Editing Dirac operator (section)

=== 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>