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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 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 D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M. If

\( D^2=\Delta, \, \)

where ∆ is the Laplacian of V, then D is called a Dirac operator.

In high-energy physics, this requirement is often relaxed: only the second-order part of D2 must equal the Laplacian.

Examples

Example 1: -i ∂x is a Dirac operator on the tangent bundle over a line.

Example 2: We now consider a simple bundle of importance in physics: The configuration space of a particle with spin ½ confined to a plane, which is also the base manifold. It's represented by a a wavefunction ψ: R2 → C2

\( \psi(x,y) = \begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix} \)

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

\( D=-i\sigma_x\partial_x-i\sigma_y\partial_y,\, \)

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].

Example 3: The most famous Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written

\( D=\gamma^\mu\partial_\mu\ \equiv \partial\!\!\!/, \)

using the Feynman slash notation.

Example 4: There is also the Dirac operator arising in Clifford analysis. In euclidean n-space this is

\( D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}} \)

where {ej: j = 1, ..., n} is an orthonormal basis for euclidean n-space, and Rn 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.

Example 5: For a spin manifold, M, the Atiyah-Singer-Dirac operator is locally defined as follows: For x ∈ M and e1(x), ..., ej(x) a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is

\( \sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}, \)

where \( \tilde{\Gamma} is \) a lifting of the Levi-Civita connection on M to the spinor bundle over M.
Generalisations

In Clifford analysis, the operator D: C∞(Rk ⊗ Rn, S) → C∞(Rk ⊗ Rn, Ck ⊗ S) acting on spinor valued functions defined by

\( f(x_1,\ldots,x_k)\mapsto \begin{pmatrix} \partial_{\underline{x_1}}f\\ \partial_{\underline{x_2}}f\\ \ldots\\ \partial_{\underline{x_k}}f\\ \end{pmatrix} \)

is sometimes called Dirac operator in k Clifford variables. In the notation, S is the space of spinors, \( x_i=(x_{i1},x_{i2},\ldots,x_{in}) \) are n-dimensional variables and \( \partial_{\underline{x_i}}=\sum_j e_j\cdot \partial_{x_{ij}} \) is the Dirac operator in the i-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 SL(k) × Spin(n). The resolution of D is known only in some special cases.
See also

Dirac equation
Clifford algebra
Clifford analysis
Connection
Dolbeault operator
Heat kernel
Spinor bundle

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

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