Mathematical inequality relating the derivative of a function to its covariant derivative
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. 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.[1] From the assumptions,
when viewed in the sense of distributions and
for almost every
such that
(and
if
). 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
.[3]
See also
Citations
- ^ a b Lieb, Elliott; Loss, Michael (2001). Analysis. Providence: American Mathematical Society. ISBN 9780821827833.
- ^ Hiroshima, Fumio (1996). "Diamagnetic inequalities for systems of nonrelativistic particles with a quantized field". Reviews in Mathematical Physics. 8 (2): 185–203. Bibcode:1996RvMaP...8..185H. doi:10.1142/S0129055X9600007X. hdl:2115/69048. MR 1383577. S2CID 115703186. Retrieved November 25, 2021.
- ^ Oh, Sung-Jin; Tataru, Daniel (2016). "Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation". Annals of PDE. 2 (1). arXiv:1503.01560. doi:10.1007/s40818-016-0006-4. S2CID 116975954.