Invertibility conditions for field transformations with derivatives: toward extensions of disformal transformation with higher derivatives

We discuss a field transformation from fields $\psi_a$ to other fields $\phi_i$ 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 two fields and involves up to their first derivatives. General field transformation of this type changes 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.

We discuss a field transformation from fields ψ a to other fields φ i that involves derivatives, φ i =φ i (ψ a , ∂ α ψ a , . . . ; x µ ), and derive conditions for this transformation to be invertible, primarily focusing on the simplest case that the transformation maps between a pair of two fields and involves up to their first derivatives. General field transformation of this type changes 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.

Introduction
Field transformations are quite ubiquitous in all of the fields of physics and mathematics. The reason is that by using suitable fields (variables), one can often get a better and more intuitive insight into physical phenomena, find a different form of equations of motion which may allow to obtain solutions more easily, and so on. For these purposes, the (local) invertibility of a field transformation is essential because, otherwise, physics would not be the same after the field transformation. In case that a field transformation does not involve the derivatives, its local invertibility can be judged by the well-known inverse function theorem. On the other hand, when a field transformation does involve derivatives, it is clear that the invertibility conditions become much more complicated. If one regards such a field transformation as differential equations for old variables (fields), one can naively expect the presence of integration constants associated with the derivatives, which breaks one-to-one correspondence between old and new variables. Thus, apparently, one might arrive at a conclusion that no invertible transformation with derivatives exists. But, of course, this is not true in general. If one assumes specific degeneracy of the derivative terms in a field transformation which prohibits the appearance of associated integration constants, one can have an invertible field transformation with the derivatives. Indeed, 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. 1 The purpose of this paper is twofold. First, we give the full and complete proof of necessary and sufficient conditions for the invertibility of field transformations with derivatives. Here we fill some gaps of our proof that has been omitted in our letter [1]. More importantly, we provide the full proof for sufficient condition, while in the letter we gave it only within the perturbative regime. 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. We are strongly motivated by pursuing the extensions of conformal and disformal transformations, which are often used in gravity, cosmology, and many other fields (see e.g. Ref. [3] for classification of new theories generated from a simpler theory [4] using disformal transformations). A disformal transformation involving the first derivative of a scalar field is a natural extension of a conformal transformation [5]. The next natural question is whether one can further extend a disformal transformation to that involving the second derivatives of a field or even its higher derivatives. As a useful application of our invertibility conditions, we explicitly prove that there is no invertible disformal transformation involving the second derivatives given byg = C(χ, X)g + D(χ, X)∇χ∇χ + E(χ, X)∇∇χ (X ≡ (∇χ) 2 ) with E = 0. To the best of our knowledge, this is the first rigorous proof of the absence of such a transformation.
The organization of this paper is as follows. In section 2, we give a complete derivation of the necessary conditions. The results in sections 2.1 apply to field transformations involving arbitrary number of fields and their first derivatives. To show the explicit form of the invertibility conditions, in section 2.2 and 2.3 we focus on the field transformation between two fields and their first derivatives. In the section 3, the complete proof of the sufficient conditions is given. As an application of our result, in section 4 we construct some examples of invertible field transformation by solving the invertibility conditions. As another application, in section 5, the no-go theorem for a class of disformal transformations of the metric with second derivatives is proven. Section 6 is devoted to conclusions and discussion. In Appendix A, we examine a field transformation that changes the number of fields and show that it cannot be invertible in our sense. In Appendices B and C, we show details of some calculations in section 2. Appendix D shows that transformations between single fields and their derivatives can never be invertible, which implies that the two-field case we focus on is the simplest nontrivial case. In Appendix E, the necessity for nonlinear analysis in the inverse function theorem is explained. Appendix F gives some technical details of the derivation in section 4, and Appendix G discusses an extension of the results given in section 5.

Derivation of the necessary conditions
In this section we present a complete derivation of necessary conditions for invertibility of the transformation φ i =φ i (ψ a , ∂ α ψ a , x µ ) that transforms fields φ i into fields ψ a . 2 We divide the derivation of the invertibility conditions into two parts. In the section 2.1, we derive necessary conditions for the invertibility. An invertible transformation preserves the number of degree of freedom, while a transformation with field derivatives typically generates additional degrees of freedom. To formulate such an idea mathematically, we employ the method of characteristics for a differential equation to count the number of propagating modes [6]. In section 2.1.1 we explain our approach to derive the necessary conditions for the invertibility based on the method of characteristics. In this approach, the transformation equation (5) given below is converted to a set of differential equations that relates old variables to new ones, and the number of independent solutions for these equations is related to the number of degrees of freedom. To establish the invertibility, the transformation must satisfy certain degeneracy conditions to remove unnecessary additional modes originating from the derivatives in the transformation equation. Such degeneracy conditions must be imposed at each order of the aforementioned differential equations, which are summarized in sections 2.1.2 and 2.1.3. This procedure should be applied iteratively until the number of independent solutions are reduced appropriately, and then we may impose the non-degeneracy conditions to ensure the number of degree of freedom is not changed by the transformation. We summarize this procedure in section 2.1.4.
The procedure in 2.1 applies to field transformation for general number of fields. For illustration of our method, we focus on the transformation between two fields in section 2.2 and 2.3. The necessary conditions for the invertibility in the two-field case are derived in section 2.2 based on the general method introduced in the previous sections. The expressions of these conditions are rather complicated, and actually they can be simplified by solving a part of the conditions explicitly. Based on such an idea, the expressions of the necessary conditions are simplified in sections 2.3.1, 2.3.2 and 2.3.3. After simplification, the necessary conditions for the invertibility of a two-field transformations are summarized in Eqs. (50), (68) and (73).
After deriving the necessary conditions in this section, in section 3 we show that these conditions are actually sufficient to guarantee the invertibility.
2.1. Method of characteristics as the key to derive the necessary conditions 2.1.1. Invertibility and number of degeneracies. Let us consider a field transformation from ψ a to φ i given by whereφ i is a function of ψ a , ∂ α ψ a and x µ . We suppose that the numbers of fields before and after transformation are the same, that is, we assume that both a and i run from 1 to N , where N is the number of field ψ a . 3 If the transformation (5) is invertible, it does not change physical properties of theories before and after the transformation, and particularly the causal structure should be invariant. If the transformation changes the number of characteristic hypersurfaces, the causal structure is changed correspondingly. Therefore, for the invertibility, the transformation should not change the number of characteristic hypersurfaces. This gives the necessary conditions for invertibility.
To examine if the transformation (5) changes the number of characteristic hypersurfaces, we employ the method of characteristics for partial differential equations. This method can be applied only to quasi-linear differential equations, while the transformation equation (5) is nonlinear in ∂ α ψ i in general. To convert Eq. (5) to a quasi-linear partial differential equation, we act a differential operator K (µ1...µn) bi ∂ µ1...µn on it to obtain where A α ia , B ia are N × N matrices defined by and n k is the binomial coefficient for k ≤ n − ⌊ n 2 ⌋ − 2. In this expression, ∂ ⌊ n 2 ⌋+2 ψ a and higher derivatives of ψ a appear linearly, while the O ∂ ⌊ n 2 ⌋+1 ψ a , ∂ ⌊ n 2 ⌋ ψ a , . . . part is a nonlinear function of lower-order derivatives of ψ a . We show the derivation of the formula (6) in Appendix B.
The operation of the differential operator K (µ1...µn) bi ∂ µ1...µn introduces additional characteristic hypersurfaces to those of Eq. (5). Hence, the characteristics of ψ a , determined by the right-hand side of Eq. (6), are comprised of the original ones inherent to the field transformation equation (5) and the additional ones generated by the operation of K (µ1...µn) bi ∂ µ1...µn . Now we can analyse the characteristics of quasi-linear differential equation (6), regarding this equation as partial differential equations on ψ a . We have N differential equations with (n + 1)-th order derivatives of ψ a . Therefore, they generically give N × (n + 1) integration constants in a solution ψ a , which corresponds N × (n + 1) characteristics. However, if the field transformation (5) is invertible, there are no characteristics inherent to (5), which implies that Eq. (6) has only the additional characteristics generated by the operation of K (µ1...µn) bi ∂ µ1...µn . The number of these additional characteristics is N × n, hence there is a mismatch between the number of derivatives in the equations, N × (n + 1), and that of the characteristics required by the invertibility, N × n.
Such a mismatch can be resolved if the structure of the highest-order derivative part is degenerate. Since the difference between them is N , the invertibility requires N degrees of degeneracies. We derive the conditions giving such N degeneracies below.
2.1.2. Degeneracy condition at leading order. The characteristic equation for Eq. (6) is given by (8) where ξ µ is a vector which is not tangent to would-be characteristics hypersurfaces. If det (A µ ia ξ µ ) does not vanish identically for any ξ µ , the characteristic equation (8) implies that there are N × (n + 1) characteristics, and then the transformation (5) is not invertible as explained above. Hence, for the invertibility A µ ia ξ µ must be degenerate for any ξ µ , that is, det (A µ ia ξ µ ) = 0 for any ξ µ .
This condition is equivalent to where N is the number of fields and the sum is computed over the set S n of all permutations a = {a 1 , . . . , a N } of {1, . . . , N }. Here, sgn(a) denotes the signature of a permutation a, which 5/38 is +1 whenever the reordering a can be achieved by successively interchanging two entries an even number of times, and −1 whenever it can be achieved by an odd number of such interchanges.
2.1.3. Degeneracy condition at subleading order. The condition (10) indicates the degeneracy of the highest-order derivatives of Eq. (6). If the number of degeneracy is N (i.e. the dimension of kernel A µ ia ξ µ is N ), it implies that A µ ia = 0, and thus transformation is independent of ∂ µ ψ a . It is a trivial case; we can directly use the implicit function theorem to Eq. (5). Therefore, we consider non-trivial cases where the number of degeneracy of (10) is less than N . Here, we discuss the case where the number of degeneracy of (10) is 1 for simplicity. Generic cases may be discussed in a similar manner.
For later use, we introduce the adjugate matrix of A µ ia , which is the transposed cofactor matrix, asĀ α1...αN−1 ai As commented above, we suppose that the dimension of kernel of A µ ia ξ µ is 1 for any ξ µ . Then the adjugate matrix is a rank-1 matrix.
The fact that the highest derivative part of Eq. (6) is degenerate in one dimension implies that Eq. (6) contains one equation that involves only lower-order derivatives of ψ a . This does not necessarily imply that one of the equations in (6) contains only lower derivative terms, instead, generically, by combining equations in (6) one should be able to find one lower-derivative equation. Let us extract this subleading equation. First, setting the operator K The first term of the right-hand side vanishes because of Eq. (12). Moreover, equation (12) givesĀ This equation shows that the last term on the right-hand side of Eq. (13) is written as (16) Then, the last term on the right-hand side of Eq. (13) can be replaced with the derivatives of φ i and O ∂ m+N −2 ψ terms. Expanding Eq. (13) up to ∂ m+N −2 ψ and eliminating some ∂ m+N −1 ψ and ∂ m+N −2 ψ terms in favor of ∂ m+N −2 φ and ∂ m+N −3 φ terms as we did above, we finally obtaiñ The right-hand side of Eq. (17) has one less derivative compared to that of Eq. (6), that is, it is the subleading equation. The combination of non-degenerate part of Eq. (6), which has N − 1 equations, and Eq. (17) gives the structure of the characteristics in the subleading order. So far we established the presence of one degeneracy, meanwhile to ensure invertibility, there must be N degeneracies. Hence the characteristics at this subleading order should be degenerate too. By assumption, the matrix A µ ia ξ µ has one degeneracy, that is, there exist only one eigenvector with zero eigenvalue, which is denoted ψ ⊥ a (ξ). The other components of ψ a are collectively defined as ψ a (ξ). (Hereinafter, we omit (ξ) of ψ ⊥ a (ξ) and ψ a (ξ) for brevity.) The characteristic matrix for the highest derivative part of Eq. (17) and non-degenerate part of Eq. (6) is written as where (K a , K) =K matrix indicating only the non-degenerate components of A µ ia ξ µ , that is, 4 Using the projection tensor E N −1,ab onto the ψ space defined by Eq. (26), the components of the vector (K a , K) may be expressed more precisely as This matrix determines the subleading characteristics, and it should be degenerate for the invertibility. Since (nd) A µ ia ξ µ is regular, the requirement of the degeneracy gives the condition that K = 0 for anyK After that, we impose the condition that the next-order characteristic matrix is not degenerate, in order for the transformation to be invertible. The last condition corresponds to that of the inverse function theorem (without derivatives). In order to implement the procedure described above, it is useful to introduce a projection matrix E N −1,ij (ξ) to the (N − 1)-dimensional field space of ψ , that is, E N −1,ij (ξ) is the identity matrix for the (N − 1)-dimensional field space and zero for the one-dimensional field space parallel to ψ ⊥ . Let us express E N −1,ij (ξ) in terms of A µ ia ξ µ . We consider a matrix A µ1...µ2N−3 ai This matrix satisfies Let us demonstrate how to obtain the lower order equations iteratively with the projection tensor (23). On the right-hand side of Eq. (17), the leading term is the first term proportional to ∂ m+N −1 ψ a . The degeneracy condition for the subleading order (21) implies that the first term does not have the ψ ⊥ component, i.e. it can be written as where we used Eq. (23). Then, by subtracting Eq. (17) from Eq. (6) with its coefficient , the subleading (∂ m ′ +3N −3 ψ) term is canceled out and the subsubleading-order equation is obtained as a result.
The degeneracy condition can be obtained applying the procedure explained around (20) to the subsubleading equation, and the condition will be similar to the condition (21) for the subleading-order equation. Then, using the projector E N −1,ab again, we can construct the lower-order equation. Applying this procedure iteratively, we can derive N degeneracy conditions and the final non-degeneracy condition, which constitute the necessary conditions for the invertibility.

Necessary conditions in two-field case
Although we have already derived the leading and subleading degeneracy conditions in the previous subsection using the general procedure, it is instructive to follow a concrete example for understanding the method. For this purpose, based on the discussion in the previous section, we demonstrate how to derive the necessary conditions for the invertibility of a field transformation of two fields involving up to first derivative. 5 That is, we consider the case where N = 2, i = 1, 2 and a = 1, 2. In this section, we will complete the iterations to derive all the degeneracy and the final non-degeneracy conditions. The obtained conditions will be simplified in section 2.3.
We apply the procedure explained in section 2.1 to the two-field case in this section. In this case,Ā µ ai , A µ 2,ab and B 2,ab appeared in the previous section are written as andÃ µ ai is simplified as Then the identity (25) becomes We operate K (µ1...µn) ∂ µ1...µn on the field transformation equation (5) to obtain For (5) to be invertible, the coefficient of the highest-order derivative on the right-hand side Eq. (31) must be degenerate, which implies Following the previous section, we assume that the matrix A µ ia ξ µ is degenerate only in one dimension. Then the kernel of A µ ia ξ µ parallel to ψ ⊥ a and the field space in the other direction parallel to ψ a are one dimension each.
The degeneracy condition at the subleading order is obtained following the procedure in section 2.  (5) as The first term on the right-hand side vanishes thanks to Eq. (32). As shown in section 2.1.3, the second term in the brackets proportional to n can be canceled by adding nK (αµ1...µn−1) ∂ αĀ µn bi ∂ µ1...µn φ i to Eq. (33). Nevertheless, without such a cancellation, we can directly obtain the same subleading condition from Eq. (33) by rewriting it as The subleading degeneracy condition is given by demanding that the coefficient of highestorder derivative for ψ ⊥ vanishes. Noting thatĀ ν ai ξ ν works as a projector onto the ψ ⊥ space, this condition is written as The second term vanishes, which is shown by using Eq. (12) and (14) as SinceK (µ1...µn) is arbitrary, the subleading condition is The non-degeneracy condition at the subsubleading order can be constructed following the procedure in section 2.1.4. In this procedure we need to see the structure of the coefficient 10/38 of highest-order derivative term in Eq. (33) decomposing the variables into the ψ ⊥ and ψ space. For this purpose we use Eq. (30). OperatingK (µ1...µn)Ā µn+1 biĀ 2,µn+2µn+3 ∂ µ1...µn+3 to Eq. (5), we havẽ Using Eq. (30) for the first term in brackets and Eq. (14) for the other two, we havẽ The ∂ n+3 ψ term in the right-hand side can be reproduced by applying the operator to Eq. (5). Hence, by subtracting it from Eq. (39), we obtain the subsubleading-order equation involving terms only up to ∂ n+2 ψ. Expanding Eqs. (6) and Eq. (39) up to the ∂ n+2 ψ terms, we find (see Appendix C for details) The coefficient of ψ ⊥ component of the highest order derivative term ∂ n+2 ψ on the right-hand side of Eq. (40) determines the (non)degeneracy of the characteristics in the subsubleading order. This coefficient is obtained by contractingĀ µn+3 ai with the coefficient of ∂ n+2 ψ a in Eq. (40). We can show that the terms proportional toK (α1α2µ1...µn−2) andK (α1µ1...µn−1) become zero (see Appendix C). Among the terms proportional toK (µ1...µn) , the first one (∂ α1α2Ā µn+1 bi term) vanishes by actingĀ µn+3 ai , and the remaining term must not vanish for the ψ ⊥ component to be non-degenerate in this order. Hence, the non-degeneracy condition is given by for any ξ. Using Eq. (30), this can be also written as 11/38 To summarize, the necessary conditions for the invertibility are given by Eqs. (32), (37) and (42).

Simplification of necessary conditions
We have derived above the necessary conditions for the invertibility (32), (37) and (42) in the two-field case with first-order derivative. We present a simpler form of these expressions in this section. The simplified form of the necessary conditions will be helpful in the proof that these conditions are also sufficient conditions shown in the next section. In addition, we will use these expressions for building examples of invertible transformations.

Simplification of condition (32).
First of all, we simplify condition (32). Equation (32) implies that the rank of A µ ia ξ µ is less than two. Since we have assumed that A µ ia ξ µ has only one eigenvector with zero eigenvalue, the rank of A µ ia ξ µ has to be unity for any ξ, that is, is a rank-one matrix for any Then, by setting ξ µ = δ µν for each 1 ≤ ν ≤ D, we find that A ν ia is a rank-one matrix for any ν. Any rank-one 2×2 matrix can be written by a product of vectors, i.e. the matrices Below, we show by the induction that, if the rank of (43) is one for any ξ, A µ ia can be written as For µ ≤ 0 (that is, µ = 0), A 0 ia is written as V 0 i U 0 a , and thus, regarding Then, what we need to show is that, for any integer k, Eq. (45) satisfied for µ ≤ k + 1 if it is satisfied for µ ≤ k. Since two choices of (45) are symmetric with respect to U a and V i , without loss of generality we assume the former Now suppose that ξ k+1 = 0. In this case, the latter of these equations uniquely fixes e a because the field space dimension is two. Then, because e a is independent of ξ µ (µ ≤ k) and 12/38 also ξ µ 's for µ ≤ k are arbitrary, Eq. (47) implies This implies that all U µ a 's (µ ≤ k + 1) are normal to e a . Since e a is non-zero and the space of index a is two-dimensional, all U µ a 's (µ ≤ k + 1) are parallel to each other. Therefore, by rescaling U a , A µ ia can be expressed in terms of the common U a as where C µ is a normalization factor. Therefore, by redefining We have seen that one of the invertibility conditions, Eq. (32), implies that A µ ia is decomposed as Eq. (45). Using the other conditions, we show that A µ ia can be decomposed further as Let us show the proof in each case of Eq. (45).
In this case, if we can show U µ a = a µ U a , A µ ia is expressed as a µ V i U a . We define a vector normal n i to V j as We also define a vector m µ a with spacetime index µ as Then,Ā µ ia andÃ µ ia are written as and condition (37) becomes Now, we show that U µ a has to be written as a µ U a . It can be proven by contradiction; the assumption that two of U µ a 's are not parallel to each other (for instance, U 0 a is not parallel to U 1 a ) leads to a contradiction. Under this assumption, two of m µ a 's are not parallel correspondingly. Since the space spanned by index "a" is two-dimensional, Eq. (54) implies Then, the left hand side of (42) becomes Here we used Eq. (55) and (56) is inconsistent with the non-degeneracy condition (42), and thus all U µ a 's are parallel. Hence, U µ a can be written as a µ U a .
ii). A µ ia = V µ i U a The proof in this case is parallel to that for the case A µ ia = V i U µ a shown above. Defining m a and n µ i similarly as the condition (37) becomes where we used As is the case other, which gives a condition that However, we can show that it is inconsistent with the non-degeneracy condition (42). The left-hand side of (42) is calculated as where we used Eqs. (60), (61) and This is inconsistent with the non-degeneracy condition (42), and thus all n µ i 's are parallel. This implies that V µ i can be written as 2.3.3. Further simplification of conditions (37) and (42). We have shown that, for the invertibility conditions to be satisfied, A µ ia should be written as Without loss of generality, we can normalize V i and U a as V i V i = 1 = U a U a . We define unit vectors n i and m a that are normal to V i and U a respectively as 15/38 Since A µ ia is written with a µ , V i and U a , the matricesĀ µ ai ,Ã µ ai , alsoĀ 2,µν , A µ 2,ab and B 2,ab are written asĀ µ ai = a µ m a n i ,Ã µ ai = a µ U a V i ,Ā 2,µν = a µ a ν , Substituting the above equations into the condition (37) we have The non-degeneracy condition (42) becomes The first term of the left-hand side can be transformed as where we use the fact that U a U b + m a m b = δ ab and Eq. (68). The last term of (69) can be transformed as where we use the fact that V i V j + n i n j = δ ij and Eq. (68). Then, condition (69) can be written as This means that both of

Sufficiency of the invertibility conditions
In the previous section, we derived the necessary conditions (50), (68) and (73) for invertibility of a field transformation involving two fields and up to first-order derivatives of them.
In this section, we show that these conditions are also sufficient conditions, that is, (50), (68) and (73) are the necessary and sufficient conditions for the invertibility in the two-field case. As a preliminary step, in section 3.1, we introduce a notion of "partial invertibility" for a field transformation whose inverse transformation is uniquely determined for a part of variables. Then, we show in section 3.2 with the partial invertibility that (50), (68) and (73) are the necessary and sufficient conditions for invertibility. 6 6 See Appendix E for the difference between standard approach based on the inverse function theorem on functional spaces and our approach based on the implicit function theorem on finitedimensional subspaces associated with the functional space. 16/38

Partial invertibility
We consider invertibility of a transformation ψ a → φ i . Suppose that the transformation can be described by where I runs from 1 to a constant N . If N is equal to the number of all the degrees of freedom in ψ a , ∂ µ ψ a , ∂ µ ∂ ν ψ a , · · · , 7 which is denoted by N ψ , we can use the implicit function theorem by regarding ψ a , ∂ µ ψ a , ∂ µ ∂ ν ψ a , · · · as independent variables at a point in spacetime. However, we may not need to have N ψ equations of the form of (74) to fix ψ a uniquely in terms of φ a , indeed, even if we have less than N ψ equations, i.e. N < N ψ ; it may be possible to prove the uniqueness of ψ a if the equations have a special structure. Once ψ a is fixed in this manner, its derivatives ∂ µ ψ a , ∂ µ ∂ ν ψ a , . . . are uniquely fixed by differentiating ψ a repeatedly.
To illustrate the procedure explained above, we first discuss a generic case, where equations are for functions of variables x a , y b , z c and w c , where x a corresponds to ψ a , while y b and z c correspond to some combinations of the derivatives ∂ µ ψ a , ∂ µ ∂ ν ψ a , · · · . 8 The difference between y b and z c is explained later. w d corresponds to the other variables φ i and their derivatives ∂ µ φ i , ∂ µ ∂ ν φ i , · · · . If the number of equations is the same as sum of those of x a , y b and z c , we can use the implicit function theorem for them, i.e. x a , y b and z c can be fixed uniquely in terms of w d , provided that the conditions for the implicit function theorem are satisfied. However, even if the number of equations is less than the sum of those of x a , y b and z c , it may be possible to fix x a in terms w d uniquely if Eq. (75) have a certain structure, as explained below. Some of y b and z c may not be fixed uniquely in this process, but the uniqueness of x a can be shown independently.
The variables x a are fixed by stepwise application of the inverse function theorem provided that the function F I has a structure given by Eq. (77), which will be introduced shortly. Suppose that we have N + M equations of the form of Eq. (75), i.e. I runs from 1 to N + M (= N ), where N denotes the number of the fields x a and M is a positive integer. The variation of (75) is 7 For instance, if F I depends on up to the second order derivative of ψ a , the number of all the degrees of freedom of (ψ a , ∂ µ ψ a , ∂ µ ∂ ν ψ a ) is N + N × D + N × D(D + 1)/2, where N and D are the number of ψ a fields and the dimension of spacetime, respectively. 8 For our purpose to prove the invertibility for the two-field case with first-order derivative given by Eq. (5), x a corresponds to ψ a , y b is a component of ∂ µ ψ a , and z c correspond to the other components of the derivative ∂ µ ψ a and ∂ µ ∂ ν ψ a .

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Here, we assume that δy b and δz c appear only as M independent combinations in any δF I , i.e. δF I is expressed as Without loss of generality, the principal components of M independent combinations are set to be y b , which means that the rank of B Ib is M . This makes the difference between y b and z c . It also fixes the number of y b to be M . Then, picking up M equations from F I 's such that y b is fixed uniquely, we use the implicit function theorem to fix only y b . Without loss of generality, we assume that these equations are given by Since B Ib is assumed to be regular, this equation implies This equation fixes the z c dependence of y b (x a , z c , w d ). Now the remaining equations can be written purely in terms of x a , z c and w d as where I runs from 1 to N . Their variation is given by The δz c term is canceled because of Eq. (79) in this equation. This implies that F I 's for I = 1, . . . , N are independent of z c , i.e. they are functions of only x a and w d . Since the number of equations is the same as that of x a , we can use the implicit function theorem; if is satisfied, x a has a unique solution locally and it is written in terms of only w d .
In the next section, we will see that the transformation (5) in the two field case behaves as Eq. (77) once the invertibility conditions (50), (68) and (73) are imposed, and then ψ i is fixed in terms of φ a uniquely as a consequence. 18/38

Sufficient conditions for invertibility of our transformation
The transformation equation (5) can be rewritten in terms of a function of ψ a , ∂ µ ψ a and φ i as We analyze this equation in the two field case imposing the necessary conditions for the invertibility, Eqs. (50), (68) and (73). Operating n i ∂ µ on Eq. (83), we have where B ib , n i and U a are defined in section 2.3 and C iµ := ∂φ i /∂x µ . The variation of F i becomes where we omit the variation with respect to φ i , because it is irrelevant to the condition for the implicit function theorem. Now let us evaluate the variation of a µ G µ , which is given as where we used ∂ µ F i = 0 at the first equality, and some formulae in section 2.3.3 such as the degeneracy conditions n i ∂φi ∂(∂αψa) = n i A α ia = 0, n i B ia m a = 0 at the fourth equality. The key point is that, in Eqs. (85), (86) and (87), δ(∂ µ ψ a ) appears only in a linear combination a µ U a δ(∂ µ ψ a ). The last term proportional to a µ U a δ(∂ µ ψ a ) in Eq. (87) corresponds to the last line of Eq. (77) for M = 1. This allows us to use the result of section 3.1 by regarding ψ a , ∂ µ ψ a and ∂ µ ∂ ν ψ a as independent variables and also identifying x a , y b and z c as ψ a , a component of ∂ µ ψ a and the other components of (∂ µ ψ a , ∂ µ ∂ ν ψ a ), respectively. Then, the condition of invertibility is 19/38 where the components of the matrix are expressed with respect to the bases (V i , n i ), (U a , m a ), and also the degenerate row and column whose components are completely zero are removed. Equation (88) is satisfied if n i B ia U a = 0 and (V j B jb − a µ ∂ µ U b )m b = 0. Therefore, the necessary conditions for the invertibility, Eqs. (50), (68) and (73), are also the sufficient conditions.

Examples of invertible transformations
Invertibility conditions for transformations φ i = φ i (ψ a , ∂ψ a ) are established in the previous sections. In this section, we propose some non-trivial examples of field transformations that satisfy the invertibility conditions. In principle, the most general invertible transformation could be constructed by finding the general solution of the invertibility conditions. Unfortunately it is not an easy task, 9 so we proceed with introducing the following ansatz of field transformations: These ansatze are chosen so that the first degeneracy condition (50) among the invertibility conditions is automatically satisfied. The second degeneracy condition (68) and the nondegeneracy conditions (73) are then used to constrain the form of the transformations. Although these transformations may not span the most general transformation satisfying the invertibility conditions, they would give a good starting point for investigating the mostgeneral invertible field transformation.
In the derivation below, we use the fact that the vector U a can be set to a constant vector (0, 1) if it depends only on ψ a . We explain this method in Appendix F.

Ansatz 1:
We first consider the following ansatz for a field transformation given by where V i is normalized as V i V i = 1. In this transformation, U a can be set to U a = (0, 1) without loss of generality. It then follows For (89), the first degeneracy condition (50) is given by The second degeneracy condition (68) is evaluated as follows. In (93), only b depends on Y µ while the other terms depend only on ψ a . Because Eq. (93) should be satisfied identically for any ψ a and Y µ , it implies Equation (94) Below we focus on the case n i ∂Vi ∂ψ1 = 0, in which the transformation depends on derivatives φ i = φ i (ψ a , ∂ψ a ). In this case, Eq. (95) implies that ∂Ṽ i /∂ψ 1 is parallel to V i , that is, for some functionsc(ψ a ), c(ψ 1 ) = c(ψ a )dψ 1 andV i (ψ 2 ). The c V i term can be absorbed into the b V i term in the ansatz (89), and the remainderV i (ψ 2 ) depends only on ψ 2 . It implies that, once Eq. (90) is imposed, we may setṼ i =Ṽ i (ψ 2 ) in the ansatz (89) without loss of generality by absorbing the c V i term. Below we continue the derivation under this assumption.
The non-degeneracy conditions (73) are given by which can be regarded as constraints on b, V i andṼ i . To summarize, a transformation given by is invertible if Eqs. (97) and (98) are satisfied. One may further apply an invertible transformation ψ a = ψ a (ψ b ) such that det(∂ψ a /∂ψ b ) = 0 on Eq. (99) to re-introduce nontrivial U a (ψ a ) and construct a transformation of the form of Eq. (89).

Ansatz 2:
The next ansatz is given by The first degeneracy condition (50) is automatically satisfied for this ansatz: The second degeneracy (68) condition is given as Equation (105) may be solved forṼ i (ψ a ) and b(ψ a , Y a ) once V i (ψ a ) and c(ψ a , Y a ) are freely specified, and each solution gives an invertible transformation as long as it satisfies the non-degeneracy conditions (73). In a special case where the function c(ψ a , Y a ) depends only on ψ a but not on Y a , one can construct invertible transformations more explicitly as follows. When c = c(ψ a ), we may apply an invertible transformation ψ a = ψ a (ψ a ) on Eq. (105) to eliminate c(ψ a ), that is, Eq. (105) transforms under ψ a = ψ a (ψ a ) as and then the functionψ a =ψ a (ψ a ) may be chosen (at least locally) so that with which Eq. (106) reduces to This is equivalent to setting c(ψ a ) = 0 in Eq. (105) by means of a transformation in ψ a space. When c(ψ a ) = 0, Eq. (104) implies that The second equation implies b = b(ψ a , Y 2 ), which gives U a = (0, 1) and m a = (1, 0). In the first equation, the first term involving b(ψ a , Y 2 ) depends on Y 2 while the second term is independent of Y a , then it follows that Equation (110) implies either V i = V i (ψ 2 ) or b = 0, for the latter of which the transformation (100) does not involve ∂ψ a . Below we focus on the former case, which gives a transformation 22/38 with derivatives. In this case, we can show thatṼ i = f (ψ a )V i (ψ 2 ) +V i (ψ 2 ) for some function f (ψ a ) andV i (ψ 2 ) using Eq. (111) as we did around Eq. (96) for the previous ansatz. Then, once setting U a = (0, 1), by absorbing the c V i term into the b V i term we may setṼ i =Ṽ i (ψ 2 ) without loss of generality.
To summarize the results above, a transformation may be invertible if Eq. (111) is satisfied. Adding to that, the non-degeneracy conditions (73) must be satisfied for the invertibility, and also an invertible transformation ψ a = ψ a (ψ b ) may be applied to construct a transformation with nontrivial U a (ψ a ). Actually this transformation is a special case of the transformation (99)

Ansatz 3:
The third ansatz we consider is for which the first degeneracy condition (50) is satisfied automatically as In this expression V i is normalized as V i V i = 1, that is, where v(ψ a , Y ) is a normalization factor. For simplicity, we assume v = 0 in the following. Following the procedure explained in Appendix F, we set U a = (0, 1), m a = (1, 0) without loss of generality. Then, using n i = v −1 ǫ ij ∂φ j /∂Y , the second degeneracy condition (68) is evaluated as Equation (117) implies that, provided v = 0, where F (ψ 2 , φ 1 ) is an arbitrary function of φ 1 and ψ 2 . 10 10 When the (ψ 1 , Y ) space is separated into connected sets by borders on which the gradient vector ∂φ1 ∂ψ1 , ∂φ1 ∂Y vanishes, the function form of F (ψ 2 , φ 1 (ψ a , Y )) may be different on each connected sets on the (ψ 1 , Y ) space. The non-degeneracy conditions however imply ∂φ 1 /∂Y = 0, and it guarantees that F (ψ 2 , φ 1 ) is given uniquely on the entire (ψ 1 , Y ) space.

No-go for disformal transformation of the metric with higher derivatives
Here we apply our approach to disformal metric transformation. The disformal transformations involving only one derivative of the scalar field χ, where X = ∂ µ χ∂ µ χ, are invertible, provided that C C − dC dX X − dD dX X 2 = 0, see e.g. [7,8]. This follows from the fact that from the above expression one can straightforwardly express the metric g µν in terms ofg µν , χ and ∇ µ χ. Thus the transformation (122) is one-to-one change of variables (χ, g µν ) ↔ (χ,g µν ) as long as the above condition is satisfied. It is useful to check that the transformation (122) is invertible by applying our method. Although the method we developed above does not apply metric transformations in general because a metric is a tensor and has more than 2 components except for one dimensional spacetime, we can use it when applied to particular ansatze that contain scalar functions only. Indeed, let us restrict ourselves to the case of the homogeneous cosmology, where a(t) is the squared of the scale factor and n(t) is the squared of the lapse function. Note that we keep n(t) in the ansatz (123), since it changes under the transformation (122). For (123) we obtain under the disformal transformation (122), where a dot denotes a t derivative, and Note that the above transformation can be considered as the change of variables {n, a → n,ã} (with the time-dependent external function χ(t)), that does not involve derivatives. Therefore, by the virtue of the standard theorem on invertibility, the transformation (124) is invertible if det B ia = 0, where B ia is given by, so that we obtain C C − dC dX X − dD dX X 2 = 0, which reproduces the result we cited above.

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On the other hand, one can consider a more general disformal transformation by including two derivatives of the scalar as follows, g µν = C(χ, X)g µν + D(χ, X)∇ µ χ∇ ν χ + E(χ, X)∇ µ ∇ ν χ. (127) For the above transformation one cannot directly express g µν in terms ofg µν , since the last term of the r.h.s. of (127) also contains the metric g µν . Therefore, it has been conjectured that the inverse transformation of (127) does not exist, see Ref. [9]. However, to the best of our knowledge, this has not yet been proven. Indeed, although the simple inverse of (127) does not exist, it does not necessarily mean that there is no more complicated inverse transformation. Using our method, however, we are able to demonstrate that the transformation (127) with non-zero E is not invertible indeed.
To do this, let us assume that the transformation (127) is invertible, and we will see that this assumption leads to a contradiction. The FRW homogeneous ansatz (G3) expresses a subspace of functional space described with g µν and χ. The invertibility limited to this subspace gives the necessary condition of invertibility for full functional space. Here, we will show the violation of this necessary condition, which leads to a contradiction. From (127) and (123) which is a generalization of (124) for the case of non-zero E. As in the case considered above, we treat (128) as a transformation relating a set of two variables n(t), a(t) with ñ(t),ã(t) with the time-dependent external function χ(t). Contrary to standard disformal transformation, in this case the transformation involves first derivatives of the variables n(t) and a(t), therefore we should apply our method to check whether the conditions for this transformation to be invertible are satisfied. We have, According to our discussion above, the leading-order condition for the transformation given by (127) to be invertible is the vanishing of the determinant of (129). As we can see, this condition is clearly violated for non-zero E, therefore we arrive to the conclusion that the transformation (127) is not invertible unless E = 0. Once we generalize the structure of the higher-derivative term as [10,11] where F i = F i (φ, X, B, Y, Z, W) and the structure of A ia is changed and the no-go result obtained above for the transformation (127) could be evaded. We examine such a possibility in Appendix G.

Discussions
In this work we focused on field transformations that involve up to first-order derivative of fields, φ i =φ i (ψ a , ∂ α ψ a , x µ ), between two fields ψ a and other two fields φ i , and showed 25/38 conditions for this transformation to be invertible. A field transformation of this type changes the number of derivatives acting on the fields, hence in general it changes number of degree of freedom. When the transformation function satisfies certain conditions, however, appearance of additional degrees of freedom is hindered and then the transformation can be invertible. We emphasize that the degeneracy conditions and the procedure to derive the complete set of the invertibility conditions given in section 2.1 apply to field transformations that involve arbitrary number of fields and arbitrary-order derivatives, though we presented expressions only for transformations with first derivatives just for simplicity. In the following part of this work, for simplicity, we limited our scope to the simplest case that the transformation maps two fields φ i (i = 1, 2) to other two fields ψ a (a = 1, 2), and then derived the conditions for this transformation to be invertible. This is because, as we have shown, there is no invertible transformation with derivatives for one field case and hence two field case is the simplest case. To derive the necessary conditions for the invertibility, we employed the method of characteristics for a partial differential equations in section 2. If a transformation is invertible, the number of characteristic surfaces, which corresponds to the number of physical degrees of freedom, must be invariant. The derivatives contained in the transformation generates extra characteristic surfaces in general, and then the necessary conditions are obtained by demanding that extra characteristic surfaces are removed so that the total number of characteristic surfaces is invariant. After deriving the necessary conditions, in section 3 we confirmed that they are actually sufficient. It turned out that the invertibility conditions are composed of two degeneracy conditions (50), (68) and two non-degeneracy conditions given in Eq. (73).
As an application of the thus derived invertibility conditions, in section 4 we showed some examples of invertible transformations satisfying the invertibility conditions. The invertibility conditions can be regarded as equations for the function of the field transformation, and if we could construct their general solution we could obtain the most-general invertible transformation. Instead of finding the general solution, we proposed some ansatz for the transformation for which a part of the invertibility conditions are automatically satisfied, and as a result we obtained three kinds of invertible transformations as non-trivial examples. Although they may not be most general, they span broad class of invertible transformations and would provide a basis for construction and classification of various invertible transformations.
As another application, in section 5 we considered a higher-derivative extension of the disformal transformation in gravity and examined its invertibility. Using our invertibility conditions, just by simple calculations we have explicitly shown that a disformal-type transformation associated with second derivative of the scalar field cannot be invertible, which is the first rigorous proof as far as we are aware of.
Several directions of the future research are in order. In this work we considered the simplest case that the transformation involves only two scalar fields up to their first derivatives. Provided that our method can be generalized to transformations involving both a scalar field and a metric, we will be able to apply our results to studies on scalar-tensor theories and various modified gravity theories. For example, invertible disformal transformations were utilized to generate and classify the so-called DHOST theories from a simpler theory, the Horndeski theory. This scheme may be generalized to incorporate higher-order derivatives if we could generalize the disformal transformation by introducing higher derivatives. Such 26/38 an applications to modified gravity theory will be an ultimate goal of this study. As a first step for such a goal, it would be useful to consider a generalization to involve more than two scalar fields and more than first-order derivatives. Such generalizations within transformations of scalar fields, and also further generalizations including more fields such as metric will be discussed in our future work.

A. Non-invertibility of field-number changing transformations
Let us consider a transformation φ i = φ i (ψ 1 , ∂ψ 1 ) (i = 1, 2), for which the number of fields decreases but the number of derivative increases. Naively thinking, this transformation preserves the degree of freedom because the product of the number of fields and the order of derivatives is invariant under this transformation. To examine this expectation, let us evaluate the invertibility conditions. For simplicity we work in the one-dimensional case where the fields depend only on one variable, that is, we work in a point particle system. The first degeneracy condition (50) is given by where a is chosen to normalize V i as V i V i = 1. Then, using n i = ǫ ij V j and m a = ǫ ab U b = (0, −1), the second degeneracy condition (68) and the non-degeneracy conditions (73) are given by While n i B ia U a = n iṼi may not hold, V i B ia − aU a m a vanishes identically, hence the nondegeneracy condition among the invertibility conditions is violated. Then we can judge that the transformation φ i = φ i (ψ 1 , ∂ψ 1 ) (i = 1, 2) is not invertible. This result can be understood as follows. If this transformation were invertible, then an inverse transformation This implies that there exists a constrainṫ f 1 = f 2 between φ 1 and φ 2 . This is in contradiction with the fact that φ 1 and φ 2 are independent variables, then it implies that the transformation cannot be invertible.

B. Derivation of Eq. (6)
In this section we sketch the derivation of the formula (6), based on which the necessary conditions for the invertibility are derived. Acting ∂ µn on φ i =φ i (ψ a , ∂ α ψ a , x µ ) gives where the third term is the partial derivative with respect to the explicit x µ dependence of φ i . Then, the highest derivative term of ψ a contained in ∂ µ1...µn φ i is generated when all the 27/38 derivatives other than ∂ µn act on ∂ µn ∂ α ψ a in Eq. (B1), that is, This term is the origin of the leading ∂ n+1 ψ a term of Eq. (6). The subleading ∂ n ψ a term is composed of the following two contributions. The first one is generated when ∂ µ1...µn−1 acts on ∂ µn ψ a in Eq. (B1), that is, The second one is generated when n − 1 derivatives among ∂ µ1...µn are consumed to generate A µn ia ∂ n ψ a and the other one derivative acts directly on φ i , that is, Contracting with a totally-symmetric coefficient K (B5) The ∂ n−k ψ a term of Eq. (6) is obtained in a similar manner. The ∂ k B ia term is obtained by using n − k derivatives to generate ∂ n−k ψ a and by acting the other derivatives on φ i directly. The ∂ k+1 A µ ia term is obtained by using n − k − 1 derivatives to generate ∂ n−k ψ a and by acting the other derivatives on φ i directly. The coefficients of each term, n k and n k+1 , correspond to the number of combinations of the derivatives among ∂ n used to generate ∂ k ψ a .

C. Derivation of Eq. (40)
We give the derivation of Eq. (40) based on Eq. (6) and using the degeneracy conditions in the two-field case, Eqs. (32) and (37). We also use results given in section 2.2.
We start the derivation from which is given by ∂ n+2 ψ terms and lower-order terms once rewritten in terms of ψ using Eq. (6) where the terms appearing on the right-hand side are classified according to the positions of the indices α, α 1 , α 2 . Likewise, the second term of Eq. (C1) is expressed as where the ∂ n+3 ψ term on the right-hand side is the same as that in Eq. (C2). Then, in Eq. (C1) the ∂ n+3 ψ terms cancel out, and the remaining terms are given by the difference between Eqs. (C2) and (C3). Reorganizing the terms according to their tensor structures, it can be expressed as At the final equality, we used Eq. (28) and the identity that follows from the second derivative of Eq. (12) in the two-field case, which is given by (C5) As argued in section 2.2, to find the last condition for the invertibility, we should focus on the coefficient of the highest derivative term in the ψ ⊥ space, and this coefficient is obtained by replacing ∂ µ1...µn+2 ψ a byĀ µ3 aj and symmetrizing over µ 1 , µ 2 , µ 3 in Eq. (C4). Below, let us evaluate each term in Eq. (C4) after this replacement. First, theK (α1α2µ1...µn−2) term vanishes because it contains A (µn+2 iaĀ µ3) aj = 0, which is enforced by Eq. (28). Next, theK (α1µ1...µn−1) term also vanishes as follows.
The first equality follows from Eq. (28). The second equality follows from Eq. (14) and the first derivative of Eq. (37), that is,

D. Absence of invertible field transformation with derivatives for one field case
In this appendix, we show that there is no invertible field transformation of one field involving up to first derivative by use of the conditions obtained for two field case. We consider the 30/38 following transformation, which essentially represents one-field transformation with derivatives. Then, the first degeneracy condition (50) is satisfied as follows, where ∂φ1 ∂(∂µψ1) is assumed to be non-zero and One the other hand, the second degeneracy condition (68) is never satisfied as follows, Note that one can also check easily that third conditions (73) are also violated because n i B ia U a = 0 and V j B jb − a β ∂ β U b m b = 0 in this case. Thus, there is no invertible field transformation of one field involving first derivatives. A similar argument applies also to a case with more derivatives, φ 1 =φ 1 (ψ 1 , ∂ α ψ 1 , ∂ α ∂ β ψ 1 , . . . ; x µ ), and we can show that it can never be invertible.
E. Inverse function theorem applied directly to the functional space, and implicit function theorem to finite-dimensional subspaces In our proof in sections 2 and 3, we do not apply the inverse function theorem to the mapping from ψ(x) to φ(x), but instead use the implicit function theorem to the mapping from (ψ, ∂ψ, . . .) to (φ, ∂φ, . . .). This is because the former indeed does not work in the cases with derivatives. In this appendix, the reason why it is not applicable is explained in section E.1 and show the idea of our proof in section E.2.

E.1. Inapplicability of inverse function theorem to the functional space
The inverse function theorem can be applied to the mapping from a Banach space X to another Banach space Y . Functional spaces that are discussed in physics are mainly Hilbert spaces, and thus one might naively think that the inverse function theorem is directly applicable to a field redefinition with derivatives from ψ(x) to φ(x) by analyzing the whole spacetime points simultaneously. However, if derivatives of fields are involved in the field redefinition, the inverse function theorem can not be applied to the functional space. This means that the linearized analysis in the functional space does not give correct statements. Let us see that the inverse function theorem in functional spaces does not work in the cases with derivatives.
We have the condition for the degeneracy of the coefficients of the highest order derivative, In the (wrong) linear analysis, this is obtained as an equation which has to be satisfied for a fixed value of ψ a . However, we know from the analysis of the necessary condition 31/38 discussed in section 2 that Eq. (E1) should be interpreted as an identity, i.e. the equality has to be satisfied for any ψ a . Since the sufficient condition, which can be obtained from the inverse function theorem, should be stronger than the necessary condition, Eq. (E1) should be obtained as an identity (or replaced by a stronger condition).
We shall see in detail why the inverse function theorem is not applicable directly to the field redefinition with derivatives. For this purpose, let us carefully inspect the statement of the inverse function theorem for a mapping between Banach spaces [12]: Let X and Y be Banach spaces. Let U be an open neighborhood of a point x 0 ∈ X and F be a continuously differentiable mapping from U to Y , F : U → Y . Suppose that there exists the Fréchet derivative dF 0 at x 0 ∈ U which gives a bounded linear isomorphism that maps U onto an open neighborhood of F (x 0 ) ∈ Y . Then, there exists an open neighborhood U ′ of x 0 ∈ X and V of F (x 0 ) ∈ Y and a continuously differentiable map G from V to U ′ , G : V → U ′ satisfying F (G(y)) = y.
The important point is that, in the application of the theorem, a mapping is required to be continuously differentiable. To see the argument clearly, we consider an example given by We denote this mapping as Φ : ψ a (x) → φ i (x). Let us apply the linear analysis to Φ (although it gives an incorrect result). We linearize Eq. (E2) as If we consider ψ 1 (x) = ψ 2 (x) = 0 for any x, a point in the phase space, the above equations become and they can be solved uniquely for δψ 1 , δψ 2 as Since the linearized equation is uniquely solved, one might think the inverse function theorem is applicable, but it is not true. This is because, if the inverse function theorem is applicable, the mapping Φ must be invertible also for ψ 1 (x) = ǫ( = 0) and ψ 2 (x) = 0, which is in the neighborhood of ψ 1 (x) = ψ 2 (x) = 0, for sufficiently small ǫ. However, it is not the case because the linearized equations for ψ 1 (x) = ǫ, ψ 2 (x) = 0 become These equations cannot be solved uniquely for δψ 1 , δψ 2 , hence the transformation Φ is not invertible.
What was wrong in this example? Actually, derivative of the mapping Φ ψ a (x) (that is, the field redefinition (E2)) can not be obtained in the linearized analysis. The Fréchet derivative of Φ is defined as follows.

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If there exists a bounded linear operator A satisfying the Fréchet derivative is defined as Otherwise, the Fréchet derivative does not exist. Here, M is an open region in spacetime that we consider and || · || M is a norm for the phase space covered by φ i (x) or ψ a (x).
For the existence of the Fréchet derivative that is necessary to establish the invertibility, the linear part of Φ ψ a (x) + δψ a (x) − Φ ψ a (x) = δψ 1ψ2 + ψ 1δ ψ 2 + δψ 1δ ψ 2 + δψ 1 ,δψ 1 + δψ 2 (E9) must be close to Aδψ a (x), where A is a bounded linear operator. One may expect that the linear part of (E9) can be approximated by Aδψ a (x). However, Eq. (E9) includes derivatives of δψ, which may not be bounded. Hence, the inverse function theorem is not applicable. The inverse functional theorem roughly means that, if it is applicable, the linear analysis works well. However, in the application to an infinite dimensional space, such as a functional space, the existence of the Fréchet derivative is required. Without confirmation of it, the result of the invertibility by the linear analysis is untrustable and, in the case of the field redefinition with derivative, it generically does not work.

E.2. Our idea: Application of the implicit function theorem to the finite-dimensional subspaces
To avoid complication due to the derivatives involved in the mapping, we take a different approach by working in the function space spanned by φ i (x 0 ), ∂ µ φ i (x 0 ), ∂ µ ∂ ν φ i (x 0 ), . . . as follows. The field redefinition (5) from ψ a to φ i is obviously unique; Eq. (5) shows that ψ a is uniquely obtained from a fixed φ i . Hence, if also the mapping from φ i to ψ a is unique, the field redefinition becomes invertible. Let us give φ i in an open region of spacetime and then its derivatives ∂ µ φ i , ∂ µ ∂ ν φ i , . . . are uniquely obtained. Thus, if we can show that ψ a is uniquely fixed for given φ i and its derivatives ∂ µ φ i , ∂ µ ∂ ν φ i , . . . , we can say the inverse mapping is unique. To show it, we use implicit function theorem.
The problem appearing in the direct application of the inverse function theorem in section E.1 stems from that the dimension of the space is infinite. Even if we use the implicit function theorem, we have a similar problem. However, if we use equations shown in section 3, we can consider the implicit function theorem at each point of spacetime separately, i.e. for ψ a (x 0 ) with fixed φ i (x 0 ) and its derivatives (∂ µ φ i (x 0 ), ∂ µ ∂ ν φ i (x 0 ), . . .) at each spacetime point x 0 . Note that now we (are trying to) apply the implicit function theorem to (φ i (x 0 ); φ i (x 0 )∂ µ φ i (x 0 ), ∂ µ ∂ ν φ i (x 0 ), . . .) with fixed x 0 , and thus the dimension of space is finite. (Here, we consider the case where the number of derivative of φ is finite.) Since the dimension of space is finite, we do not bother the infiniteness of the dimension and the implicit function theorem can be easily applied. Note that the neighborhood in the implicit function theorem here means that for the values of φ i (x 0 ), ∂ µ φ i (x 0 ), ∂ µ ∂ ν φ i (x 0 ), . . ., not spacetime point. Hence, the neighborhood or "locality" in our argument means a 33/38 local region of functional space, where the deviations of φ i (x), ∂ µ φ i (x), ∂ µ ∂ ν φ i (x), . . . from a reference value should be small. It would be acceptable in physics, which usually has the UV cutoff scale.
Note that "neighborhood" in the implicit (or inverse) function theorem is not necessarily small. Let us take a simple example f (x) = x 2 . The inverse function theorem is applicable except x = 0. At x = 3, one might take an open neighborhood 2 < x < 4. Then, we take the inverse function theorem again near x = 2 and the "neighborhood" can be extended finally to 0 < x. This extension of "neighborhood" is general and it is done just before the condition is violated.
F. Setting U a = (0, 1) in φ i = φ i (ψ a , U a (ψ a )∂ µ ψ a ) In this appendix, we consider a transformation in which derivative of ψ a appears only in a combination U a (ψ a )∂ µ ψ a , that is, and show that U a (ψ a ) can be set to a constant vector U a = (0, 1) without loss of generality by a field transformation ψ a = ψ a (Ψ a ). This technique is used in section 4 to simplify examples of invertible transformations.
F.1. Field transformation to set U a = (0, 1) As a first step to set U a (ψ a ) to a constant vector, we rewrite this transformation as whereŨ a ≡ c(ψ a )U a (ψ a ) (F3) is a local rescaling of U a such thatŨ a is an irrotational vector in the ψ a space, that is, ThisŨ a can be obtained by choosing the rescaling factor c(ψ a ) appropriately. We show the construction method of c(ψ a ) andŨ a (ψ a ) in the next section. When (F4) is satisfied, due to the Poincaré lemma there exists a scalar function Ψ 1 (ψ k ) satisfyingŨ Let us also introduce another scalar function Ψ 1 (ψ a ) that is functionally independent from Ψ 1 (ψ a ) (i.e. det(∂Ψ a /∂ψ b ) = 0). Then there exists are one-to-one mapping between ψ a and Ψ a , and the transformation (F2) may be expressed in terms of Ψ a as Below we assume that the rank of the matrix A ia is 1. When the rank of A ia is zero, all the components of A ia vanishes and then the transformation (G6) does not involveṅ andȧ. In this case, the invertibility condition is simply given by det B ia = 0. When the degeneracy condition (G8) is satisfied, there exists zero eigenvectors n i , m a and their dual vectors V i , U a satisfying n i A ia = 0 = A ia m a , n i n i = 1 = m a m a , n i = ǫ ij V j , m a = ǫ ab U b . (G9) They are explicitly given as (G12) where we eliminated F a,Y using the first degeneracy condition (G8).

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The non-degeneracy conditions (73) in the invertibility conditions are given as follows.
where we used the first and second degeneracy conditions (G8), (G12) to simplify the expression. The other non-degeneracy condition can be derived after some calculations as ′ − F n + XF n,X + BF n,B + YF n,Y , (G14) where a prime ( ′ ) denotes a derivative with respect to the proper time, that is, f ′ ≡ n −1/2ḟ . To simplify this equation, we used the degeneracy conditions (G8), (G12) and their t derivatives, that is, we assumed the degeneracy conditions (G8), (G12) are satisfied identically at any t.
To summarize, the transformation (G6) becomes invertible when the conditions (G8), (G12), (G13) and (G14) are satisfied. These conditions should be regarded as necessary conditions for the invertibility of the generalized disformal transformation (G1) for a general metric, because the above results are derived only for the FRW ansatz (G3), in which the degrees of freedom of the metric is reduced to two functions n(t), a(t) depending only on t. The invertibility conditions for a general metric should encompass the conditions (G8), (G12), (G13) and (G14), while they may contain more stringent conditions in general. An obvious next step is to derive the invertibility conditions for a more general metric, and also it would be interesting to construct examples of invertible transformation based on the invertibility conditions obtained above. We reserve those issues for future works.
Let us briefly mention the case that either M or N given by Eq. (G11) vanishes. In this case, it turns out that the degeneracy conditions imply that F n,B = F n,Y = F a,B = 0, that is, F n = F n (X) and F a = F a (X, Y). For these functions it follows that and V i = (0, 1), U a = (1, 0). It can be shown that the first and second degeneracy conditions are automatically satisfied, and the non-degeneracy conditions are given by F n − XF n,X = 0, F a = 0. (G16)