Editing Pauli matrices (section)

=== Classical mechanics ===
{{Main|Quaternions and spatial rotation}}

In [[classical mechanics]], Pauli matrices are useful in the context of the Cayley-Klein parameters.<ref name=Goldstein-1959>
{{cite book
 |last=Goldstein |first=Herbert
 |year=1959
 |title=Classical Mechanics
 |pages=109–118
 |publisher=Addison-Wesley
}}
</ref> The matrix {{mvar|P}} corresponding to the position <math>\vec{x}</math> of a point in space is defined in terms of the above Pauli vector matrix,
:<math>P = \vec{x} \cdot \vec{\sigma} = x\,\sigma_x + y\,\sigma_y + z\,\sigma_z .</math>

Consequently, the transformation matrix {{math|''Q{{sub|θ}}''}} for rotations about the {{mvar|x}}-axis through an angle {{mvar|θ}} may be written in terms of Pauli matrices and the unit matrix as<ref name=Goldstein-1959/>
:<math>Q_\theta = \boldsymbol{1}\,\cos\frac{\theta}{2} + i\,\sigma_x \sin\frac{\theta}{2} .</math>

Similar expressions follow for general Pauli vector rotations as detailed above.