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. ^{[1] [2]}

To precisely state the inequality, let ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ denote the usual Hilbert space of square-integrable functions, and ${\displaystyle H^{1}(\mathbb {R} ^{n})}$ the Sobolev space of square-integrable functions with square-integrable derivatives. Let ${\displaystyle f,A_{1},\dots ,A_{n}}$ be measurable functions on ${\displaystyle \mathbb {R} ^{n}}$ and suppose that ${\displaystyle A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})}$ is real-valued, ${\displaystyle f}$ is complex-valued, and ${\displaystyle f,(\partial _{1}+iA_{1})f,\dots ,(\partial _{n}+iA_{n})f\in L^{2}(\mathbb {R} ^{n})}$. Then for almost every ${\displaystyle x\in \mathbb {R} ^{n}}$,

$${\displaystyle |\nabla |f|(x)|\leq |(\nabla +iA)f(x)|.}$$

In particular, ${\displaystyle |f|\in H^{1}(\mathbb {R} ^{n})}$.

Proof

For this proof we follow Elliott H. Lieb and Michael Loss. ^[1] From the assumptions, ${\displaystyle \partial _{j}|f|\in L_{\text{loc}}^{1}(\mathbb {R} ^{n})}$ when viewed in the sense of distributions and

$${\displaystyle \partial _{j}|f|(x)=\operatorname {Re} \left({\frac {{\overline {f}}(x)}{|f(x)|}}\partial _{j}f(x)\right)}$$

for almost every ${\displaystyle x}$ such that ${\displaystyle f(x)\neq 0}$ (and ${\displaystyle \partial _{j}|f|(x)=0}$ if ${\displaystyle f(x)=0}$). Moreover,

$${\displaystyle \operatorname {Re} \left({\frac {{\overline {f}}(x)}{|f(x)|}}iA_{j}f(x)\right)=\operatorname {Im} (A_{j}f)=0.}$$

So

$${\displaystyle \nabla |f|(x)=\operatorname {Re} \left({\frac {{\overline {f}}(x)}{|f(x)|}}\mathbf {D} f(x)\right)\leq \left|{\frac {{\overline {f}}(x)}{|f(x)|}}\mathbf {D} f(x)\right|=|\mathbf {D} f(x)|}$$

for almost every ${\displaystyle x}$ such that ${\displaystyle f(x)\neq 0}$. The case that ${\displaystyle f(x)=0}$ is similar.

Application to line bundles

Let ${\displaystyle p:L\to \mathbb {R} ^{n}}$ be a U(1) line bundle, and let ${\displaystyle A}$ be a connection 1-form for ${\displaystyle L}$. In this situation, ${\displaystyle A}$ is real-valued, and the covariant derivative ${\displaystyle \mathbf {D} }$ satisfies ${\displaystyle \mathbf {D} f_{j}=(\partial _{j}+iA_{j})f}$ for every section ${\displaystyle f}$. Here ${\displaystyle \partial _{j}}$ are the components of the trivial connection for ${\displaystyle L}$. If ${\displaystyle A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})}$ and ${\displaystyle f,(\partial _{1}+iA_{1})f,\dots ,(\partial _{n}+iA_{n})f\in L^{2}(\mathbb {R} ^{n})}$, then for almost every ${\displaystyle x\in \mathbb {R} ^{n}}$, it follows from the diamagnetic inequality that

$${\displaystyle |\nabla |f|(x)|\leq |\mathbf {D} f(x)|.}$$

The above case is of the most physical interest. We view ${\displaystyle \mathbb {R} ^{n}}$ as Minkowski spacetime. Since the gauge group of electromagnetism is ${\displaystyle U(1)}$, connection 1-forms for ${\displaystyle L}$ are nothing more than the valid electromagnetic four-potentials on ${\displaystyle \mathbb {R} ^{n}}$. If ${\displaystyle F=dA}$ is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section ${\displaystyle \phi }$ of ${\displaystyle L}$ are

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

and the energy of this physical system is

$${\displaystyle {\frac {||F(t)||_{L_{x}^{2}}^{2}}{2}}+{\frac {||\mathbf {D} \phi (t)||_{L_{x}^{2}}^{2}}{2}}.}$$

The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus ${\displaystyle A=0}$. ^[3]

See also

• Diamagnetism  – Magnetic property of ordinary materials

Citations

1. Lieb, Elliott; Loss, Michael (2001). Analysis. Providence: American Mathematical Society. ISBN   9780821827833.
2. 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.
3. 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.