Diamagnetic inequality

Template:Short description 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 $L^2({ℝ}^n)$ denote the usual Hilbert space of square-integrable functions, and $H^1({ℝ}^n)$ the Sobolev space of square-integrable functions with square-integrable derivatives. Let $f,A_1,\dots ,A_n$ be measurable functions on ${ℝ}^n$ and suppose that $A_j\in L_{\text{loc}}^2({ℝ}^n)$ is real-valued, $f$ is complex-valued, and $f,({\partial }_1+i{A}_1)f,\dots ,({\partial }_n+i{A}_n)f\in L^2({ℝ}^n)$. Then for almost every $x\in {ℝ}^n$, $$|\mathrm{\nabla }|f|(x)|\le |(\mathrm{\nabla }+iA)f(x)|.$$ In particular, $|f|\in H^1({ℝ}^n)$.

Proof

For this proof we follow Elliott H. Lieb and Michael Loss. From the assumptions, ${\partial }_j|f|\in L_{\text{loc}}^1({ℝ}^n)$ when viewed in the sense of distributions and $${\partial }_j|f|(x)=\mathrm{Re}(\frac{\bar{f}(x)}{|f(x)|}{\partial }_{j}f(x))$$ for almost every $x$ such that $f(x)\ne 0$ (and ${\partial }_j|f|(x)=0$ if $f(x)=0$). Moreover, $$\mathrm{Re}(\frac{\bar{f}(x)}{|f(x)|}i{A}_{j}f(x))=\mathrm{Im}(A_j)=0.$$ So $$\mathrm{\nabla }|f|(x)=\mathrm{Re}(\frac{\bar{f}(x)}{|f(x)|}𝐃f(x))\le |\frac{\bar{f}(x)}{|f(x)|}𝐃f(x)|=|𝐃f(x)|$$ for almost every $x$ such that $f(x)\ne 0$. The case that $f(x)=0$ is similar.

Application to line bundles

Let $p:L\to {ℝ}^n$ 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 $𝐃$ satisfies $𝐃f_j=({\partial }_j+i{A}_j)f$ for every section $f$. Here ${\partial }_j$ are the components of the trivial connection for $L$. If $A_j\in L_{\text{loc}}^2({ℝ}^n)$ and $f,({\partial }_1+i{A}_1)f,\dots ,({\partial }_n+i{A}_n)f\in L^2({ℝ}^n)$, then for almost every $x\in {ℝ}^n$, it follows from the diamagnetic inequality that $$|\mathrm{\nabla }|f|(x)|\le |𝐃f(x)|.$$

The above case is of the most physical interest. We view ${ℝ}^n$ 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 ${ℝ}^n$. If $F=dA$ is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section $\varphi$ of $L$ are $$\{\begin{array}{c}{\partial }^{\mu }{F}_{\mu \nu }=\mathrm{Im}(\varphi {𝐃}_{\nu }\varphi )\\ {𝐃}^{\mu }{𝐃}_{\mu }\varphi =0\end{array}$$ and the energy of this physical system is $$\frac{||F(t)|{|}_{L_x^2}^2}{2}+\frac{||𝐃\varphi (t)|{|}_{L_x^2}^2}{2}.$$ The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus $A=0$.

See also

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