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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 [[Paul Dirac]] was to factorise formally an operator for [[ |
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 [[Paul Dirac]] was to factorise formally an operator for [[Meinkowski 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 |
In general, let <math>D</math> be a first-order differential operator acting on a |
Revision as of 19:21, 29 July 2011
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 Paul Dirac was to factorise formally an operator for Meinkowski 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
with 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
- is a Dirac operator on the tangent bundle over a line.
- We now consider a simple bundle of importance in physics: The configuration space of a particle with spin 1⁄2 confined to a plane, which is also the base manifold. It's represented by a a wavefunction ψ: R2 → C2
- ,
- ,
- The most famous Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written
- There is also the Dirac operator arising in Clifford analysis. In euclidean n-space this is
- For a spin manifold, M, the Atiyah-Singer-Dirac operator is locally defined as follows: For and a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is
- ,