Editing Four-vector (section)

===Four-vectors in spacetime algebra===

In [[spacetime algebra]], another example of Clifford algebra, the [[gamma matrices]] can also form a [[basis (linear algebra)|basis]]. (They are also called the Dirac matrices, owing to their appearance in the [[Dirac equation]]). There is more than one way to express the gamma matrices, detailed in that main article.

The [[Feynman slash notation]] is a shorthand for a four-vector '''A''' contracted with the gamma matrices:
<math display="block">\mathbf{A}\!\!\!\!/ = A_\alpha \gamma^\alpha = A_0 \gamma^0 + A_1 \gamma^1 + A_2 \gamma^2 + A_3 \gamma^3 </math>

The four-momentum contracted with the gamma matrices is an important case in [[relativistic quantum mechanics]] and [[relativistic quantum field theory]]. In the Dirac equation and other [[relativistic wave equation]]s, terms of the form:
<math display="block">\begin{align}
\mathbf{P}\!\!\!\!/ = P_\alpha \gamma^\alpha
&= P_0 \gamma^0 + P_1 \gamma^1 + P_2 \gamma^2 + P_3 \gamma^3 \\[4pt]
&= \dfrac{E}{c} \gamma^0 - p_x \gamma^1 - p_y \gamma^2 - p_z \gamma^3 \\
\end{align} </math>
appear, in which the energy {{mvar|E}} and momentum components {{math|(''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'')}} are replaced by their respective [[operator (physics)|operator]]s.