homogeneous operators to homogeneous ones through volume element adjustments, with implications for quantum observables.

Contribution

It establishes a framework for classifying differential operators via unitary equivalence and connects non-homogeneous and homogeneous operators through volume element modifications.

Findings

Operators based on the same vector field are isomorphic.

Replacing a non-homogeneous operator with a homogeneous one involves changing the volume element.

Results have implications for symmetric operators and quantum momentum observables.

Abstract

The elements of the class of non-homogeneous differential operators which are based on the same vector field, when viewed as acting on appropriate Hilbert spaces, are shown to be isomorphic to each other. It shown that the replacement of a non-homogeneous operator by a homogeneous one amouts to appropriately changing the volume element in the manifold. An emphasize is given to the case of symmetric operators and the corresponding quantum momentum observables.