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Revision as of 12:20, 8 December 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 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
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
- ,
Generalisations
In Clifford analysis, the operator acting on spinor valued functions defined by
is sometimes called Dirac operator in k Clifford variables. In the notation, S is the space of spinors, are n-dimensional variables and is the Dirac operator in the -th variable. This is a common generalization of the Dirac operator (k=1) and the Dolbeault operator (n=2, k arbitrary). It is an invariant differential operator, invariant to the action of the group . The resolution of D is known only in some special cases.
See also
References
- Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1
- Colombo, F., I.; Sabadini, I. (2004), Analysis of Dirac Systems and Computational Algebra, Birkhauser Verlag AG, ISBN 978-3764342555