A duality of fields

Wen-Du Li; Wu-Sheng Dai

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arXiv · 1905.06805

A duality of fields

Wen-Du Li·Wu-Sheng Dai

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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.

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Wen-Du Li, Wu-Sheng Dai. 2019-05-08. A duality of fields. https://arxiv.org/abs/1905.06805

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