Editing Dirac operator (section)

== Examples ==
=== Example 1 ===
''D'' = −''i'' ∂<sub>''x''</sub> is a Dirac operator on the [[tangent bundle]] over a line.

=== Example 2 ===
Consider a simple bundle of notable importance in physics: the configuration space of a particle with spin {{sfrac|1|2}} confined to a plane, which is also the base manifold.  It is represented by a wavefunction {{nowrap|''ψ''  : '''R'''<sup>2</sup> → '''C'''<sup>2</sup>}}
: <math>\psi(x,y) = \begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix}</math>
where ''x'' and ''y'' are the usual coordinate functions on '''R'''<sup>2</sup>. ''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
: <math>D=-i\sigma_x\partial_x-i\sigma_y\partial_y ,</math>
where ''σ''<sub>''i''</sub> 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''.<ref>{{SpringerEOM|id=Spinor_structure&oldid=33893 |title=Spinor structure }}</ref>

=== Example 3 ===
Feynman's Dirac operator describes the propagation of a free [[fermion]] in three dimensions and is elegantly written 
: <math>D=\gamma^\mu\partial_\mu\ \equiv \partial\!\!\!/,</math>
using the [[Feynman slash notation]]. In introductory textbooks to [[quantum field theory]], this will appear in the form

:<math>D = c\vec\alpha \cdot (-i\hbar\nabla_x) + mc^2\beta</math>

where <math>\vec\alpha = (\alpha_1, \alpha_2, \alpha_3)</math> are the off-diagonal [[Dirac matrices]] <math>\alpha_i=\beta\gamma_i</math>, with <math>\beta=\gamma_0</math> and the remaining constants are <math>c</math> the [[speed of light]], <math>\hbar</math> being the [[Planck constant]], and <math>m</math> the [[mass]] of a fermion (for example, an [[electron]]). It acts on a four-component wave function <math>\psi(x) \in L^2(\mathbb{R}^3, \mathbb{C}^4)</math>, the [[Sobolev space]] of smooth, square-integrable functions. It can be extended to a [[self-adjoint operator]] on that domain. The square, in this case, is not the Laplacian, but instead <math>D^2=\Delta+m^2</math> (after setting <math>\hbar=c=1.</math>)

=== Example 4 ===
Another Dirac operator arises in [[Clifford analysis]]. In euclidean ''n''-space this is
: <math>D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}</math>
where {''e<sub>j</sub>'': ''j'' = 1, ..., ''n''} is an orthonormal basis for euclidean ''n''-space, and '''R'''<sup>''n''</sup> 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 {{nowrap|''x'' ∈ ''M''}} and ''e<sub>1</sub>''(''x''), ..., ''e<sub>j</sub>''(''x'') a local orthonormal basis for the tangent space of ''M'' at ''x'', the Atiyah–Singer–Dirac operator is
:<math>D=\sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)} ,</math>
where <math>\tilde{\Gamma}</math> is the [[spin connection]], a lifting of the [[Levi-Civita connection]] on ''M'' to the [[spinor bundle]] over ''M''.  The square in this case is not the Laplacian, but instead <math>D^2=\Delta+R/4</math> where <math>R</math> is the [[scalar curvature]] of the connection.<ref> Jurgen Jost, (2002) "Riemannian Geometry ang Geometric Analysis (3rd edition)", Springer. ''See section 3.4 pages 142 ff.''</ref>

=== Example 6 ===
On [[Riemannian manifold]] <math>(M, g)</math> of dimension <math>n=dim(M)</math> with [[Levi-Civita connection]] <math>\nabla</math>and an [[orthonormal basis]] <math>\{e_{a}\}_{a=1}^{n}</math>, we can define [[exterior derivative]] <math>d</math> and [[Codifferential|coderivative]] <math>\delta</math> as
: <math>d= e^{a}\wedge \nabla_{e_{a}}, \quad \delta =e^{a} \lrcorner \nabla_{e_{a}}</math>.

Then we can define a Dirac-Kähler operator<ref name=":0">{{Cite journal |last=Graf |first=Wolfgang |date=1978 |title=Differential forms as spinors |url=http://www.numdam.org/item/?id=AIHPA_1978__29_1_85_0 |journal=Annales de l'Institut Henri Poincaré A |language=en |volume=29 |issue=1 |pages=85–109 |issn=2400-4863}}</ref><ref name=":1">{{Cite book |last1=Benn |first1=Ian M. |url=https://books.google.com/books?id=FzcbAQAAIAAJ |title=An Introduction to Spinors and Geometry with Applications in Physics |last2=Tucker |first2=Robin W. |date=1987 |publisher=A. Hilger |isbn=978-0-85274-169-6 |language=en}}</ref><ref name=":2">{{Cite journal |last=Kycia |first=Radosław Antoni |date=2022-07-29 |title=The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics |url=https://doi.org/10.1007/s00025-022-01646-z |journal=Results in Mathematics |language=en |volume=77 |issue=5 |pages=182 |doi=10.1007/s00025-022-01646-z |arxiv=2009.08542 |s2cid=221802588 |issn=1420-9012}}</ref> <math>D</math>, as follows
: <math>D = e^{a}\nabla_{e_{a}}=d-\delta</math>.

The operator acts on sections of [[Clifford bundle]] in general, and it can be restricted to spinor bundle, an ideal of Clifford bundle, only if the projection operator on the ideal is parallel.<ref name=":0" /><ref name=":1" /><ref name=":2" />