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 ( R n ) {\displaystyle L^{2}(\mathbb {R} ^{n})} denote the usual Hilbert space of square-integrable functions, and H 1 ( R n ) {\displaystyle H^{1}(\mathbb {R} ^{n})} 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 ( R n ) {\displaystyle A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})} is real-valued, f {\displaystyle f} is complex-valued, and f , ( ∂ 1 + i A 1 ) f , … , ( ∂ n + i A n ) f ∈ L 2 ( R n ) {\displaystyle f,(\partial _{1}+iA_{1})f,\dots ,(\partial _{n}+iA_{n})f\in L^{2}(\mathbb {R} ^{n})} . Then for almost every x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} ,
| ∇ | f | ( x ) | ≤ | ( ∇ + i A ) f ( x ) | . {\displaystyle |\nabla |f|(x)|\leq |(\nabla +iA)f(x)|.}
In particular, | f | ∈ H 1 ( R n ) {\displaystyle |f|\in H^{1}(\mathbb {R} ^{n})} .
Proof For this proof we follow Elliott H. Lieb and Michael Loss. From the assumptions, ∂ j | f | ∈ L loc 1 ( R n ) {\displaystyle \partial _{j}|f|\in L_{\text{loc}}^{1}(\mathbb {R} ^{n})} when viewed in the sense of distributions and
∂ j | f | ( x ) = Re ( f ¯ ( x ) | f ( x ) | ∂ j f ( x ) ) {\displaystyle \partial _{j}|f|(x)=\operatorname {Re} \left({\frac {{\overline {f}}(x)}{|f(x)|}}\partial _{j}f(x)\right)}
for almost every x {\displaystyle x} such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} (and ∂ j | f | ( x ) = 0 {\displaystyle \partial _{j}|f|(x)=0} if f ( x ) = 0 {\displaystyle f(x)=0} ). Moreover,
Re ( f ¯ ( x ) | f ( x ) | i A j f ( x ) ) = Im ( A j ) = 0. {\displaystyle \operatorname {Re} \left({\frac {{\overline {f}}(x)}{|f(x)|}}iA_{j}f(x)\right)=\operatorname {Im} (A_{j})=0.}
So
∇ | f | ( x ) = Re ( f ¯ ( x ) | f ( x ) | D f ( x ) ) ≤ | f ¯ ( x ) | f ( x ) | D f ( x ) | = | D f ( x ) | {\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 x {\displaystyle x} such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} . The case that f ( x ) = 0 {\displaystyle f(x)=0} is similar.
Application to line bundles Let p : L → R n {\displaystyle p:L\to \mathbb {R} ^{n}} be a U(1) line bundle, and let A {\displaystyle A} be a connection 1-form for L {\displaystyle L} . In this situation, A {\displaystyle A} is real-valued, and the covariant derivative D {\displaystyle \mathbf {D} } satisfies D f j = ( ∂ j + i A j ) f {\displaystyle \mathbf {D} f_{j}=(\partial _{j}+iA_{j})f} for every section f {\displaystyle f} . Here ∂ j {\displaystyle \partial _{j}} are the components of the trivial connection for L {\displaystyle L} . If A j ∈ L loc 2 ( R n ) {\displaystyle A_{j}\in L_{\text{loc}}^{2}(\mathbb {R} ^{n})} and f , ( ∂ 1 + i A 1 ) f , … , ( ∂ n + i A n ) f ∈ L 2 ( R n ) {\displaystyle f,(\partial _{1}+iA_{1})f,\dots ,(\partial _{n}+iA_{n})f\in L^{2}(\mathbb {R} ^{n})} , then for almost every x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , it follows from the diamagnetic inequality that
| ∇ | f | ( x ) | ≤ | D f ( x ) | . {\displaystyle |\nabla |f|(x)|\leq |\mathbf {D} f(x)|.}
The above case is of the most physical interest. We view R n {\displaystyle \mathbb {R} ^{n}} as Minkowski spacetime. Since the gauge group of electromagnetism is U ( 1 ) {\displaystyle U(1)} , connection 1-forms for L {\displaystyle L} are nothing more than the valid electromagnetic four-potentials on R n {\displaystyle \mathbb {R} ^{n}} . If F = d A {\displaystyle F=dA} is the electromagnetic tensor, then the massless Maxwell–Klein–Gordon system for a section ϕ {\displaystyle \phi } of L {\displaystyle L} are
{ ∂ μ F μ ν = Im ( ϕ D ν ϕ ) D μ D μ ϕ = 0 {\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
| | F ( t ) | | L x 2 2 2 + | | D ϕ ( t ) | | L x 2 2 2 . {\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 A = 0 {\displaystyle A=0} .
See also Diamagnetism – Magnetic property of ordinary materials
Citations