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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[[Image:Helloasdf.jpg]] |
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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]] |
Revision as of 00:49, 12 December 2005
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 spinors.
In general, let be a first-order differential operator acting on a vector bundle over a Riemannian manifold .
If
being the Laplacian of , is called a Dirac operator.
In high-energy physics, this requirement is often relaxed: only the second-order part of must equal the Laplacian.
Examples
1: is a Dirac operator on the tangential bundle over a line.
2: We now consider a simple bundle of importance in physics: The configuration space of a particle with spin confined to a plane, which is also the base manifold. Physicists generally think of wavefunctions which they write
and are the usual coordinate functions on . specifies the probability amplitude for the particle to be in the spin-up state, similarly for . The so-called spin-Dirac operator can then be written
where are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a Clifford algebra.
3: The most famous Dirac operator describes the propagation of a free electron in three dimensions and is elegantly written
using Einstein's summation convention and even more elegantly as
using the Feynman slash notation.