Editing Four-vector (section)

==Other formulations==

===Four-vectors in the algebra of physical space===

A four-vector ''A'' can also be defined in using the [[Pauli matrices]] as a [[basis (linear algebra)|basis]], again in various equivalent notations:<ref>{{cite book |pages= 1142–1143|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne | title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref>
<math display="block"> \begin{align}  
  \mathbf{A} & = \left(A^0, \, A^1, \, A^2, \, A^3\right) \\
  & = A^0\boldsymbol{\sigma}_0 + A^1 \boldsymbol{\sigma}_1 + A^2 \boldsymbol{\sigma}_2 + A^3 \boldsymbol{\sigma}_3 \\
  & = A^0\boldsymbol{\sigma}_0 + A^i \boldsymbol{\sigma}_i \\
  & = A^\alpha\boldsymbol{\sigma}_\alpha\\
\end{align}</math>
or explicitly:
<math display="block">\begin{align}  
  \mathbf{A} & = A^0\begin{pmatrix} 1 &  0 \\ 0 &  1 \end{pmatrix} +
                 A^1\begin{pmatrix} 0 &  1 \\ 1 &  0 \end{pmatrix} +
                 A^2\begin{pmatrix} 0 & -i \\ i &  0 \end{pmatrix} +
                 A^3\begin{pmatrix} 1 &  0 \\ 0 & -1 \end{pmatrix} \\
             & = \begin{pmatrix}
                   A^0 +   A^3 & A^1 - i A^2 \\
                   A^1 + i A^2 & A^0 -   A^3
                 \end{pmatrix}
\end{align}</math>
and in this formulation, the four-vector is represented as a [[Hermitian matrix]] (the [[matrix transpose]] and [[complex conjugate]] of the matrix leaves it unchanged), rather than a real-valued column or row vector. The [[determinant]] of the matrix is the modulus of the four-vector, so the determinant is an invariant:
<math display="block"> \begin{align}  
  |\mathbf{A}|
  & = \begin{vmatrix}
          A^0 +   A^3 & A^1 - i A^2 \\
          A^1 + i A^2 & A^0 -   A^3
      \end{vmatrix} \\[1ex]
  & = \left(A^0 + A^3\right)\left(A^0 - A^3\right) - \left(A^1 -i A^2\right)\left(A^1 + i A^2\right) \\[1ex]
  & = \left(A^0\right)^2 - \left(A^1\right)^2 - \left(A^2\right)^2 - \left(A^3\right)^2
\end{align}</math>

This idea of using the Pauli matrices as [[basis vector]]s is employed in the [[algebra of physical space]], an example of a [[Clifford algebra]].

===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.