the Dirac gamma matrices - Maple Programming Help

Physics[Dgamma] - the Dirac gamma matrices

 Calling Sequence Dgamma[mu]

Parameters

 mu - an integer between 1 and the dimension, or any algebraic expression generically representing it (when the dimension is equal to 4, mu = 5 is also allowed)

Description

 • The Dgamma[mu] command is used to represent the Dirac ${\mathrm{gamma}}_{\mathrm{mu}}$ matrix, where $\mathrm{mu}$ ranges from 1 to the dimension d of spacetime; these are noncommutative objects satisfying

${\gamma }_{\mu }{\gamma }_{\nu }+{\gamma }_{\nu }{\gamma }_{\mu }=2{\mathrm{g_}}_{\mu ,\nu }$

 where the products in the above are noncommutative, constructed by using the * operator of the Physics package, and ${\mathrm{g_}}_{\mathrm{mu},\mathrm{nu}}$ is the metric tensor. The properties of the Dirac matrices are derived from the defining relation above, which is invariant under a unitary transformation, thus these matrices are determined up to a transformation of that kind, and conventions are necessary to construct their representations. The Physics package's commands know about the properties of the Dirac matrices, and in particular the Trace command computes traces of products of gammas taking these properties into account.
 • The most common representations for the Dirac matrices are the standard (also known as Dirac), the chiral (also known as Weyl), and the Majorana representations. The conventions for the standard representation are uniform in the literature: in a Minkowski  spacetime, with signature - - - +, the contravariant Dirac matrices are:

${{\gamma }}_{}^{0}=\left(\begin{array}{cc}{{\sigma }}_{}^{0}& 0\\ 0& -{{\sigma }}_{}^{0}\end{array}\right),{{\gamma }}_{}^{k}=\left(\begin{array}{cc}0& {{\sigma }}_{}^{k}\\ -{{\sigma }}_{}^{k}& 0\end{array}\right)$

 where ${{\sigma }}_{}^{0}$ is the 2 x 2 identity matrix and ${{\sigma }}_{}^{k}$, with $1\le k$ $\le 3$ are the three Pauli matrices. All of ${{\sigma }}_{}^{0}$ and ${{\sigma }}_{}^{k}$ are represented in the Physics package with Psigma and, as is the case for all spacetime tensors, the value 0 of a spacetime index can also be represented by the number $d$, the spacetime dimension.
 The conventions for the chiral and Majorana representations are not uniform in the literature. The conventions adopted here are the same ones shown in Wikipedia, so that in the chiral representation, the ${{\gamma }}_{}^{k}$ are the same as in the standard representation, while ${{\gamma }}_{}^{0}$ changes to

${{\gamma }}_{}^{0}=\left(\begin{array}{cc}0& {{\sigma }}_{}^{0}\\ {{\sigma }}_{}^{0}& 0\end{array}\right)$

 The convention implemented for the Majorana representation, that is, a representation where all the components of the Dirac matrices are imaginary, is

${{\gamma }}_{}^{0}=\left(\begin{array}{cc}0& {{\sigma }}_{}^{2}\\ {{\sigma }}_{}^{2}& 0\end{array}\right),{{\gamma }}_{}^{1}=\left(\begin{array}{cc}I{{\sigma }}_{}^{3}& 0\\ 0& I{{\sigma }}_{}^{3}\end{array}\right),{{\gamma }}_{}^{2}=\left(\begin{array}{cc}0& -{{\sigma }}_{}^{2}\\ {{\sigma }}_{}^{2}& 0\end{array}\right),{{\gamma }}_{}^{3}=\left(\begin{array}{cc}-I{{\sigma }}_{}^{1}& 0\\ 0& -I{{\sigma }}_{}^{1}\end{array}\right)$

 • Note that the form of the Dirac matrices depends on the signature of spacetime: in a Euclidean spacetime, for the standard, chiral, and Majorana representations, that form is obtained from the formulas above by performing a Wick rotation, equivalent to multiplying the ${{\gamma }}_{}^{k}$ by $-I$, while ${{\gamma }}_{}^{0}$ remains unchanged.
 • In all of these representations, in a four dimensional spacetime, a matrix ${{\gamma }}_{}^{5}$ satisfying

${{\gamma }}_{}^{\mu }{{\gamma }}_{}^{5}+{{\gamma }}_{}^{5}{{\gamma }}_{}^{\mu }=0,{{\gamma }}^{5}{{\gamma }}_{}^{5}=1$

 is given in the case of a Minkowski spacetime by:

${{\gamma }}_{}^{5}=I{{\gamma }}_{}^{0}{{\gamma }}_{}^{1}{{\gamma }}_{}^{2}{{\gamma }}_{}^{3}$

 where $I$ is the imaginary unit (to represent it with a lowercase $i$, see interface). In a four dimensional Euclidean spacetime, ${{\gamma }}_{}^{5}$ is given by

${{\gamma }}_{}^{5}={{\gamma }}_{}^{1}{{\gamma }}_{}^{2}{{\gamma }}_{}^{3}{{\gamma }}_{}^{0}$

 • When the Physics package is loaded, no particular representation is enforced, and the symbol ${{\gamma }}_{}^{k}$ just represents the corresponding Dirac matrix. To set a representation and concretely make ${{\gamma }}_{}^{k}$ be the corresponding matrix, use the Setup command. For example, enter Physics[Setup](Dgammarepresentation = standard); at the Maple prompt.
 • The following are some representation-free frequently used identities for the Dirac matrices, valid provided the dimension, $d$, is greater than 1, expressed by using the sum rule for repeated indices:

${\gamma }_{\mu }{\gamma }_{\mu }=d$

${\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\mu }=\left(2-d\right){\gamma }_{\nu }$

${\gamma }_{\mu }{\gamma }_{\alpha }{\gamma }_{\beta }{\gamma }_{\mu }=4{\mathrm{g_}}_{\alpha ,\beta }+\left(d-4\right){\gamma }_{\alpha }{\gamma }_{\beta }$

${\gamma }_{\mu }{\gamma }_{\alpha }{\gamma }_{\beta }{\gamma }_{\rho }{\gamma }_{\mu }=-2{\gamma }_{\rho }{\gamma }_{\beta }{\gamma }_{\alpha }+\left(4-d\right){\gamma }_{\alpha }{\gamma }_{\beta }{\gamma }_{\rho }$

 where in these formulas, Trace is the Physics command to compute traces, g_ is the metric, epsilon is the Levi-Civita totally antisymmetric symbol, and $I$ is the imaginary unit.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)

As is conventional in the Physics package, the value "0" for a spacetime index is mapped into the value $d$, the dimension, so that in the input, you can use Dgamma[0] and it will be interpreted as

 > ${\mathrm{Dgamma}}_{0}$
 ${{\mathrm{\gamma }}}_{{4}}$ (2)

Note that even after the package is loaded, no representation for the Dirac matrices is set, so Dgamma[$\mathrm{mu}$] is just a symbol representing these matrices, not even a matrix. So this input is just echoed

 > ${{\mathrm{Dgamma}}_{0}}_{\mathrm{matrix}}$
 ${\left({{\mathrm{\gamma }}}_{{4}}\right)}_{{\mathrm{matrix}}}$ (3)

The Dirac matrices have representation-free properties; for example, for the trace of the product of two of them,

 > ${\mathrm{Dgamma}}_{\mathrm{μ}}{\mathrm{Dgamma}}_{\mathrm{ν}}$
 ${{\mathrm{\gamma }}}_{{\mathrm{\mu }}}{}{{\mathrm{\gamma }}}_{{\mathrm{\nu }}}$ (4)
 > $\mathrm{Trace}\left(\right)$
 ${4}{}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}$ (5)
 > ${\mathrm{Dgamma}}_{1}{\mathrm{Dgamma}}_{2}+{\mathrm{Dgamma}}_{2}{\mathrm{Dgamma}}_{1}$
 ${{\mathrm{\gamma }}}_{{1}}{}{{\mathrm{\gamma }}}_{{2}}{+}{{\mathrm{\gamma }}}_{{2}}{}{{\mathrm{\gamma }}}_{{1}}$ (6)
 > $\mathrm{Trace}\left(\right)$
 ${0}$ (7)

To set a representation, for example the chiral one (which is only valid in 4 dimensions), use the Setup command.

 > $\mathrm{Setup}\left(\mathrm{Dgammarepresentation}=\mathrm{chiral}\right):$
 ${\mathrm{Setting lowercaselatin letters to represent spinor indices}}$
 ${\mathrm{Defined Dirac gamma matrices \left(Dgamma\right) in chiral representation}}{,}{{\mathrm{\gamma }}}_{{1}}{,}{{\mathrm{\gamma }}}_{{2}}{,}{{\mathrm{\gamma }}}_{{3}}{,}{{\mathrm{\gamma }}}_{{4}}$
 ${\mathrm{__________________________________________________}}$ (8)

After setting the representation, the symbol Dgamma[mu] for mu ranging from 1 to the dimension of spacetime, can be manipulated as a tensor with two spinor indices, or as a matrix by using the corresponding matrix keyword

 > ${\mathrm{Dgamma}}_{0}$
 ${{\mathrm{\gamma }}}_{{4}}$ (9)
 > ${{\mathrm{Dgamma}}_{0}}_{\mathrm{matrix}}$
 ${\left({{\mathrm{\gamma }}}_{{4}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]$ (10)
 > ${{\mathrm{Dgamma}}_{0}}_{1,3}$
 ${1}$ (11)

With the conventions used, among ${\mathrm{gamma}}_{\mathrm{mu}}$, only ${\mathrm{gamma}}_{0}$ (consequently, when $d=4,{\mathrm{gamma}}_{5}$ too) changes in form between the chiral and standard representations. Note below that Dgamma[0] and Dgamma[1] are symbols representing matrices, which also evaluate to matrices, and so can be referenced in the usual way.

 > $\mathrm{Setup}\left(\mathrm{Dgammarepresentation}=\mathrm{standard}\right):$
 ${\mathrm{Defined Dirac gamma matrices \left(Dgamma\right) in standard representation}}{,}{{\mathrm{\gamma }}}_{{1}}{,}{{\mathrm{\gamma }}}_{{2}}{,}{{\mathrm{\gamma }}}_{{3}}{,}{{\mathrm{\gamma }}}_{{4}}$
 ${\mathrm{__________________________________________________}}$ (12)
 > ${\mathrm{Dgamma}}_{0}+{\mathrm{Dgamma}}_{1}$
 ${{\mathrm{\gamma }}}_{{4}}{+}{{\mathrm{\gamma }}}_{{1}}$ (13)

As with any tensor of the Physics package, to see its components, in this case the matrix for of ${\mathrm{gamma}}_{\mathrm{mu}}$, you can index ${\mathrm{gamma}}_{\mathrm{mu}}$ with the keyword matrix or just with no indices, both show the matrix form:

 > ${{\mathrm{Dgamma}}_{0}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{{4}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {-1}\end{array}\right]$ (14)
 > ${{\mathrm{Dgamma}}_{5}}_{\mathrm{matrix}}$
 ${\left({{\mathrm{\gamma }}}_{{5}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]$ (15)
 > ${{\mathrm{Dgamma}}_{1}}_{3,2}$
 ${1}$ (16)

In a Minkowski spacetime with signature - - - +, the matrix representation of the spacial part of the covariant ${\mathrm{gamma}}_{\mathrm{mu}}$ and contravariant $\mathrm{^}\left(\mathrm{gamma},\mathrm{mu}\right)$ changes sign, while ${\mathrm{gamma}}_{5}$ and $\mathrm{^}\left(\mathrm{gamma},5\right)$ are equal

 > ${{\mathrm{Dgamma}}_{2}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{{2}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{0}& {0}& {0}& {I}\\ {0}& {0}& {-I}& {0}\\ {0}& {-I}& {0}& {0}\\ {I}& {0}& {0}& {0}\end{array}\right]$ (17)
 > ${{\mathrm{Dgamma}}_{\mathrm{~2}}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{2}}}^{\phantom{{}}{2}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{0}& {0}& {0}& {-I}\\ {0}& {0}& {I}& {0}\\ {0}& {I}& {0}& {0}\\ {-I}& {0}& {0}& {0}\end{array}\right]$ (18)
 > ${{\mathrm{Dgamma}}_{\mathrm{~5}}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{5}}}^{\phantom{{}}{5}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]$ (19)

Note that the representation also depends on the signature (Euclidean or Minkowski) of spacetime; with the conventions used, among ${\mathrm{gamma}}_{\mathrm{mu}}$, only ${\mathrm{gamma}}_{0}$ does not change when the signature of spacetime is changed.

 > $\mathrm{Setup}\left(\mathrm{signature}=\mathrm{+}\right)$
 ${\mathrm{Changing the signature of the tensor spacetime to: + + + +}}$
 $\left[{\mathrm{signature}}{=}{\mathrm{+ + + +}}\right]$ (20)
 > ${{\mathrm{Dgamma}}_{\mathrm{~0}}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{4}}}^{\phantom{{}}{4}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {-1}\end{array}\right]$ (21)
 > ${{\mathrm{Dgamma}}_{\mathrm{~1}}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{1}}}^{\phantom{{}}{1}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cccc}{0}& {0}& {0}& {-I}\\ {0}& {0}& {-I}& {0}\\ {0}& {I}& {0}& {0}\\ {I}& {0}& {0}& {0}\end{array}\right]$ (22)

The standard representation is also defined in lower dimensions. For example, if you change the dimension and signature again:

 > $\mathrm{Setup}\left(\mathrm{dimension}=\left[3,\mathrm{-}\right]\right)$
 ${\mathrm{The dimension and signature of the tensor space are set to: \left[3, - - +\right]}}$
 $\left[{\mathrm{dimension}}{=}{3}{,}{\mathrm{signature}}{=}{\mathrm{- - +}}\right]$ (23)
 > ${{\mathrm{Dgamma}}_{\mathrm{~1}}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{1}}}^{\phantom{{}}{1}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cc}{0}& {I}\\ {I}& {0}\end{array}\right]$ (24)
 > $\mathrm{Setup}\left(\mathrm{dimension}=2\right)$
 ${\mathrm{The dimension and signature of the tensor space are set to: \left[2, - +\right]}}$
 $\left[{\mathrm{dimension}}{=}{2}\right]$ (25)
 > ${{\mathrm{Dgamma}}_{\mathrm{~1}}}_{[]}$
 ${\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{1}}}^{\phantom{{}}{1}}\right)}_{{a}{,}{b}}{=}\left[\begin{array}{cc}{0}& {1}\\ {-1}& {0}\end{array}\right]$ (26)
 >