Dirac operator: Difference between revisions
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dirac operator in many variables |
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== Generalisations == |
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In Clifford analysis, the operator |
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<math>D: C^\infty(\R^k\otimes \R^n,S)\to C^\infty(\R^k\otimes\R^n,\C^k\otimes S)</math> |
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acting on spinor valued functions defined by |
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:<math>f(x_1,\ldots,x_k)\mapsto |
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\begin{pmatrix} |
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\partial_{\underline{x_1}f}\\ |
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\partial_{\underline{x_2}f}\\ |
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\ldots\\ |
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\partial_{\underline{x_k}f}\\ |
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\end{pmatrix}</math> |
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is sometimes called Dirac operator in ''k'' Clifford variables. In the notation, ''S'' is the space of spinors, <math>x_i=(x_{i1},x_{i2},\ldots,x_{in})</math> are ''n''-dimensional variables and <math>\partial_{\underline{x_i}}=\sum_j e_j\cdot \partial_{x_{ij}}</math> is the Dirac operator in the <math>i</math>-th variable and ''S'' is the space of spinors. This is a common generalization of the Dirac operator (''k=1'') and the [[Dolbeault_cohomology|Dolbeault operator]] (''n=2''). This is an example of an [[invariant differential operator]] invariant to the action of the group <math>SL(k)\times SO(n)</math>. The [[injective_resolution#Injective_resolutions|resolution]] is known only in some special cases. |
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==See also== |
==See also== |
Revision as of 21:43, 14 September 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 and S is the space of spinors. This is a common generalization of the Dirac operator (k=1) and the Dolbeault operator (n=2). This is an example of an invariant differential operator invariant to the action of the group . The resolution 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