Diamagnetic inequality

In mathematics and physics, the diamagnetic inequality relates the Sobolev norm of the absolute value of a section of a line bundle to its covariant derivative. The diamagnetic inequality has an important physical interpretation, that a charged particle in a magnetic field has more energy in its ground state than it would in a vacuum.

To precisely state the inequality, let $$L^2(\mathbb R^n)$$ denote the usual Hilbert space of square-integrable functions, and $$H^1(\mathbb R^n)$$ the Sobolev space of square-integrable functions with square-integrable derivatives. Let $$f, A_1, \dots, A_n$$ be measurable functions on $$\mathbb R^n$$ and suppose that $$A_j \in L^2_{\text{loc}} (\mathbb R^n)$$ is real-valued, $$f$$ is complex-valued, and $$f, (\partial_1 + iA_1)f, \dots, (\partial_n + iA_n)f \in L^2(\mathbb R^n)$$. Then for almost every $$x \in \mathbb R^n$$, $$|\nabla |f|(x)| \leq |(\nabla + iA)f(x)|.$$ In particular, $$|f| \in H^1(\mathbb R^n)$$.

Proof
For this proof we follow Elliott H. Lieb and Michael Loss. From the assumptions, $$\partial_j |f| \in L^1_{\text{loc}}(\mathbb R^n)$$ when viewed in the sense of distributions and $$\partial_j |f|(x) = \operatorname{Re}\left(\frac{\overline f(x)}{|f(x)|} \partial_j f(x)\right)$$ for almost every $$x$$ such that $$f(x) \neq 0$$ (and $$\partial_j |f|(x) = 0$$ if $$f(x) = 0$$). Moreover, $$\operatorname{Re}\left(\frac{\overline f(x)}{|f(x)|} i A_j f(x)\right) = \operatorname{Im}(A_jf) = 0.$$ So $$\nabla |f|(x) = \operatorname{Re}\left(\frac{\overline f(x)}{|f(x)|} \mathbf D f(x)\right) \leq \left|\frac{\overline f(x)}{|f(x)|} \mathbf D f(x)\right| = |\mathbf D f(x)|$$ for almost every $$x$$ such that $$f(x) \neq 0$$. The case that $$f(x) = 0$$ is similar.

Application to line bundles
Let $$p: L \to \mathbb R^n$$ be a U(1) line bundle, and let $$A$$ be a connection 1-form for $$L$$. In this situation, $$A$$ is real-valued, and the covariant derivative $$\mathbf D$$ satisfies $$\mathbf Df_j = (\partial_j + iA_j)f$$ for every section $$f$$. Here $$\partial_j$$ are the components of the trivial connection for $$L$$. If $$A_j \in L^2_{\text{loc}} (\mathbb R^n)$$ and $$f, (\partial_1 + iA_1)f, \dots, (\partial_n + iA_n)f \in L^2(\mathbb R^n)$$, then for almost every $$x \in \mathbb R^n$$, it follows from the diamagnetic inequality that $$|\nabla |f|(x)| \leq |\mathbf Df(x)|.$$

The above case is of the most physical interest. We view $$\mathbb R^n$$ as Minkowski spacetime. Since the gauge group of electromagnetism is $$U(1)$$, connection 1-forms for $$L$$ are nothing more than the valid electromagnetic four-potentials on $$\mathbb R^n$$. If $$F = dA$$ is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section $$\phi$$ of $$L$$ are $$\begin{cases} \partial^\mu F_{\mu\nu} = \operatorname{Im}(\phi \mathbf D_\nu \phi) \\ \mathbf D^\mu \mathbf D_\mu \phi = 0\end{cases}$$ and the energy of this physical system is $$\frac{||F(t)||_{L^2_x}^2}{2} + \frac{||\mathbf D \phi(t)||_{L^2_x}^2}{2}.$$ The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus $$A = 0$$.