Diamagnetic inequality explained

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

L ^{2(R} ^{n)} denote the usual Hilbert space of square-integrable functions, and H ^{1(R} ^{n)} the Sobolev space of square-integrable functions with square-integrable derivatives.Let f, A _{1,} ... , A _{n} be measurable functions on R ^{n} and suppose that A _{j} \in L ^{2} _{loc} ( R ^{n)} is real-valued, f is complex-valued, and f , (\partial _{1} + iA _{1)f,} ... , (\partial _{n} + iA _{n)f} \in L ^{2(R} ^{n)} .Then for almost every x \in R ^{n} ,$$|\backslash nabla\; |f|(x)|\; \backslash leq\; |(\backslash nabla\; +\; iA)f(x)|.$$In particular, |f| \in H ^{1(R} ^{n)} .

Proof

For this proof we follow Elliott H. Lieb and Michael Loss.^[1] From the assumptions,

\partial _{j} |f| \in L ^{1} _{loc} ( R ^{n)} when viewed in the sense of distributions and $$\backslash partial\_j\; |f|(x)\; =\; \backslash operatorname\backslash left(\backslash frac$$ \partial_j f(x)\right)for almost every x such that f(x) ≠ 0 (and \partial _{j} |f|(x) = 0 if f(x) = 0 ).Moreover,$$\backslash operatorname\backslash left(\backslash frac$$ i A_j f(x)\right) = \operatorname(A_jf) = 0.So$$\backslash nabla\; |f|(x)\; =\; \backslash operatorname\backslash left(\backslash frac$$ \mathbf D f(x)\right) \leq \left|\frac \mathbf D f(x)\right| = |\mathbf D f(x)|for almost every x such that f(x) ≠ 0 . The case that f(x) = 0 is similar.

Application to line bundles

Let

p: L \to 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 D satisfies 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} _{loc} ( R ^{n)} and f , (\partial _{1} + iA _{1)f,} ... , (\partial _{n} + iA _{n)f} \in L ^{2(R} ^{n)} , then for almost every x \in R ^{n} , it follows from the diamagnetic inequality that$$|\backslash nabla\; |f|(x)|\; \backslash leq\; |\backslash mathbf\; Df(x)|.$$

The above case is of the most physical interest. We view

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 R ^{n} .If F = dA is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section \phi of L are $$\backslash begin\; \backslash partial^\backslash mu\; F\_\; =\; \backslash operatorname(\backslash phi\; \backslash mathbf\; D\_\backslash nu\; \backslash phi)\; \backslash \backslash \backslash mathbf\; D^\backslash mu\; \backslash mathbf\; D\_\backslash mu\; \backslash phi\; =\; 0\backslash end$$and the energy of this physical system is $$\backslash frac$$

F(t)

_^2

+ \frac

\mathbf D \phi(t)

_^2

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

Notes and References

1. Book: Lieb. Elliott. Loss. Michael. 2001. Analysis . Providence. American Mathematical Society. 9780821827833.
2. Hiroshima . Fumio . 1996 . Diamagnetic inequalities for systems of nonrelativistic particles with a quantized field.. Reviews in Mathematical Physics. 8. 2 . 185–203. 10.1142/S0129055X9600007X . 1996RvMaP...8..185H . 2115/69048 . 1383577 . 115703186 . November 25, 2021. free .
3. Oh . Sung-Jin . Tataru . Daniel . 2016 . Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation. Annals of PDE. 2. 1 . 10.1007/s40818-016-0006-4 . 1503.01560 . 116975954 .