# Contents

## Physics

The first relativistic Schroedinger 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. 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).

## Mathematics

The tangent bundle of an oriented Riemannian $n$-dimensional manifold $M$ is an $\mathrm{SO}\left(n\right)$-bundle. Orientation means that the first Stiefel-Whitney class ${w}_{1}\left(M\right)$ is zero. If ${w}_{2}\left(M\right)$ is zero than the $\mathrm{SO}\left(n\right)$ bundle can be lifted to a $\mathrm{Spin}\left(n\right)$-bundle. A choice of connection on such a $\mathrm{Spin}\left(n\right)$-bundle is a $\mathrm{Spin}$-structure on $M$. There is a standard $n/2$-dimensional representation of $\mathrm{Spin}\left(n\right)$-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 $\mathrm{Spin}\left(n\right)$ bundle $P$ with respect to these representations. Thus we get the spinor bundles ${E}_{±}:=P{×}_{\mathrm{Spin}\left(n\right)}{S}_{±}\to M$ 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
${\gamma }_{5}={i}^{n\left(n+1\right)/2}{\gamma }_{1}\cdots {\gamma }_{n},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\gamma }_{5}^{2}=I$\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 $\Gamma \left(E\right)\to \Gamma \left({E}_{-}\right)$; there are several versions, in mathematics is pretty important the chiral Dirac operator

$\Gamma \left(M,{E}_{+}\right)\to \Gamma \left(M,{E}_{-}\right)$\Gamma(M,E_+)\to \Gamma(M,E_-)

given by local formula

$\sum _{a}{\gamma }^{a}{e}_{a}^{\mu }\left(x\right){\nabla }_{\mu }\frac{1+{\gamma }_{5}}{2}$\sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}

where ${e}_{a}^{\mu }\left(x\right)$ 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.

## 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.

Revised on November 7, 2012 19:47:38 by Urs Schreiber (82.169.65.155)