Editing Dirac operator

{{short description|First-order differential linear operator on spinor bundle, whose square is the Laplacian}}
In [[mathematics]] and in [[quantum mechanics]], a '''Dirac operator''' is a first-order [[differential operator]] that is a formal square root, or [[half-iterate]], of a second-order differential operator such as a [[Laplacian]]. It was introduced  in 1847 by [[William Rowan Hamilton|William Hamilton]]<ref name="Hamilton1847" /> and in 1928 by [[Paul Dirac]].<ref name="Dirac1928">{{ cite journal
|author=Dirac, P. A. M.
|title=The Quantum Theory of the Electron
|journal= Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character
|year=1928
|volume=117
|number=778
|pages=610−624
|doi=10.1098/rspa.1928.0023
|doi-access=free
}}
</ref> The question which concerned Dirac was to factorise formally the [[Laplace operator]] of the [[Minkowski space]], to get an equation for the [[wave function]] which would be compatible with [[special relativity]].

== Formal definition ==

In general, let ''D'' be a first-order differential operator acting on a [[vector bundle]] ''V'' over a [[Riemannian manifold]] ''M''. If 
: <math>D^2=\Delta, \,</math>
where ∆ is the [[Laplace_operator#Analytic_and_geometric_Laplacians|(positive, or geometric) Laplacian]] of ''V'', then ''D'' is called a '''Dirac operator'''.

Note that there are two different conventions as to how the Laplace operator is defined: the "analytic" Laplacian, which could be characterized in <math>\R^n</math> as <math>\Delta=\nabla^2=\sum_{j=1}^n\Big(\frac{\partial}{\partial x_j}\Big)^2</math> (which is [[Definite_quadratic_form|negative-definite]], in the sense that <math>\int_{\R^n}\overline{\varphi(x)}\Delta\varphi(x)\,dx=-\int_{\R^n}|\nabla\varphi(x)|^2\,dx<0</math> for any [[Smoothness|smooth]] [[Support_(mathematics)#Compact_support|compactly supported]] function <math>\varphi(x)</math> which is not identically zero), and the "geometric", [[Definite_quadratic_form|positive-definite]] Laplacian defined by <math>\Delta=-\nabla^2=-\sum_{j=1}^n\Big(\frac{\partial}{\partial x_j}\Big)^2</math>.

== History ==

[[W.R. Hamilton]] defined "the square root of the Laplacian" in 1847<ref name="Hamilton1847">{{ cite journal
|author=Hamilton, William Rowan
|title=On quaternions; or on a new system of imaginaries in Algebra
|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
|volume=xxxi
|issue=208
|year=1847
|pages=278–283
|url=
|doi=10.1080/14786444708562643
}}
</ref> in his series of articles about [[quaternion|quaternions]]:
<blockquote>
<...> if we
introduce a new characteristic of operation, <math>\triangleleft</math>, defined with relation to these three symbols
<math>ijk,</math> and to the known operation of partial differentiation, performed with respect to three
independent but real variables <math>xyz,</math> as follows:
<math display="block">
\triangleleft=\frac{i\mathrm{d}}{\mathrm{d}x}+\frac{j\mathrm{d}}{\mathrm{d}y}+\frac{k\mathrm{d}}{\mathrm{d}z};
</math>
''this new characteristic'' <math>\triangleleft</math> ''will have the negative of its symbolic square expressed by the following formula'' :
<math display="block">
-\triangleleft^2=\Big(\frac{\mathrm{d}}{\mathrm{d}x}\Big)^2+\Big(\frac{\mathrm{d}}{\mathrm{d}y}\Big)^2+\Big(\frac{\mathrm{d}}{\mathrm{d}z}\Big)^2;
</math>
of which it is clear that the applications to analytical physics must be extensive in a high degree.
</blockquote>

== 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" />

== Generalisations ==
In Clifford analysis, the operator {{nowrap|''D'' : ''C''<sup>∞</sup>('''R'''<sup>''k''</sup> ⊗ '''R'''<sup>''n''</sup>, ''S'') → ''C''<sup>∞</sup>('''R'''<sup>''k''</sup> ⊗ '''R'''<sup>''n''</sup>, '''C'''<sup>''k''</sup> ⊗ ''S'')}} acting on spinor valued functions defined by 
:<math>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}</math>
is sometimes called Dirac operator in ''k'' Clifford variables. In the notation, ''S'' is the space of spinors, <math>x_i=(x_{i1},x_{i2},\ldots,x_{in})</math> are ''n''-dimensional variables and <math>\partial_{\underline{x_i}}=\sum_j e_j\cdot \partial_{x_{ij}}</math> is the Dirac operator in the ''i''-th variable. This is a common generalization of the Dirac operator ({{nowrap|1=''k'' = 1}}) and the [[Dolbeault cohomology|Dolbeault operator]] ({{nowrap|1=''n'' = 2}}, ''k'' arbitrary). It is an [[invariant differential operator]], invariant under the action of the group {{nowrap|SL(''k'') × Spin(''n'')}}. The [[injective_resolution#Injective_resolutions|resolution]] of ''D'' is known only in some special cases.

== See also ==
{{colbegin}}
* [[AKNS system|AKNS hierarchy]]
* [[Dirac equation]]
* [[Clifford algebra]]
* [[Clifford analysis]]
* [[connection (mathematics)|Connection]]
* [[Dolbeault operator]]
* [[Heat kernel]]
* [[Spinor bundle]]
{{colend}}

== References ==
{{reflist}}
{{refbegin}}
* {{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}}
* {{citation | last1=Colombo, F.|first1=I.| last2=Sabadini |first2=I. |author2-link=Irene Sabadini| title = Analysis of Dirac Systems and Computational Algebra| publisher=Birkhauser Verlag AG | year=2004|isbn=978-3-7643-4255-5}}
{{refend}}

[[Category:Differential operators]]
[[Category:Quantum operators]]
[[Category:Mathematical physics]]