Dirac operator: Difference between revisions
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In [[mathematics]] and [[quantum mechanics]], a '''Dirac operator''' is a [[differential operator]] that is a formal square root, or [[half-iterate]], of a second-order operator such as a [[Laplacian]]. The original case which concerned [[Dirac]] was to factorise formally an operator for [[Minkowski space]], to get a form of quantum theory compatible with [[special relativity]]; to get the relevant Laplacian as a product of first-order operators he introduced [[spinor]]s. |
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In general, let <math>D</math> be a first-order differential operator acting on a |
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[[vector bundle]] <math>V</math> over a [[Riemannian manifold]] <math>M</math>. |
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If |
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:<math>D^2=\triangle,</math> |
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<math>\triangle</math> being the Laplacian of <math>V</math>, <math>D</math> is |
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called a '''Dirac operator'''. |
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In [[high-energy physics]], this requirement is often relaxed: only the second-order part |
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of <math>D^2</math> must equal the Laplacian. |
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==Examples== |
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1: <math>-i\partial_x</math> is a Dirac operator on the tangential bundle over a line. |
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2: We now consider a simple bundle of importance in physics: The configuration space of a particle |
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with spin <math>\begin{matrix}\frac{1}{2}\end{matrix}</math> confined to a plane, which is also the base manifold. |
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Physicists generally think of wavefunctions <math>\psi:\mathbb{R}^2\to\mathbb{C}^2</math> |
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which they write |
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:<math>\begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix}.</math> |
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<math>x</math> and <math>y</math> are the usual coordinate functions on <math>\mathbb{R}^2</math>. |
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<math>\chi</math> specifies the [[probability amplitude]] for the particle to be in the |
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spin-up state, similarly for <math>\eta</math>. The so-called [[spin-Dirac operator]] |
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can then be written |
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:<math>D=-i\sigma_x\partial_x-i\sigma_y\partial_y,</math> |
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where <math>\sigma_i</math> are the [[Pauli matrices]]. Note that the anticommutation relations |
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for the Pauli matrices make the proof of the above defining property trivial. Those |
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relations define the notion of a [[Clifford algebra]]. |
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3: The most famous Dirac operator describes the propagation of a free electron in |
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three dimensions and is elegantly written |
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:<math>D=\gamma^\mu\partial_\mu</math> |
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using [[Einstein's summation convention]] and even more elegantly as |
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:<math>D=\partial\!\!\!/</math> |
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using the [[Feynman slash notation]]. |
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==See also== |
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*[[Clifford algebra]] |
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*[[connection (mathematics)|Connection]] |
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*[[Dolbeault operator]] |
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*[[Heat kernel]]. |
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[[Category:Differential operators]] |
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[[Category:Quantum mechanics]] |
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[[Category:Quantum field theory]] |
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