Editing Clifford algebra (section)

=== Physics ===
Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra that has a basis that is generated by the matrices {{math|''γ''<sub>0</sub>, ..., ''γ''<sub>3</sub>}}, called [[Dirac matrices]], which have the property that
<math display="block">\gamma_i\gamma_j + \gamma_j\gamma_i = 2\eta_{ij} ,</math>
where {{math|''η''}} is the matrix of a quadratic form of signature {{math|(1, 3)}} (or {{math|(3, 1)}} corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra {{math|Cl{{su|b=1,3}}('''R''')}}, whose [[complexification]] is {{math|Cl{{su|b=1,3}}('''R''')<sub>'''C'''</sub>}}, which, by the [[classification of Clifford algebras]], is isomorphic to the algebra of {{math|4 × 4}} complex matrices {{math|Cl{{sub|4}}('''C''') ≈ M<sub>4</sub>('''C''')}}. However, it is best to retain the notation {{math|Cl{{su|b=1,3}}('''R''')<sub>'''C'''</sub>}}, since any transformation that takes the bilinear form to the canonical form is ''not'' a Lorentz transformation of the underlying spacetime.

The Clifford algebra of spacetime used in physics thus has more structure than {{math|Cl<sub>4</sub>('''C''')}}. It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebra {{math|'''so'''(1, 3)}} sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by
<math display="block">\begin{align}
  \sigma^{\mu\nu}
    &= -\frac{i}{4}\left[\gamma^\mu,\, \gamma^\nu\right], \\
  \left[\sigma^{\mu\nu},\, \sigma^{\rho\tau}\right]
    &= i\left(\eta^{\tau\mu}\sigma^{\rho\nu} + \eta^{\nu\tau}\sigma^{\mu\rho} - \eta^{\rho\mu}\sigma^{\tau\nu} - \eta^{\nu\rho} \sigma^{\mu\tau}\right).
\end{align}</math>

This is in the {{math|(3, 1)}} convention, hence fits in {{math|Cl{{su|b=3,1}}('''R''')<sub>'''C'''</sub>}}.{{sfn|Weinberg|2002|ps=none}}

The Dirac matrices were first written down by [[Paul Dirac]] when he was trying to write a relativistic first-order wave equation for the [[electron]], and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the [[Dirac equation]] and introduce the [[Dirac operator]]. The entire Clifford algebra shows up in [[quantum field theory]] in the form of [[Dirac field bilinear]]s.

The use of Clifford algebras to describe quantum theory has been advanced among others by [[Mario Schönberg]],{{efn|See the references to Schönberg's papers of 1956 and 1957 as described in section "The Grassmann–Schönberg algebra {{math|''G''{{sub|''n''}}}}" of {{harvnb|Bolivar|2001}}}} by [[David Hestenes]] in terms of [[geometric calculus]], by [[David Bohm]] and [[Basil Hiley]] and co-workers in form of a [[Basil Hiley#Hierarchy of Clifford algebras|hierarchy of Clifford algebras]], and by Elio Conte et al.{{sfn|Conte|2007|ps=none}}{{sfn|Conte|2012|ps=none}}