Consider a functional $F[f]$ where $f=f(x)$. What is the definition of its Legendre transform?
I know about the Legendre transforma of functions. For example, $f=f(x)$ will have the Legendre transform $$g(m)=f-mx\tag{1}$$ where $m=\frac{df}{dx}$. By the same analogy, the Legendre transform of the functional $S[f]$ can be writeen as $$G[\frac{\delta F}{\delta f(x)}]=F[f]-\frac{\delta F}{\delta f(x)}f(x).\tag{2}$$
- Is Eq. (2) correct generalization of Legendre transform for functionals?
- If yes, how can we simplify (2) to express the second term as an integral over $x$? I know that $$\frac{\delta F[f]}{\delta f(y)}=\lim\limits_{\epsilon\to 0}\frac{F[f(x)+\epsilon\delta(x-y)]-F[f]}{\epsilon}.$$ But I can't understand how to proceed next.
Context I'm trying to define the effective action functional $\Gamma[\phi_{c}]=W[J]-\int d^4x J(x)\phi_{c}(x)$ where the symbols have their usual meaning in Quantum Field Theory.
In one dimension we have $f^*(p) = f(x) - p x,$ where $x = (f')^{-1}(p).$
In several dimensions we have $f^*(p) = f(x) - p \cdot x,$ where $x = (\nabla f)^{-1}(p).$
In continuous dimensions we then should have $f^*(p) = f(x) - \langle p, x \rangle,$ where $x = (\delta f)^{-1}(p)$ with $\delta f$ being the functional derivative of $f.$
In other notation: $$F^*[g] = F[f]- \int g(x) \, f(x) \, dx,$$ where $f(x) = \left(\frac{\delta F}{\delta f(x)}\right)^{-1}[g].$