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
,
where x and y are the usual coordinate functions on R2. χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written
where σi 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.
Solutions to the Dirac equation for spinor fields are often called harmonic spinors[1].
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
where
is an orthonormal basis for euclidean n-space, and is considered to be embedded in a Clifford algebra.
This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a spinor bundle.
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
,
where is a lifting of the Levi-Civita connection on M to the spinor bundle over M.
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.