Dirac operator

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This article has been nominated for deletion. See Conservapedia:AFD Dirac_operator

Let $M 2 n$ denote an oriented Riemannian manifold. Then the structure group of $M$ reduces to $S O (2 n)$ , and the associated transition functions of $M$ give rise to a principle $S O (2 n)$ bundle. Given any representation, $\rho: SO(2n)\rightarrow GL(V)$ , there is an associated vectorbundle $P_{SO(2n)}\times_{\rho}V$ over $M$ . In fact, a tensor is nothing more than a section of a vectorbundle constructed in this fashion.

Now $S O (2 n)$ has a simply connected double cover, which we denote by $S p i n (2 n)$ . Thus, on contractible coordinate patches, we can always lift the transition functions $g_{\alpha\beta}:U_{\alpha\beta}\rightarrow SO(2n)$ to maps $\tilde{g}_{\alpha\beta}:U_{\alpha\beta}\rightarrow Spin(2n)$ . However, there is an ambiguity in the choice of the lift given by which covering sheet some initial point is sent to. Thus, the lifted transition maps may no longer satisfy the cocycle condition, and therefore do not give rise necessarily to a principle $S p i n (2 n)$ bundle. In fact, the obstruction to being able to lift $g αβ$ so that $\tilde{g}_{\alpha\beta}$ satisfy the cocycle condition is given by a class $w_2 \in H^2(M,\mathbb{Z}_2)$ called the second Stiefel-Whitney class.

It is not hard to see why the obstruction to lifting $P S O (2 n)$ to a principle spin bundle is given by such a class. Consider the following exact sequence of sheaves:

$0\to \mathbb{Z}_2 \to Spin(2n) \to SO(2n) \to 0$

From the cohomology long exact sequence we get:

$H^1(M,Spin(2n))\rightarrow H^1(M,SO(2n))\rightarrow H^2(M,\mathbb{Z}_2)$

This says that the principle $S O (2 n)$ bundle of $M$ comes from a principle spin bundle if and only if a certain class associated to the $S O (2 n)$ bundle vanishes. When this happens, we say that $M$ is a spin manifold.

The spin group $S p i n (2 n)$ has two irreducible complex representations $\Delta^{\pm}$ . A spinor is a section of the vectorbundle constructed out of these representations via $P S p i n (2 n)$ . Let $\Delta = \Delta^+\oplus\Delta^-$ , viewed now as vectorbundles. There is a canonical pairing $\Delta\otimes TM\rightarrow \Delta$ given by Clifford multiplication. Moreover, by pulling back the Levi-Civita connection from $P S O (2 n)$ to $P S p i n (2 n)$ , $Δ$ inherits a canonical connection. For $ψ$ a section of $Δ$ (i.e., a spinor), the Dirac operator acting on $ψ$ is given by the formula:

$D\psi = \sum e_i\cdot \nabla_{e_i}\psi$

where $e i$ form an oriented orthonormal basis for the tangent space of $M$ . Note that this definition is invariant of the choice of oriented orthonormal basis.

Dirac operator

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