diamagnetic inequality

{{Short description|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.{{cite book |last1= Lieb|first1= Elliott|last2= Loss|first2= Michael|date= 2001|title= Analysis |location= Providence|publisher= American Mathematical Society|isbn= 9780821827833}}{{cite journal |last1= Hiroshima |first1=Fumio |date=1996 |title= Diamagnetic inequalities for systems of nonrelativistic particles with a quantized field.|url= https://mathscinet.ams.org/mathscinet-getitem?mr=1383577|journal= Reviews in Mathematical Physics|volume= 8|issue=2 |pages= 185–203|doi=10.1142/S0129055X9600007X |bibcode=1996RvMaP...8..185H |hdl=2115/69048 |mr=1383577 |s2cid=115703186 |access-date=November 25, 2021|hdl-access=free }}

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 $A_j \in L^2_{\text{loc}} (\mathbb R^n)$ 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, $\partial_j |f| \in L^1_{\text{loc}}(\mathbb R^n)$ when viewed in the sense of distributions and

$\partial_j |f|(x) = \operatorname{Re}\left(\frac{\overline f(x)}$

f(x)

\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,

$\operatorname{Re}\left(\frac{\overline f(x)}$

f(x)

i A_j f(x)\right) = \operatorname{Im}(A_j) = 0.

$\nabla |f|(x) = \operatorname{Re}\left(\frac{\overline f(x)}$

f(x)

\mathbf D f(x)\right) \leq \left|\frac{\overline f(x)}

f(x)

\mathbf D f(x)\right| = |\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 $A_j \in L^2_{\text{loc}} (\mathbb R^n)$ 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

$\begin{cases} \partial^\mu F_{\mu\nu} = \operatorname{Im}(\phi \mathbf D_\nu \phi) \\
\mathbf D^\mu \mathbf D_\mu \phi = 0\end{cases}$

and the energy of this physical system is

$\frac{||F(t)||_{L^2_x}^2}{2} + \frac{||\mathbf D \phi(t)||_{L^2_x}^2}{2}.$

The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus $A = 0$ .{{cite journal |last1= Oh |first1=Sung-Jin |last2=Tataru |first2=Daniel |date=2016 |title= Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation|journal= Annals of PDE|volume= 2|issue=1 |doi=10.1007/s40818-016-0006-4 |arxiv=1503.01560 |s2cid=116975954 }}

Citations

Category:Inequalities (mathematics)

Category:Electromagnetism

diamagnetic inequality

Proof

Application to line bundles

See also

Citations