Abstract
Abstract We discuss a field transformation from fields ψa to other fields ϕi that involves derivatives, $\phi _i = \bar{\phi }_i(\psi _a, \partial _\alpha \psi _a, \ldots ;x^\mu )$, and derive conditions for this transformation to be invertible, primarily focusing on the simplest case that the transformation maps between a pair of fields and involves up to their first derivatives. General field transformation of this type changes the number of degrees of freedom; hence, for the transformation to be invertible, it must satisfy certain degeneracy conditions so that additional degrees of freedom do not appear. Our derivation of necessary and sufficient conditions for invertible transformation is based on the method of characteristics, which is used to count the number of independent solutions of a given differential equation. 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.
Highlights
Field transformations are quite ubiquitous in all of the fields of physics and mathematics
As a useful application of our invertibility conditions, we explicitly prove that there is no invertible disformal transformation involving the second derivatives given by g = C(χ, X)g + D(χ, X)∇χ∇χ + E(χ, X)∇∇χ (X ≡ (∇χ)2) with E = 0
In Appendix A, we examine a field transformation that changes the number of fields and show that it cannot be invertible in our sense
Summary
Field transformations are quite ubiquitous in all of the fields of physics and mathematics. On the other hand, when a field transformation does involve derivatives, it is clear that the invertibility conditions become much more complicated. In our letter [1] we have explicitly given necessary and sufficient conditions for the invertibility of a field transformation involving two-fields and first derivatives.. We give the full and complete proof of necessary and sufficient conditions for the invertibility of field transformations with derivatives. The second goal of this paper is to prove the no-go theorem of disformal transformation of the metric that involves second derivatives of a scalar field. Throughout this work, we do not distinguish the lower and upper indices for the field space indices a, b, . . . and i, j, . . . , while in some parts upper/lower indices are used for clarity of the notation
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