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In general, let be a first-order differential operator acting on a

If

called a Dirac operator.

In high-energy physics, this requirement is often relaxed: only the second-order part of must equal the Laplacian.

1 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

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.

2 See also

• Clifford algebra
• ConnectionDifferential geometry In differential geometry, a connection (also connexion or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. That is an application to tangent bundles; there are more
• Dolbeault operator
• Heat kernel .