Quantum Field Theory for the Gifted Amateur Exercise 1.3 Part 3 The Elegance of Functional Derivatives

Exploring the depths of Quantum Field Theory (QFT) is akin to navigating an enigmatic realm where the conventional rules of physics blur into abstractions. Among the many mathematical tools that prove indispensable in this journey, functional derivatives are like a compass, guiding us through the mysterious terrain of functionals and their variations. Today, let’s delve into one such fascinating exercise from the book Quantum Field Theory for the Gifted Amateur and see what it reveals about the interplay of functionals and derivatives.

Table of Contents

The Problem Statement

Consider a functional \(J[f]\) defined as:

\[J[f] = \int g\left(y, f, f^{\prime}, f^{\prime \prime}\right)\,\mathrm{d}y.\]

We are tasked with showing that:

\[\frac{\delta J[f]}{\delta f(x)}= \frac{\partial g}{\partial f} -\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial g}{\partial f^{\prime}} +\frac{\mathrm{d}^2}{\mathrm{d}x^2}\frac{\partial g}{\partial f^{\prime \prime}}.\]

Here, \(f^{\prime \prime}\) is the second derivative of \(f\) with respect to \(y\), and the partial derivatives of \(g\) are evaluated at \(y=x\) after the delta-function integrations are performed.

The Solution Journey

Step 1: Variational Notation

Let us vary the function by a sharply localized amount,

\[f(y) \mapsto f(y)+\epsilon\delta(y-x).\]

Then its first and second derivatives vary as

\[f^{\prime}(y) \mapsto f^{\prime}(y)+\epsilon\frac{\mathrm{d}}{\mathrm{d}y}\delta(y-x),\]

and

\[f^{\prime\prime}(y) \mapsto f^{\prime\prime}(y)+\epsilon\frac{\mathrm{d}^2}{\mathrm{d}y^2}\delta(y-x).\]

The functional derivative is defined by the coefficient of this infinitesimal localized variation:

\[\frac{\delta J[f]}{\delta f(x)} =\left.\frac{\mathrm{d}}{\mathrm{d}\epsilon}J\left[f+\epsilon\delta(\cdot-x)\right]\right|_{\epsilon=0}.\]

Step 2: Expanding the Functional

Expanding \(g\) to first order in \(\epsilon\), we obtain

\[\begin{aligned} J\left[f+\epsilon\delta(\cdot-x)\right]-J[f] =&\ \epsilon\int \bigg[ \frac{\partial g}{\partial f}\delta(y-x) +\frac{\partial g}{\partial f^{\prime}}\frac{\mathrm{d}}{\mathrm{d}y}\delta(y-x) \\ &\qquad\qquad +\frac{\partial g}{\partial f^{\prime\prime}}\frac{\mathrm{d}^2}{\mathrm{d}y^2}\delta(y-x) \bigg]\,\mathrm{d}y+O(\epsilon^2). \end{aligned}\]

Therefore,

\[\frac{\delta J[f]}{\delta f(x)} =\int \bigg[ \frac{\partial g}{\partial f}\delta(y-x) +\frac{\partial g}{\partial f^{\prime}}\delta^{\prime}(y-x) +\frac{\partial g}{\partial f^{\prime\prime}}\delta^{\prime\prime}(y-x) \bigg] \mathrm{d}y.\]

Step 3: The Final Derivation

Now use the distribution identities

\[\int A(y)\delta(y-x)\,\mathrm{d}y=A(x),\]

\[\int A(y)\delta^{\prime}(y-x)\,\mathrm{d}y=-A^{\prime}(x),\]

and

\[\int A(y)\delta^{\prime\prime}(y-x)\,\mathrm{d}y=A^{\prime\prime}(x),\]

assuming that boundary terms vanish or that the variations are fixed at the boundary. Applying these identities gives

\[\frac{\delta J[f]}{\delta f(x)} =\frac{\partial g}{\partial f} -\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial g}{\partial f^{\prime}} +\frac{\mathrm{d}^2}{\mathrm{d}x^2}\frac{\partial g}{\partial f^{\prime\prime}}.\]

This is the Euler-Lagrange functional derivative for a functional whose integrand depends on the function, its first derivative, and its second derivative.

The Quantum Epilogue

What we’ve accomplished here is akin to deciphering a piece of the quantum code that describes the universe. The equation we’ve derived is more than a mere mathematical curiosity: it is a lens that allows us to glimpse the interplay of functionals and derivatives in the realm of QFT. It tells us how a small change in \(f(x)\) can propagate through the entire functional landscape.

The same methodology extends to many other problems in quantum field theory, solidifying our understanding of the universe, one functional derivative at a time.