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.^[1]^[2]

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 $A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})$ 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.^[1] From the assumptions, $\partial _{j}|f|\in L_{\text{loc}}^{1}(\mathbb {R} ^{n})$ 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, $\operatorname {Re} \left({\frac {{\overline {f}}(x)}{|f(x)|}}iA_{j}f(x)\right)=\operatorname {Im} (A_{j}f)=0.$ 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 $\mathbf {D} f_{j}=(\partial _{j}+iA_{j})f$ for every section $f$ . Here $\partial _{j}$ are the components of the trivial connection for $L$ . If $A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})$ 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 $|\nabla |f|(x)|\leq |\mathbf {D} f(x)|.$

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 ${\frac {||F(t)||_{L_{x}^{2}}^{2}}{2}}+{\frac {||\mathbf {D} \phi (t)||_{L_{x}^{2}}^{2}}{2}}.$ The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus $A=0$ .^[3]