A duality of fields

TL;DR

The paper introduces a duality concept among fields, showing how solutions of one field can be derived from its dual, and classifies fields into duality families for efficient solution finding.

Contribution

It presents a general framework for dual fields, classifies fields into duality families, and proposes a high-efficiency method for solving field equations using duality transformations.

Findings

Dual fields can be systematically classified based on their potentials.

Solutions of one field in a duality family determine solutions of all others.

Examples include φ^n-fields, polynomial potentials, and sine-Gordon field.

Abstract

It is shown that there exists a duality among fields. If a field is dual to another field, the solution of the field can be obtained from the dual field by the duality transformation. We give a general result on the dual fields. Different fields may have different numbers of dual fields, e.g., the free field and the $\phi^{4}$-field are self-dual, the $\phi^{n}$-field has one dual field, a field with an $n$-term polynomial potential has $n+1$ dual fields, and a field with a nonpolynomial potential may have infinite number of dual fields. All fields which are dual to each other form a duality family. This implies that the field can be classified in the sense of duality, or, the duality family defines a duality class. Based on the duality relation, we can construct a high-efficiency approach for seeking the solution of field equations: solving one field in the duality family, all solutions of other fields in the family are obtained immediately by the duality transformation. As examples, we consider some $\phi^{n}$-fields, general polynomial-potential fields, and the sine-Gordon field.