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 denote the usual Hilbert space of square-integrable functions, and the Sobolev space of square-integrable functions with square-integrable derivatives.
Let be measurable functions on and suppose that is real-valued, is complex-valued, and.
Then for almost every,
In particular,.

Proof

For this proof we follow Elliott H. Lieb and Michael Loss.
From the assumptions, when viewed in the sense of distributions and
for almost every such that .
Moreover,
So
for almost every such that. The case that is similar.

Application to line bundles

Let be a U(1) line bundle, and let be a connection 1-form for.
In this situation, is real-valued, and the covariant derivative satisfies for every section. Here are the components of the trivial connection for.
If and, then for almost every, it follows from the diamagnetic inequality that
The above case is of the most physical interest. We view as Minkowski spacetime. Since the gauge group of electromagnetism is, connection 1-forms for are nothing more than the valid electromagnetic four-potentials on.
If is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section of are
and the energy of this physical system is
The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus.