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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 L 2 {\displaystyle L^{2}} denote the usual Hilbert space of square-integrable functions, and H 1 {\displaystyle H^{1}} the Sobolev space of square-integrable functions with square-integrable derivatives.Let f , A 1 , … , A n {\displaystyle f,A_{1},\dots ,A_{n}} be measurable functions on R n {\displaystyle \mathbb {R} ^{n}} and suppose that A j ∈ L loc 2 {\displaystyle A_{j}\in L_{\text{loc}}^{2}} is real-valued, f {\displaystyle f} is complex-valued, and f , f , … , f ∈ L 2 {\displaystyle f,f,\dots ,f\in L^{2}}.Then for almost every x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} ,