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The Dirac algebra arises from Dirac's solution [[#References|[a3]]] to the relativistic electron equation:
 
The Dirac algebra arises from Dirac's solution [[#References|[a3]]] to the relativistic electron equation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301101.png" /></td> </tr></table>
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\begin{equation*} H ^ { 2 } = ( \mathbf{p} _ { x } ^ { 2 } + \mathbf{p} _ { y } ^ { 2 } + \mathbf{p} _ { z } ^ { 2 } ) c ^ { 2 } + m _ { 0 } ^ { 2 } c ^ { 4 }. \end{equation*}
  
Dirac found hypercomplex elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301105.png" /> (cf. also [[Hypercomplex number|Hypercomplex number]]) such that
+
Dirac found hypercomplex elements $\alpha _ { x }$, $\alpha_y$, $\alpha _ { z }$ and $\beta$ (cf. also [[Hypercomplex number|Hypercomplex number]]) such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301106.png" /></td> </tr></table>
+
\begin{equation*} ( {\bf p} _ { x } ^ { 2 } + {\bf p} _ { y } ^ { 2 } + {\bf p} _ { z } ^ { 2 } ) + m _ { 0 } ^ { 2 } c ^ { 2 } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301107.png" /></td> </tr></table>
+
\begin{equation*} = ( \alpha _ { x } \mathbf{p} _ { x } + \alpha _ { y } \mathbf{p}_ y + \alpha _ { z } \mathbf{p} _ { z } + \beta m _ { 0 } c ) ^ { 2 }. \end{equation*}
  
 
The Hamiltonian is given, after the usual substitutions for the linear momentum components, by
 
The Hamiltonian is given, after the usual substitutions for the linear momentum components, by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301108.png" /></td> </tr></table>
+
\begin{equation*} H = c \frac { \hbar } { i } \overset{\rightharpoonup} { \alpha } . \overset{\rightharpoonup} { \nabla } + \overset{\rightharpoonup} { \beta } m _ { 0 } c ^ { 2 }. \end{equation*}
  
 
The time-dependent equation is known as Dirac's equation:
 
The time-dependent equation is known as Dirac's equation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d1301109.png" /></td> </tr></table>
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\begin{equation*} \left( c \frac { \hbar } { c } \vec { \alpha } \cdot \vec { \nabla } + \vec { \beta } m _{0} c ^ { 2 } \right) \Phi = i \hbar \frac { \partial \Phi } { \partial t }. \end{equation*}
  
The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011013.png" /> satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011014.png" /> and the anti-commutativity relations:
+
The elements $\alpha _ { x }$, $\alpha_y$, $\alpha _ { z }$ and $\overset{\rightharpoonup} { \beta }$ satisfy $\alpha _ { x } ^ { 2 } = \alpha _ { y } ^ { 2 } = \alpha _ { z } ^ { 2 } = \beta ^ { 2 } = 1$ and the anti-commutativity relations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011015.png" /></td> </tr></table>
+
\begin{equation*} \left\{ \begin{array} { l l } { \alpha _ { i } \alpha _ { j } + \alpha _ { j } \alpha _ { i } = 0 } &amp; { \text { for } i, j \in \{ x , y , z \} , i \neq j, } \\ { \alpha _ { i } \beta + \beta \alpha _ { i } = 0 } &amp; { \text { for } i , j \in \{ x , y , z \}. } \end{array} \right. \end{equation*}
  
The Dirac representation of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011017.png" /> is
+
The Dirac representation of the matrices $\alpha$ and $\beta$ is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011018.png" /></td> </tr></table>
+
\begin{equation*} \alpha _ { X } = \left( \begin{array} { l l l l } { 0 } &amp; { 0 } &amp; { 0 } &amp; { 1 } \\ { 0 } &amp; { 0 } &amp; { 1 } &amp; { 0 } \\ { 0 } &amp; { 1 } &amp; { 0 } &amp; { 0 } \\ { 1 } &amp; { 0 } &amp; { 0 } &amp; { 0 } \end{array} \right) = \left( \begin{array} { l l } {\bf 0 } &amp; { \sigma _ { x } } \\ { \sigma _ { x } } &amp; \bf{ 0 } \end{array} \right), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011019.png" /></td> </tr></table>
+
\begin{equation*} \alpha_{y} = \left( \begin{array} { c c c c } { 0 } &amp; { 0 } &amp; { 0 } &amp; { - i } \\ { 0 } &amp; { 0 } &amp; { i } &amp; { 0 } \\ { 0 } &amp; { - i } &amp; { 0 } &amp; { 0 } \\ { i } &amp; { 0 } &amp; { 0 } &amp; { 0 } \end{array} \right) = \left( \begin{array} { c c } { \mathbf{0} } &amp; { \sigma_ y } \\ { \sigma_ y } &amp; { \mathbf{0}  } \end{array} \right) , \alpha _ { z } = \left( \begin{array} { c c c c } { 0 } &amp; { 0 } &amp; { 1 } &amp; { 0 } \\ { 0 } &amp; { 0 } &amp; { 0 } &amp; { - 1 } \\ { 1 } &amp; { 0 } &amp; { 0 } &amp; { 0 } \\ { 0 } &amp; { - 1 } &amp; { 0 } &amp; { 0 } \end{array} \right) = \left( \begin{array} { c c } { \mathbf{0}  } &amp; { \sigma _ { z } } \\ { \sigma _ { z } } &amp; { \mathbf{0}  } \end{array} \right), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011022.png" /> are the Pauli spin matrices (cf. also [[Pauli matrices|Pauli matrices]]; [[Dirac matrices|Dirac matrices]]). This choice is not unique; pre-multiplying by any unitary matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011023.png" /> and post-multiplying by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011024.png" /> will produce a new set of matrices satisfying the conditions. The defining relations are often expressed more abstractly by the Dirac gamma matrices
+
where $\sigma _ { x }$, $\sigma_y$ and $\sigma _ { z }$ are the Pauli spin matrices (cf. also [[Pauli matrices|Pauli matrices]]; [[Dirac matrices|Dirac matrices]]). This choice is not unique; pre-multiplying by any unitary matrix $S$ and post-multiplying by $S ^ { - 1 }$ will produce a new set of matrices satisfying the conditions. The defining relations are often expressed more abstractly by the Dirac gamma matrices
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011025.png" /></td> </tr></table>
+
\begin{equation*} \gamma _ { i } ^ { 2 } = 1 , i = 1,2,3,4, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011026.png" /></td> </tr></table>
+
\begin{equation*} \gamma _ { i } \gamma _ { j } + \gamma _ { j } \gamma _ { i } = 0 , i \neq j , i , j = 1,2,3,4. \end{equation*}
  
The Dirac algebra is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011027.png" />-dimensional complex [[Clifford algebra|Clifford algebra]] generated by the gamma matrices under the usual matrix operations and is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011028.png" />, the ring of four-by-four matrices over the complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011029.png" />. The use of the complex numbers as scalars apparently is motivated by the fact that complex numbers are use to express solutions to the Schrödinger wave equations (cf. also [[Schrödinger equation|Schrödinger equation]]).
+
The Dirac algebra is the $2 ^ { 4 }$-dimensional complex [[Clifford algebra|Clifford algebra]] generated by the gamma matrices under the usual matrix operations and is isomorphic to ${\bf C} ( 4 )$, the ring of four-by-four matrices over the complex numbers $\mathbf{C}$. The use of the complex numbers as scalars apparently is motivated by the fact that complex numbers are use to express solutions to the Schrödinger wave equations (cf. also [[Schrödinger equation|Schrödinger equation]]).
  
Two other  "Dirac algebras"  commonly appear in the literature [[#References|[a4]]]; in each the relation among the squares of the generating elements, the metric, has been modified and the scalar field is the real numbers. Upon complexification, both become the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011030.png" />. The modified metrics are the metrics of Minkowski space-time (cf. also [[Minkowski space|Minkowski space]]) and more easily illustrate the physics or the geometry.
+
Two other  "Dirac algebras"  commonly appear in the literature [[#References|[a4]]]; in each the relation among the squares of the generating elements, the metric, has been modified and the scalar field is the real numbers. Upon complexification, both become the algebra ${\bf C} ( 4 )$. The modified metrics are the metrics of Minkowski space-time (cf. also [[Minkowski space|Minkowski space]]) and more easily illustrate the physics or the geometry.
  
 
In one case (see, for example [[#References|[a2]]]) the metric is given by
 
In one case (see, for example [[#References|[a2]]]) the metric is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011031.png" /></td> </tr></table>
+
\begin{equation*} \gamma _ { 1 } ^ { 2 } = 1 , \gamma _ { 2 } ^ { 2 } = \gamma _ { 3 } ^ { 2 } = \gamma _ { 4 } ^ { 2 } = - 1, \end{equation*}
  
the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011032.png" /> generate a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011033.png" />-dimensional real Clifford algebra that is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011034.png" />, the ring of two-by-two matrices over the real quaternion division ring. Every Clifford algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011035.png" /> admits a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011036.png" />-grading,
+
the $\gamma _ { i }$ generate a $2 ^ { 4 }$-dimensional real Clifford algebra that is isomorphic to $H ( 2 )$, the ring of two-by-two matrices over the real quaternion division ring. Every Clifford algebra $C$ admits a $\mathbf{Z}_{2}$-grading,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011037.png" /></td> </tr></table>
+
\begin{equation*} C = C _ { 0 } \oplus C _ { 1 }, \end{equation*}
  
such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011038.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011041.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011042.png" />, spanned by the identity element and all products of an even number, is, in the present case, a subalgebra isomorphic to to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011043.png" />-dimensional Pauli algebra.
+
such that for all $r_i$ $\in C_i$ and $s _ { j } \in C _ { j }$, $r _ { i } s _ { j } \in C _ {  ( i + j ) \operatorname { mod } 2}$. The subspace $C _ { 0 }$, spanned by the identity element and all products of an even number, is, in the present case, a subalgebra isomorphic to to the $2 ^ { 3 }$-dimensional Pauli algebra.
  
The Majorana representation is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011044.png" />-dimensional real Clifford algebra with metric (as in [[#References|[a1]]])
+
The Majorana representation is the $2 ^ { 4 }$-dimensional real Clifford algebra with metric (as in [[#References|[a1]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011045.png" /></td> </tr></table>
+
\begin{equation*} \gamma _ { 1 } ^ { 2 } = - 1 , \gamma _ { 2 } ^ { 2 } = \gamma _ { 3 } ^ { 2 } = \gamma _ { 4 } ^ { 2 } = 1, \end{equation*}
  
that is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011046.png" />. Here, the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011049.png" /> generate a subalgebra isomorphic to to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130110/d13011050.png" />-dimensional Pauli algebra.
+
that is isomorphic to $\operatorname { Re } ( 4 )$. Here, the elements $\gamma_2$, $\gamma_3$ and $\gamma_4$ generate a subalgebra isomorphic to to the $2 ^ { 3 }$-dimensional Pauli algebra.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Cartan,  "The theory of spinors" , Dover  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.M. Corson,  "Introduction to tensors, spinors, and relativistic wave-equations" , Chelsea  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.A.M. Dirac,  "The quantum theory of the electron"  ''Proc. Royal Soc. London'' , '''A117'''  (1928)  pp. 610–624</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N.A. Salingaros,  G.P. Wene,  "The Clifford algebra of differential forms"  ''Acta Applic. Math.'' , '''4'''  (1985)  pp. 271–191</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  E. Cartan,  "The theory of spinors" , Dover  (1966)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  E.M. Corson,  "Introduction to tensors, spinors, and relativistic wave-equations" , Chelsea  (1953)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P.A.M. Dirac,  "The quantum theory of the electron"  ''Proc. Royal Soc. London'' , '''A117'''  (1928)  pp. 610–624</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N.A. Salingaros,  G.P. Wene,  "The Clifford algebra of differential forms"  ''Acta Applic. Math.'' , '''4'''  (1985)  pp. 271–191</td></tr></table>

Revision as of 17:01, 1 July 2020

The Dirac algebra arises from Dirac's solution [a3] to the relativistic electron equation:

\begin{equation*} H ^ { 2 } = ( \mathbf{p} _ { x } ^ { 2 } + \mathbf{p} _ { y } ^ { 2 } + \mathbf{p} _ { z } ^ { 2 } ) c ^ { 2 } + m _ { 0 } ^ { 2 } c ^ { 4 }. \end{equation*}

Dirac found hypercomplex elements $\alpha _ { x }$, $\alpha_y$, $\alpha _ { z }$ and $\beta$ (cf. also Hypercomplex number) such that

\begin{equation*} ( {\bf p} _ { x } ^ { 2 } + {\bf p} _ { y } ^ { 2 } + {\bf p} _ { z } ^ { 2 } ) + m _ { 0 } ^ { 2 } c ^ { 2 } = \end{equation*}

\begin{equation*} = ( \alpha _ { x } \mathbf{p} _ { x } + \alpha _ { y } \mathbf{p}_ y + \alpha _ { z } \mathbf{p} _ { z } + \beta m _ { 0 } c ) ^ { 2 }. \end{equation*}

The Hamiltonian is given, after the usual substitutions for the linear momentum components, by

\begin{equation*} H = c \frac { \hbar } { i } \overset{\rightharpoonup} { \alpha } . \overset{\rightharpoonup} { \nabla } + \overset{\rightharpoonup} { \beta } m _ { 0 } c ^ { 2 }. \end{equation*}

The time-dependent equation is known as Dirac's equation:

\begin{equation*} \left( c \frac { \hbar } { c } \vec { \alpha } \cdot \vec { \nabla } + \vec { \beta } m _{0} c ^ { 2 } \right) \Phi = i \hbar \frac { \partial \Phi } { \partial t }. \end{equation*}

The elements $\alpha _ { x }$, $\alpha_y$, $\alpha _ { z }$ and $\overset{\rightharpoonup} { \beta }$ satisfy $\alpha _ { x } ^ { 2 } = \alpha _ { y } ^ { 2 } = \alpha _ { z } ^ { 2 } = \beta ^ { 2 } = 1$ and the anti-commutativity relations:

\begin{equation*} \left\{ \begin{array} { l l } { \alpha _ { i } \alpha _ { j } + \alpha _ { j } \alpha _ { i } = 0 } & { \text { for } i, j \in \{ x , y , z \} , i \neq j, } \\ { \alpha _ { i } \beta + \beta \alpha _ { i } = 0 } & { \text { for } i , j \in \{ x , y , z \}. } \end{array} \right. \end{equation*}

The Dirac representation of the matrices $\alpha$ and $\beta$ is

\begin{equation*} \alpha _ { X } = \left( \begin{array} { l l l l } { 0 } & { 0 } & { 0 } & { 1 } \\ { 0 } & { 0 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 0 } & { 0 } \end{array} \right) = \left( \begin{array} { l l } {\bf 0 } & { \sigma _ { x } } \\ { \sigma _ { x } } & \bf{ 0 } \end{array} \right), \end{equation*}

\begin{equation*} \alpha_{y} = \left( \begin{array} { c c c c } { 0 } & { 0 } & { 0 } & { - i } \\ { 0 } & { 0 } & { i } & { 0 } \\ { 0 } & { - i } & { 0 } & { 0 } \\ { i } & { 0 } & { 0 } & { 0 } \end{array} \right) = \left( \begin{array} { c c } { \mathbf{0} } & { \sigma_ y } \\ { \sigma_ y } & { \mathbf{0} } \end{array} \right) , \alpha _ { z } = \left( \begin{array} { c c c c } { 0 } & { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } \\ { 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 0 } & { 0 } \end{array} \right) = \left( \begin{array} { c c } { \mathbf{0} } & { \sigma _ { z } } \\ { \sigma _ { z } } & { \mathbf{0} } \end{array} \right), \end{equation*}

where $\sigma _ { x }$, $\sigma_y$ and $\sigma _ { z }$ are the Pauli spin matrices (cf. also Pauli matrices; Dirac matrices). This choice is not unique; pre-multiplying by any unitary matrix $S$ and post-multiplying by $S ^ { - 1 }$ will produce a new set of matrices satisfying the conditions. The defining relations are often expressed more abstractly by the Dirac gamma matrices

\begin{equation*} \gamma _ { i } ^ { 2 } = 1 , i = 1,2,3,4, \end{equation*}

\begin{equation*} \gamma _ { i } \gamma _ { j } + \gamma _ { j } \gamma _ { i } = 0 , i \neq j , i , j = 1,2,3,4. \end{equation*}

The Dirac algebra is the $2 ^ { 4 }$-dimensional complex Clifford algebra generated by the gamma matrices under the usual matrix operations and is isomorphic to ${\bf C} ( 4 )$, the ring of four-by-four matrices over the complex numbers $\mathbf{C}$. The use of the complex numbers as scalars apparently is motivated by the fact that complex numbers are use to express solutions to the Schrödinger wave equations (cf. also Schrödinger equation).

Two other "Dirac algebras" commonly appear in the literature [a4]; in each the relation among the squares of the generating elements, the metric, has been modified and the scalar field is the real numbers. Upon complexification, both become the algebra ${\bf C} ( 4 )$. The modified metrics are the metrics of Minkowski space-time (cf. also Minkowski space) and more easily illustrate the physics or the geometry.

In one case (see, for example [a2]) the metric is given by

\begin{equation*} \gamma _ { 1 } ^ { 2 } = 1 , \gamma _ { 2 } ^ { 2 } = \gamma _ { 3 } ^ { 2 } = \gamma _ { 4 } ^ { 2 } = - 1, \end{equation*}

the $\gamma _ { i }$ generate a $2 ^ { 4 }$-dimensional real Clifford algebra that is isomorphic to $H ( 2 )$, the ring of two-by-two matrices over the real quaternion division ring. Every Clifford algebra $C$ admits a $\mathbf{Z}_{2}$-grading,

\begin{equation*} C = C _ { 0 } \oplus C _ { 1 }, \end{equation*}

such that for all $r_i$ $\in C_i$ and $s _ { j } \in C _ { j }$, $r _ { i } s _ { j } \in C _ { ( i + j ) \operatorname { mod } 2}$. The subspace $C _ { 0 }$, spanned by the identity element and all products of an even number, is, in the present case, a subalgebra isomorphic to to the $2 ^ { 3 }$-dimensional Pauli algebra.

The Majorana representation is the $2 ^ { 4 }$-dimensional real Clifford algebra with metric (as in [a1])

\begin{equation*} \gamma _ { 1 } ^ { 2 } = - 1 , \gamma _ { 2 } ^ { 2 } = \gamma _ { 3 } ^ { 2 } = \gamma _ { 4 } ^ { 2 } = 1, \end{equation*}

that is isomorphic to $\operatorname { Re } ( 4 )$. Here, the elements $\gamma_2$, $\gamma_3$ and $\gamma_4$ generate a subalgebra isomorphic to to the $2 ^ { 3 }$-dimensional Pauli algebra.

References

[a1] E. Cartan, "The theory of spinors" , Dover (1966)
[a2] E.M. Corson, "Introduction to tensors, spinors, and relativistic wave-equations" , Chelsea (1953)
[a3] P.A.M. Dirac, "The quantum theory of the electron" Proc. Royal Soc. London , A117 (1928) pp. 610–624
[a4] N.A. Salingaros, G.P. Wene, "The Clifford algebra of differential forms" Acta Applic. Math. , 4 (1985) pp. 271–191
How to Cite This Entry:
Dirac algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirac_algebra&oldid=50416
This article was adapted from an original article by G.P. Wene (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article