Physics: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.

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

Proof

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

Application to line bundles

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

The above case is of the most physical interest. We view [math]\displaystyle{ \mathbb R^n }[/math] as Minkowski spacetime. Since the gauge group of electromagnetism is [math]\displaystyle{ U(1) }[/math], connection 1-forms for [math]\displaystyle{ L }[/math] are nothing more than the valid electromagnetic four-potentials on [math]\displaystyle{ \mathbb R^n }[/math]. If [math]\displaystyle{ F = dA }[/math] is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section [math]\displaystyle{ \phi }[/math] of [math]\displaystyle{ L }[/math] are [math]\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} }[/math] and the energy of this physical system is [math]\displaystyle{ \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]\displaystyle{ A = 0 }[/math].

See also

• Physics:Diamagnetism – Magnetic property of ordinary materials