View source for Dirac algebra

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

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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

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

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

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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:

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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

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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

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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]]).

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.

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

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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,

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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.

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]]])

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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.

====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>