Duality family of scalar field

TL;DR

This paper introduces a duality family of self-interacting scalar fields related by transformations that preserve their equations, enabling the construction of multiple solvable models from a single solution, inspired by classical mechanics dualities.

Contribution

The paper establishes a duality framework for scalar fields, extending classical Newton-Hooke duality to quantum field theory, and demonstrates how to generate new solvable models from known solutions.

Findings

Duality transformation relates different scalar field models.

Exactly solvable models can be generated within the duality family.

Analogies with classical and quantum mechanics dualities are shown.

Abstract

We show that there exists a duality family of self-interacting massive scalar fields. The scalar field in a duality family are related by a duality transformation. Such a duality of scalar fields is a field version of the Newton-Hooke duality in classical mechanics. The duality transformation preserves the type of the field equation: transforming a Klein-Gordon type equation to another Klein-Gordon type equation with a different self-interacting potential. Once a field in a duality family is solved, all other family members are solved by the transformation. That is, a series of exactly solvable models can be constructed from one exactly solvable model. The dual field of the power-interaction field, the sine-Gordon field, etc., are considered. Moreover, as a comparison, we show an analogue of the duality in classical and quantum mechanics.