12 May 2022 Thu 3 pm

Field transformations are ubiquitous in modern physics and mathematics. However, as far as we know, nobody has yet discussed the invertibility of field transformations with derivatives. We formulate explicitly the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms. Our approach is to apply the method of characteristics of differential equations, by treating such a transformation as differential equations that give new variables in terms of original ones. The obtained results generalize the well-known and widely used inverse function theorem.  As applications of the invertibility conditions, we show some non-trivial examples of the invertible field transformations with derivatives, and also give a rigorous proof that a simple extension of the disformal transformation involving a second derivative of the scalar field is not invertible.

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