nLab
Dirac operator

Idea

For $S\to X$ a spinor bundle over a Riemannian manifold $(X,g)$, a Dirac operator on $S$ is an differential operator on (sections of) $S$ whose principal symbol is that of $c\circ d$, where $d$ is the exterior derivative and $c$ is the symbol map.

More abstractly, for $D$ a Dirac operator, its normalization $D(1+D^2{)}^{-1/2}$ is a Fredholm operator, hence defines an element in K-homology.

Origin and role in Physics

The first relativistic Schrödinger type equation found was Klein-Gordon. At first it did not look that K-G equation could be interpreted physically because of negative energy states and other paradoxes. Paul Dirac proposed to take a square root of Laplace operator within the matrix-valued differential operators and obtained a Dirac equation; matrix valued generators involved representations of a Clifford algebra. It also had negative energy solutions, but with half-integer spin interpretation which was appropriate the Pauli exclusion principle together with the Dirac sea picture came at rescue (Klein-Gordon is now also useful with more modern formalisms).

(…)

Definition

In components

The tangent bundle of an oriented Riemannian $n$-dimensional manifold $M$ is an $\mathrm{SO}(n)$-bundle. Orientation means that the first Stiefel-Whitney class $w_1(M)$ is zero. If $w_2(M)$ is zero than the $\mathrm{SO}(n)$ bundle can be lifted to a $\mathrm{Spin}(n)$-bundle. A choice of connection on such a $\mathrm{Spin}(n)$-bundle is a $\mathrm{Spin}$-structure on $M$. There is a standard $n/2$-dimensional representation of $\mathrm{Spin}(n)$-group, so called Spin representation, which is depending, if $n$ is odd irreducible, and if $n$ is even it decomposes into the sum of two irreducible representations of equal dimension $S_{+}$ and $S_{-}$. Thus we can associate associated bundles to the original $\mathrm{Spin}(n)$ bundle $P$ with respect to these representations. Thus we get the spinor bundles $E_{\pm }:=P{\times }_{\mathrm{Spin}(n)}{S}_{\pm }\to M$ and $E=E_{+}\oplus E_{-}$.

Gamma matrices, which are the representations of the Clifford algebra

$${\gamma }_a{\gamma }_b+{\gamma }_b{\gamma }_a=-2{\delta }_{\mathrm{ab}}I$$\gamma_a \gamma_b + \gamma_b \gamma_a = -2\delta_{ab} I

$${\gamma }_5=i^{n(n+1)/2}{\gamma }_1\cdots {\gamma }_n,{\gamma }_5^2=I$$\gamma_5 = i^{n(n+1)/2}\gamma_1\cdots\gamma_n, \,\,\,\,\gamma^2_5 = I

thus act on such a space; certain combinations of products of gamma matrices with partial derivatives define a first order Dirac operator $\Gamma (E)\to \Gamma (E_{-})$; there are several versions, in mathematics is pretty important the chiral Dirac operator

$$\Gamma (M,E_{+})\to \Gamma (M,E_{-})$$\Gamma(M,E_+)\to \Gamma(M,E_-)

given by local formula

$$\sum _a{\gamma }^a{e}_a^{\mu }(x){\nabla }_{\mu }\frac{1+{\gamma }_5}{2}$$\sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}

where $e_a^{\mu }(x)$ are orthonormal frames of tangent vectors and ${\nabla }_{\mu }$ is the covariant derivative with respect to the Levi-Civita spin connection. The expression $\frac{1+{\gamma }_5}{2}$ is the chirality operator.

In Euclidean space the Dirac operator is elliptic, but not in Minkowski space.

The Dirac operator is involved in approaches to the Atiyah-Singer index theorem about the index of an elliptic operator: namely the index can be easier calculated for Dirac operator and the deformation to the Dirac operator does not change the index. An appropriate version of a Dirac operator is a part of a concept of the spectral triple in noncommutative geometry a la Alain Connes.

Examples

• Spin^c Dirac operator

• Kähler-Dirac operator

Related concepts

• index of a Dirac operator

• Fredholm module, K-homology, KK-theory

References

• C. Nash, Differential topology and quantum field theory, Acad. Press 1991.

• H. Blaine Lawson Jr. , Marie-Louise Michelson, Spin geometry, Princeton Univ. Press 1989.

• Dan Freed, Geometry of Dirac operators (pdf)

• Michael Atiyah, Raoul Bott, V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330.

• N. Berline, Ezra Getzler, M. Vergne, Heat kernels and Dirac operators, Grundlehren 298, Springer 1992, “Text Edition” 2003.

• Eckhard Meinrenken, Clifford algebras and Lie groups, Lecture Notes, University of Toronto, Fall 2009.

• Jing-Song Huang, Pavle Pandžić, J.-S. Huang, P. Pandzic, Dirac Operators in Representation Theory,. Birkhäuser, Boston, 2006, 199 pages; short version Dirac operators in representation theory, 48 pp. pdf

• J.-S. Huang, Pavle Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185—202.

• R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1-30.