Legendre transformations and Clairaut-type equations

It is noted that the Legendre transformations in the standard formulation of quantum field theory have the form of functional Clairaut-type equations. It is shown that in presence of composite fields the Clairaut-type form holds after loop corrections are taken into account. A new solution to the functional Clairaut-type equation appearing in field theories with composite fields is found.


Introduction
In quantum field theory the main object containing all possible information about a given dynamical system in quantum field theory is the generating functional of vertex functions or, in other words, the effective action. The usual way to introduce the effective action is by means of the Legendre transformation applied to the generating functional of connected Green functions. The relation between the effective action and the generating functional of connected Green functions has the form of functional Clairaut-type equation (see recent discussion in [1]). However, the perturbative (loop) expansion in the effective action does not preserve the mentioned Clairaut-type form of the Legendre transformation. This may explain why in the quantum field theory this specific feature of the Legendre transformation has been ignored.
In the present paper we are motivated by recent works [2,3] devoted to the development of a new method to the functional renormalization group approach [4,5,6] and the study of the average effective action with composite fields. The approach to quantum field theory with composite fields has been developed by Cornwall, Jackiw and Tomboulis [7] in attempts to study physical phenomena (spontaneous symmetry breakdown, bound states, etc.) which cannot be easily considered in the perturbation (loop) expansion. Generalization of this method to gauge theories [8] detected a special form of gauge dependence of the effective action with composite fields which in turn allowed to formulate the approach to functional renormalization group [2] being free of the gauge dependence problem inherent to the standard one [5,6]. Introduction of the effective action with composite fields requires to use the double Legendre transformations which as compared with the standard case cannot be presented in the form of functional Clairaut-type equation. Nevertheless, the perturbation series in the effective action with composite fields leads exactly to the functional Clairaut-type equation.
The paper is organized as follows. In Section 2 the relations existing between the effective actions without composite fields and with composite fields and functional Clairaut-type equations are derived. In Section 3 we study in detail solutions to the first-order partial Clairaut-type equations with a special form of the right-hand side and then generalize this result to the case of functional Clairaut-type equations. In Section 4 the way to find the one-loop correction to the effective action with composite fields by solving an appropriate functional Clairaut-type equation is shown. The remarkable result is that the effective action without composite fields does not offer such a solution. In Appendix A the simplest example of a set of matrices playing a very important role in solving the functional Clairaut-type equation which appears in field theories with composite fields is given.

Effective actions and Clairaut-type equations
Eliminating the source J form (2.2) one obtains the equation  [7]. Indeed, let us consider the effective action with composite fields. The starting point of such an approach is the generating functional of Green functions Z[J, K], is the generating functional of connected Green functions. In Eq. (2.9) J = J(x) and K = K(x, y) are sources to field φ = φ(x) and composite field L(φ) = L(φ)(x, y), respectively, and the notation The effective action with composite field, Γ = Γ[Φ, F ], is defined by using the double Legendre transformation [7] Γ Eliminating the sources J and K from the (2.11) one obtains the equation (2.14) Since the right-hand side of Eq. (2.14) depends on the fields Φ not only through derivatives of functional Γ = Γ[Φ, F ], the Eq. (2.14) does not belong to the Clairaut-type equation. But, in contrast with Eq. (2.5), the one-loop approximation for the effective action with composite field, Γ (1) = Γ (1) [Φ, F ] by itself satisfies the equation being exactly the Clairaut-type with respect to field F wherein the variable Φ should be considered as parameter.

Remarks on the solutions of Clairaut-type equations
A Clairaut equation is a differential equation of the form where y = y(x), y ′ = dy/dx and ψ = ψ(z) is a real function of z. It is well-known that the general solution of the Clairaut equation is the family of straight line functions given by where C is a real constant. The so-called singular solution is defined by the equation if a solution to Eq.
Now let us examine a first-order partial differential equation [10] y which is also known as the Clairaut equation [10]. Here y = y(x) is the real function of variables .., z n } and the notation Differentiation of Eq. (3.7) with respect to x i leads to a system of differential equations In the case of the Hessian matrix vanishing H ij = 0 where

9)
z i = C i = const. Therefore the solution to Eq.(3.5) is the family of linear functions If det H ij = 0 then the equations (3.8) are reduced to the following system ∂ψ ∂z j + x j = 0, j = 1, 2, ..., n. z j = ϕ j (x), j = 1, 2, ..., n , (3.12) then the solution to Eq. (3.5) is reduced to the system of partial first-order differential equations resolved with respect to derivatives. If the conditions of integrability are fulfilled then in any simply connected domain G ⊂ R n the system (3.13) is solvable and the solution to this system can be presented in the form In case when det H ij = 0 the equations (3.11) for z i = z i (x) have the form a j + x j = 0 , j = 1, 2, ..., n.
In turn multiplying the Eq. (3.18) by z j and summing the results we have The results obtained above for the partial differential equations can be immediately extended to the case of functional Clairaut-type equations. Let Γ = Γ[F ] be a functional of fields F m = F m (x) , m = 1, 2, ..., N, which are real integrable functions of real variables x ∈ R n . We use the notion of functional Clairaut-type equations for the equations of the form

The effective action with composite fields
In this Section we generalize the results obtained in [7] to the case of a field model described by a set of scalar bosonic fields φ A (x) , A = 1, .., N, with a classical non-degenerate action S[φ]. Let L i (φ) = L i (φ)(x, y), i = 1, 2, ..., M, be composite non-local fields, where A i AB = A i BA are constants. The generating functional of Green functions, Z[J, K], is given by the following path integral where K i = K i (x, y), i = 1, 2, ..., M, are sources to composite fields L i (φ)(x, y). Here the notations are used. From Eq. (4.2) we can construct the following relations , (4.5) or, in terms of the functional W [J, K], We define the average fields Φ A (x) and composite fields F i (x, y) as follows One can eliminate the sources from Eq. (4.9) using The relation (4.6) rewritten in terms of Γ[Φ, F ] reads where (G −1 ) is the matrix inverse to G, and we have used the notation . (4.14) In the one-loop approximation, Γ[Φ, F ] = S[Φ] + Γ (1) [Φ, F ], the equation for one-loop contribution, Γ (1) , to the effective action can be found using procedure similar to [3]. It has the form being the exact functional Clairaut-type equation.
According to the general scheme which is discussed in the previous section, to solve the Eq.
and substituting them to the Eq. (4.15) we obtain where the matrix Q = {Q AB (x, y)} is defined as Then varying the functional (4.17) with respect to F i we obtain Thus the equation defining non-trivial functions Z i (x, y) reads Note that in the approximation considered here this equation coincides with (4.12).
To solve the equation (4.22) we introduce a set of matrices B j = {B AB j } by the relations Then we have (4.24) or, due to the symmetry property of Q AB (x, y) = Q BA (y, x), We can express the Z i F i as a functional of Φ, F multiplying Eq. (4.22) by Z j and using (4.28) with the result Finally substituting (4.28) into (4.17) and using (4.30) we find the one-loop effective action, The expression for Γ (1) (4.31) generalizes the known result of [7] in the cases A = B = 1 and j = 1 when A 1 11 = 1 and B 11 1 = 1. This generalization involves the introduction of the set of matrices B j obeying the properties (4.23). In Appendix A we give a simple example of matrices A j and B j satisfying all required properties.

Discussions
In the present article we have studied relations existing between the Legendre transformations in quantum field theory and the functional differential equation for effective action which has the form of functional Clairaut-type equation. We have found that specific features of this equation do not hold within the perturbation theory in a quantum field theory without composite operators. But it is not the case within the approach to the quantum field theory based on composite fields when perturbation expansion of the effective action leads exactly to a functional Clairaut-type equation with a special type of the right-hand side. Partial first-order differential equations of Clairaut-type were our preliminary step in the study of solutions to the problem. It was shown that in case when the right-hand side of the equation has the form inspired by the real situation in quantum field theory with composite fields the solution to that functional Clairaut-type equation can be found with the help of algebraic manipulations only. In our knowledge the solution (3.21) to the equations (3.5) and (3.16) can be considered as a new result in the theory of partial first-order differential equations of Clairaut-type. This result has been easily extended (see (3.26)) to the case of functional Clairaut-type equation (3.22) with the special right-hand side (3.25). We have found an explicit solution to the functional Clairaut-type equation appearing in the quantum field theory with composite fields to define one loop contribution to the corresponding effective action (4.31).
We have studied the case with maximum number of composite fields, M = 1 2 N(N +1), being quadratic in the given scalar fields φ A , A = 1, ..., N. In a similar manner one can consider the situation when the number of composite fields is less the maximum one, L i (φ)(x, y) = Here the matrixes B ab i are introduced in the same way as in (4.23) but for the A i ab ones.
Extension of the results obtained above to gauge theories can be easily performed in a way used in papers [2,3] on the basis of supermathematics [11,12]. We are going to present such kind of generalizations in our further studies.