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
denote the usual Hilbert space of square-integrable functions, and
the Sobolev space of square-integrable functions with square-integrable derivatives.
be measurable functions on
loc
{\displaystyle A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})}
For this proof we follow Elliott H. Lieb and Michael Loss.
loc
{\displaystyle \partial _{j}|f|\in L_{\text{loc}}^{1}(\mathbb {R} ^{n})}
when viewed in the sense of distributions and
= Im (
{\displaystyle \operatorname {Re} \left({\frac {{\overline {f}}(x)}{|f(x)|}}iA_{j}f(x)\right)=\operatorname {Im} (A_{j})=0.}
be a U(1) line bundle, and let
is real-valued, and the covariant derivative
are the components of the trivial connection for
loc
{\displaystyle A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})}
, it follows from the diamagnetic inequality that
The above case is of the most physical interest.
as Minkowski spacetime.
Since the gauge group of electromagnetism is
are nothing more than the valid electromagnetic four-potentials on
is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section
μ ν
= Im ( ϕ
{\displaystyle {\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
The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus