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.

## Formal definition

In general, let *D* be a first-order differential operator acting on a vector bundle *V* over a Riemannian manifold *M*. If

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 *D*^{2} must equal the Laplacian.

## Examples

**Example 1:** *D* = −*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 1/2 confined to a plane, which is also the base manifold. It is represented by a wavefunction *ψ* : **R**^{2} → **C**^{2}

where *x* and *y* are the usual coordinate functions on **R**^{2}. *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 *σ*_{i} 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:** Feynman's Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written

using the Feynman slash notation.

**Example 4:** Another Dirac operator arises in Clifford analysis. In euclidean *n*-space this is

where {*e _{j}*:

*j*= 1, ...,

*n*} is an orthonormal basis for euclidean

*n*-space, and

**R**

^{n}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 *e _{1}*(

*x*), ...,

*e*(

_{j}*x*) 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 *D* : *C*^{∞}(**R**^{k} ⊗ **R**^{n}, *S*) → *C*^{∞}(**R**^{k} ⊗ **R**^{n}, **C**^{k} ⊗ *S*) 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 *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 under the action of the group SL(*k*) × Spin(*n*). 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 978-3-7643-4255-5