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 $A_j \in L^2_ (\mathbb R^n)$ 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, $\partial_j , f, \in L^1_(\mathbb R^n)$ when viewed in the sense of distributions and
$\partial_j , f, (x) = \operatorname\left(\frac \partial_j f(x)\right)$
for almost every $x$ such that $f(x) \neq 0$ (and $\partial_j , f, (x) = 0$ if $f(x) = 0$).
Moreover,
$\operatorname\left(\frac i A_j f(x)\right) = \operatorname(A_j) = 0.$
So
$\nabla , f, (x) = \operatorname\left(\frac \mathbf D f(x)\right) \leq \left, \frac \mathbf D f(x)\ = , \mathbf D f(x),$
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 $A_j \in L^2_ (\mathbb R^n)$ 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
$\begin \partial^\mu F_ = \operatorname(\phi \mathbf D_\nu \phi) \\ \mathbf D^\mu \mathbf D_\mu \phi = 0\end$
and the energy of this physical system is
$\frac + \frac.$
The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus $A = 0$.

See also

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Citations

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Inequalities (mathematics)
Electromagnetism