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.<ref name="Lieb">Lieb, Elliott; Loss, Michael (2001). Analysis. Providence: American Mathematical Society. ISBN 9780821827833.</ref><ref name="Hiroshima">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.</ref>

To precisely state the inequality, let <math>L^2(\mathbb R^n)</math> denote the usual Hilbert space of square-integrable functions, and <math>H^1(\mathbb R^n)</math> the Sobolev space of square-integrable functions with square-integrable derivatives. Let <math>f, A_1, \dots, A_n</math> be measurable functions on <math>\mathbb R^n</math> and suppose that <math>A_j \in L^2_{\text{loc}} (\mathbb R^n)</math> is real-valued, <math>f</math> is complex-valued, and <math>f , (\partial_1 + iA_1)f, \dots, (\partial_n + iA_n)f \in L^2(\mathbb R^n)</math>. Then for almost every <math>x \in \mathbb R^n</math>, <math display="block">|\nabla |f|(x)| \leq |(\nabla + iA)f(x)|.</math> In particular, <math>|f| \in H^1(\mathbb R^n)</math>.

Proof

For this proof we follow Elliott H. Lieb and Michael Loss.<ref name="Lieb"/> From the assumptions, <math>\partial_j |f| \in L^1_{\text{loc}}(\mathbb R^n)</math> when viewed in the sense of distributions and <math display="block">\partial_j |f|(x) = \operatorname{Re}\left(\frac{\overline f(x)}{|f(x)|} \partial_j f(x)\right)</math> for almost every <math>x</math> such that <math>f(x) \neq 0</math> (and <math>\partial_j |f|(x) = 0</math> if <math>f(x) = 0</math>). Moreover, <math display="block">\operatorname{Re}\left(\frac{\overline f(x)}{|f(x)|} i A_j f(x)\right) = \operatorname{Im}(A_jf) = 0.</math> So <math display="block">\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(x)|</math> for almost every <math>x</math> such that <math>f(x) \neq 0</math>. The case that <math>f(x) = 0</math> is similar.

Application to line bundles

Let <math>p: L \to \mathbb R^n</math> be a U(1) line bundle, and let <math>A</math> be a connection 1-form for <math>L</math>. In this situation, <math>A</math> is real-valued, and the covariant derivative <math>\mathbf D</math> satisfies <math>\mathbf Df_j = (\partial_j + iA_j)f</math> for every section <math>f</math>. Here <math>\partial_j</math> are the components of the trivial connection for <math>L</math>. If <math>A_j \in L^2_{\text{loc}} (\mathbb R^n)</math> and <math>f , (\partial_1 + iA_1)f, \dots, (\partial_n + iA_n)f \in L^2(\mathbb R^n)</math>, then for almost every <math>x \in \mathbb R^n</math>, it follows from the diamagnetic inequality that <math display="block">|\nabla |f|(x)| \leq |\mathbf Df(x)|.</math>

The above case is of the most physical interest. We view <math>\mathbb R^n</math> as Minkowski spacetime. Since the gauge group of electromagnetism is <math>U(1)</math>, connection 1-forms for <math>L</math> are nothing more than the valid electromagnetic four-potentials on <math>\mathbb R^n</math>. If <math>F = dA</math> is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section <math>\phi</math> of <math>L</math> are <math display="block">\begin{cases} \partial^\mu F_{\mu\nu} = \operatorname{Im}(\phi \mathbf D_\nu \phi) \\ \mathbf D^\mu \mathbf D_\mu \phi = 0\end{cases}</math> and the energy of this physical system is <math display="block">\frac{||F(t)||_{L^2_x}^2}{2} + \frac{||\mathbf D \phi(t)||_{L^2_x}^2}{2}.</math> The diamagnetic inequality guarantees that the energy is minimized in the absence of electromagnetism, thus <math>A = 0</math>.<ref name="Oh">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.</ref>

See also

• Diamagnetism – Magnetic property of ordinary materials