Dirac operator

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

: $D^2=\backslash Delta,\; \backslash ,$

where ∆ is the (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 $\backslash R^n$ as $\backslash Delta=\backslash nabla^2=\backslash sum\_\{j=1\}^n\backslash Big(\backslash frac\{\backslash partial\}\{\backslash partial\; x\_j\}\backslash Big)^2$ (which is negative-definite, in the sense that for any smooth compactly supported function $\backslash varphi(x)$ which is not identically zero), and the "geometric", positive-definite Laplacian defined by $\backslash Delta=-\backslash nabla^2=-\backslash sum\_\{j=1\}^n\backslash Big(\backslash frac\{\backslash partial\}\{\backslash partial\; x\_j\}\backslash Big)^2$.

W.R. Hamilton defined "the square root of the Laplacian" in 1847{{ 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

}}

in his series of articles about quaternions:

<...> if we

introduce a new characteristic of operation, $\backslash triangleleft$, defined with relation to these three symbols

$ijk,$ and to the known operation of partial differentiation, performed with respect to three

independent but real variables $xyz,$ as follows:

$$$$

\triangleleft=\frac{i\mathrm{d}}{\mathrm{d}x}+\frac{j\mathrm{d}}{\mathrm{d}y}+\frac{k\mathrm{d}}{\mathrm{d}z};

this new characteristic $\backslash triangleleft$ will have the negative of its symbolic square expressed by the following formula :

$$$$

-\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;

of which it is clear that the applications to analytical physics must be extensive in a high degree.

D = −i ∂_x is a Dirac operator on the tangent bundle over a line.

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 : R² → C²}}

: $\backslash psi(x,y)\; =\; \backslash begin\{bmatrix\}\backslash chi(x,y)\; \backslash \backslash \; \backslash eta(x,y)\backslash end\{bmatrix\}$

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

: $D=-i\backslash sigma\_x\backslash partial\_x-i\backslash sigma\_y\backslash partial\_y\; ,$

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.{{SpringerEOM|id=Spinor_structure&oldid=33893 |title=Spinor structure }}

Feynman's Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written

: $D=\backslash gamma^\backslash mu\backslash partial\_\backslash mu\backslash \; \backslash equiv\; \backslash partial\backslash !\backslash !\backslash !/,$

using the Feynman slash notation. In introductory textbooks to quantum field theory, this will appear in the form

:$D\; =\; c\backslash vec\backslash alpha\; \backslash cdot\; (-i\backslash hbar\backslash nabla\_x)\; +\; mc^2\backslash beta$

where $\backslash vec\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash alpha\_2,\; \backslash alpha\_3)$ are the off-diagonal Dirac matrices $\backslash alpha\_i=\backslash beta\backslash gamma\_i$, with $\backslash beta=\backslash gamma\_0$ and the remaining constants are $c$ the speed of light, $\backslash hbar$ being the Planck constant, and $m$ the mass of a fermion (for example, an electron). It acts on a four-component wave function $\backslash psi(x)\; \backslash in\; L^2(\backslash mathbb\{R\}^3,\; \backslash mathbb\{C\}^4)$, 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 $D^2=\backslash Delta+m^2$ (after setting $\backslash hbar=c=1.$)

Another Dirac operator arises in Clifford analysis. In euclidean n-space this is

: $D=\backslash sum\_\{j=1\}^\{n\}e\_\{j\}\backslash frac\{\backslash partial\}\{\backslash partial\; x\_\{j\}\}$

where {e_j: 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.

For a spin manifold, M, the Atiyah–Singer–Dirac operator is locally defined as follows: For {{nowrap|x ∈ M}} and e₁(x), ..., e_j(x) a local orthonormal basis for the tangent space of M at x, the Atiyah–Singer–Dirac operator is

:$D=\backslash sum\_\{j=1\}^\{n\}e\_\{j\}(x)\backslash tilde\{\backslash Gamma\}\_\{e\_\{j\}(x)\}\; ,$

where $\backslash tilde\{\backslash Gamma\}$ 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 $D^2=\backslash Delta+R/4$ where $R$ is the scalar curvature of the connection. Jurgen Jost, (2002) "Riemannian Geometry ang Geometric Analysis (3rd edition)", Springer. See section 3.4 pages 142 ff.

On Riemannian manifold $(M,\; g)$ of dimension $n=dim(M)$ with Levi-Civita connection $\backslash nabla$and an orthonormal basis $\backslash \{e\_\{a\}\backslash \}\_\{a=1\}^\{n\}$, we can define exterior derivative $d$ and coderivative $\backslash delta$ as

: $d=\; e^\{a\}\backslash wedge\; \backslash nabla\_\{e\_\{a\}\},\; \backslash quad\; \backslash delta\; =e^\{a\}\; \backslash lrcorner\; \backslash nabla\_\{e\_\{a\}\}$.

Then we can define a Dirac-Kähler operator{{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}}{{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}}{{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}} $D$, as follows

: $D\; =\; e^\{a\}\backslash nabla\_\{e\_\{a\}\}=d-\backslash delta$.

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.

In Clifford analysis, the operator {{nowrap|D : C^∞(R^k ⊗ Rⁿ, S) → C^∞(R^k ⊗ Rⁿ, C^k ⊗ S)}} acting on spinor valued functions defined by

:$f(x\_1,\backslash ldots,x\_k)\backslash mapsto$

\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, $x\_i=(x\_\{i1\},x\_\{i2\},\backslash ldots,x\_\{in\})$ are n-dimensional variables and $\backslash partial\_\{\backslash underline\{x\_i\}\}=\backslash sum\_j\; e\_j\backslash cdot\; \backslash partial\_\{x\_\{ij\}\}$ 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 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 resolution of D is known only in some special cases.

{{colbegin}}

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

{{colend}}

{{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