The DiracConjugate command represents and computes the Dirac conjugate of its argument; the returned result is built as follows:

- If $\mathrm{\psi }=z$ is a scalar, return its conjugate, $\stackrel{\&conjugate0;}{z}$.

- If $\mathrm{\psi }$ is a spinor, so defined with one spinor index using the Define command, or $\mathrm{\psi }$ is an anticommutative quantum operator (see Setup) with a spinor index, then return unevaluated, as DiracConjugate(psi), displayed as $\overline{\mathrm{\psi }}$, representing $\overline{\mathrm{\psi }}={\mathrm{\psi }}^{†}·{\mathrm{\gamma }}^0$, where ${\mathrm{\psi }}^{†}$ is the Hermitian conjugate Dagger(psi) and ${\mathrm{\gamma }}^0$ is the contravariant $0^{\mathrm{th}}$ Dirac matrix Dgamma[~0].

- If $\mathrm{\psi }=\overline{A}$ is the Dirac conjugate of - say -  $A$, then return $A$.

- If $\mathrm{\psi }={\mathrm{\gamma }}^{\mathrm{\mu }}$ is a Dirac matrix (represented by the Dgamma command), then return the matrix ${\mathrm{\gamma }}^{\mathrm{\mu }}$ itself, also when $\mathrm{\mu }=5$.

- If $\mathrm{\psi }=M$ is a Matrix - say $M$ - return the matrix product ${\mathrm{\gamma }}^0·M^{†}·{\mathrm{\gamma }}^0$, where $M^{†}$ is the Dagger(M), the Hermitian conjugate of $M$.

- If $\mathrm{\psi }$ is a sum of terms, return the sum of the DiracConjugate of each term.

- If $\mathrm{\psi }=A\mathrm{...}B$ is a product, return $\overline{B}\mathrm{...}\overline{A}$, that is the product of the DiracConjugate of each of the factors with the ordering reversed.

- If $\mathrm{\psi }$ is one of the d_ or dAlembertian operators, return the operator applied to the DiracConjugate of the first operand of $\mathrm{\psi }$.

- Otherwise, return the operation unevaluated, DiracConjugate(A).

The %DiracConjugate command is the inert form of DiracConjugate; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{DiracConjugate}\left(z\right)$

$\stackrel{\&conjugate0;}{z}$

$\mathrm{DiracConjugate}\left(\right)$

$z$

$\left(\mathrm{\%DiracConjugate}=\mathrm{DiracConjugate}\right)\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\right)$

$\overline{{\mathrm{\gamma }}_{\mathrm{\mu }}}={\mathrm{\gamma }}_{\mathrm{\mu }}$

$\left(\mathrm{\%DiracConjugate}=\mathrm{DiracConjugate}\right)\left(\mathrm{Dgamma}\left[5\right]\right)$

$\overline{{\mathrm{\gamma }}_5}={\mathrm{\gamma }}_5$

$\mathrm{Setup}\left(\mathrm{coordinates}=\mathrm{cartesian}\,\mathrm{quantumoperator}=A\,\mathrm{anticommutativeprefix}=B\,\mathrm{spinorindices}=\mathrm{lowercaselatin}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{quantumoperator}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{B\right\}\,\mathrm{coordinatesystems}=\left\{X\right\}\,\mathrm{quantumoperators}=\left\{A\right\}\,\mathrm{spinorindices}=\mathrm{lowercaselatin}\right]$

$\mathrm{Define}\left(A\left[j\right]\,B\left[j\right]\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_j\,B_j\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$A\left[j\right]B\left[j\right]$

$A_j{B}_j$

$\mathrm{SumOverRepeatedIndices}\left(\right)$

$A_1{B}_1+A_2{B}_2+A_3{B}_3+A_4{B}_4$

$\mathrm{DiracConjugate}\left(\right)$

$\stackrel{\&conjugate0;}{B_1}\stackrel{\&conjugate0;}{A_1}+\stackrel{\&conjugate0;}{B_2}\stackrel{\&conjugate0;}{A_2}+\stackrel{\&conjugate0;}{B_3}\stackrel{\&conjugate0;}{A_3}+\stackrel{\&conjugate0;}{B_4}\stackrel{\&conjugate0;}{A_4}$

$\mathrm{DiracConjugate}\left(\right)$

${\overline{B}}_j{\overline{A}}_j$

$\mathrm{SumOverRepeatedIndices}\left(\right)$

$\stackrel{\&conjugate0;}{B_1}\stackrel{\&conjugate0;}{A_1}+\stackrel{\&conjugate0;}{B_2}\stackrel{\&conjugate0;}{A_2}+\stackrel{\&conjugate0;}{B_3}\stackrel{\&conjugate0;}{A_3}+\stackrel{\&conjugate0;}{B_4}\stackrel{\&conjugate0;}{A_4}$

$\mathrm{DiracConjugate}\left(B\right)B$

$\overline{B}B$

$\mathrm{DiracConjugate}\left(\right)$

$\overline{B}B$

$\mathrm{DiracConjugate}\left(B\left[j\right]\right)B\left[j\right]$

${\overline{B}}_j{B}_j$

$\mathrm{DiracConjugate}\left(\right)$

${\overline{B}}_j{B}_j$

$M≔\mathrm{Matrix}\left(4\,\mathrm{symbol}=m\right)$

$M≔\left[\begin{array}{cccc}{m}_{1,1}& m_{1,2}& m_{1,3}& m_{1,4}\\ m_{2,1}& m_{2,2}& m_{2,3}& m_{2,4}\\ m_{3,1}& m_{3,2}& m_{3,3}& m_{3,4}\\ m_{4,1}& m_{4,2}& m_{4,3}& m_{4,4}\end{array}\right]$

$\mathrm{DiracConjugate}\left(M\right)$

$\left[\begin{array}{cccc}\stackrel{\&conjugate0;}{m_{1,1}}& \stackrel{\&conjugate0;}{m_{2,1}}& -\stackrel{\&conjugate0;}{m_{3,1}}& -\stackrel{\&conjugate0;}{m_{4,1}}\\ \stackrel{\&conjugate0;}{m_{1,2}}& \stackrel{\&conjugate0;}{m_{2,2}}& -\stackrel{\&conjugate0;}{m_{3,2}}& -\stackrel{\&conjugate0;}{m_{4,2}}\\ -\stackrel{\&conjugate0;}{m_{1,3}}& -\stackrel{\&conjugate0;}{m_{2,3}}& \stackrel{\&conjugate0;}{m_{3,3}}& \stackrel{\&conjugate0;}{m_{4,3}}\\ -\stackrel{\&conjugate0;}{m_{1,4}}& -\stackrel{\&conjugate0;}{m_{2,4}}& \stackrel{\&conjugate0;}{m_{3,4}}& \stackrel{\&conjugate0;}{m_{4,4}}\end{array}\right]$

$\mathrm{DiracConjugate}\left(M\right)\mathbf{assuming}\mathrm{real}$

$\left[\begin{array}{cccc}{m}_{1,1}& m_{2,1}& -m_{3,1}& -m_{4,1}\\ m_{1,2}& m_{2,2}& -m_{3,2}& -m_{4,2}\\ -m_{1,3}& -m_{2,3}& m_{3,3}& m_{4,3}\\ -m_{1,4}& -m_{2,4}& m_{3,4}& m_{4,4}\end{array}\right]$

$M≔\mathrm{Matrix}\left(4\,\left(i\,j\right)\mapsto A\left[i\right]\cdot B\left[j\right]\right)$

$M≔\left[\begin{array}{cccc}{A}_1{B}_1& A_1{B}_2& A_1{B}_3& A_1{B}_4\\ A_2{B}_1& A_2{B}_2& A_2{B}_3& A_2{B}_4\\ A_3{B}_1& A_3{B}_2& A_3{B}_3& A_3{B}_4\\ A_4{B}_1& A_4{B}_2& A_4{B}_3& A_4{B}_4\end{array}\right]$

$\mathrm{DiracConjugate}\left(M\right)$

$\left[\begin{array}{cccc}\stackrel{\&conjugate0;}{B_1}\stackrel{\&conjugate0;}{A_1}& \stackrel{\&conjugate0;}{B_1}\stackrel{\&conjugate0;}{A_2}& -\stackrel{\&conjugate0;}{B_1}\stackrel{\&conjugate0;}{A_3}& -\stackrel{\&conjugate0;}{B_1}\stackrel{\&conjugate0;}{A_4}\\ \stackrel{\&conjugate0;}{B_2}\stackrel{\&conjugate0;}{A_1}& \stackrel{\&conjugate0;}{B_2}\stackrel{\&conjugate0;}{A_2}& -\stackrel{\&conjugate0;}{B_2}\stackrel{\&conjugate0;}{A_3}& -\stackrel{\&conjugate0;}{B_2}\stackrel{\&conjugate0;}{A_4}\\ -\stackrel{\&conjugate0;}{B_3}\stackrel{\&conjugate0;}{A_1}& -\stackrel{\&conjugate0;}{B_3}\stackrel{\&conjugate0;}{A_2}& \stackrel{\&conjugate0;}{B_3}\stackrel{\&conjugate0;}{A_3}& \stackrel{\&conjugate0;}{B_3}\stackrel{\&conjugate0;}{A_4}\\ -\stackrel{\&conjugate0;}{B_4}\stackrel{\&conjugate0;}{A_1}& -\stackrel{\&conjugate0;}{B_4}\stackrel{\&conjugate0;}{A_2}& \stackrel{\&conjugate0;}{B_4}\stackrel{\&conjugate0;}{A_3}& \stackrel{\&conjugate0;}{B_4}\stackrel{\&conjugate0;}{A_4}\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{spinorindices}=\mathrm{none}\right)\:$

$\mathrm{with}\left(\mathrm{StandardModel}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{Dirac\; spinor}\mathrm{indices}$

$\mathrm{Setting}\mathrm{lowercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{SU(3)\; adjoint\; representation,\; (1..8)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{SU(3)\; fundamental\; representation,\; (1..3)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{SU(2)\; adjoint\; representation,\; (1..3)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercasegreek}\mathrm{letters\; to\; represent}\mathrm{SU(2)\; fundamental\; representation,\; (1..2)}\mathrm{indices}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Defined\; as\; the\; electron,\; muon\; and\; tau\; leptons\; and\; corresponding\; neutrinos:}{\mathrm{e}}_i,{\mathrm{\mu }}_i,{\mathrm{\tau }}_i,{{\mathrm{\nu }}^{\left(\mathrm{e}\right)}}_i,{{\mathrm{\nu }}^{\left(\mathrm{\mu }\right)}}_i,{{\mathrm{\nu }}^{\left(\mathrm{\tau }\right)}}_i$

$\mathrm{Defined\; as\; the\; up,\; charm,\; top,\; down,\; strange\; and\; bottom\; quarks:}{\mathrm{u}}_{A,i},{\mathrm{c}}_{A,i},{\mathrm{t}}_{A,i},{\mathrm{d}}_{A,i},{\mathrm{s}}_{A,i},{\mathrm{b}}_{A,i}$

$\mathrm{Defined\; as\; gauge\; tensors:}{\mathrm{B}}_{\mathrm{\mu }},{\mathrm{𝔹}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{A}}_{\mathrm{\mu }},{\mathrm{𝔽}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{W}}_{\mathrm{\mu },J},{\mathrm{𝕎}}_{\mathrm{\mu },\mathrm{\nu },J},{{\mathrm{W}}^{\mathrm{+}}}_{\mathrm{\mu }},{{\mathrm{𝕎}}^{\mathrm{+}}}_{\mathrm{\mu },\mathrm{\nu }},{{\mathrm{W}}^{\mathrm{-}}}_{\mathrm{\mu }},{{\mathrm{𝕎}}^{\mathrm{-}}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{Z}}_{\mathrm{\mu }},{\mathrm{\mathbb{Z}}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{G}}_{\mathrm{\mu },a},{\mathrm{𝔾}}_{\mathrm{\mu },\mathrm{\nu },a}$

$\mathrm{Defined\; as\; Gell-Mann\; (Glambda),\; Pauli\; (Psigma)\; and\; Dirac\; (Dgamma)\; matrices:}{\mathrm{\lambda }}_a,{\mathrm{\sigma }}_J,{\mathrm{\gamma }}_{\mathrm{\mu }}$

$\mathrm{Defined\; as\; the\; electric,\; weak\; and\; strong\; coupling\; constants:}\mathrm{g\_\_e},\mathrm{g\_\_w},\mathrm{g\_\_s}$

$\mathrm{Defined\; as\; the\; charge\; in\; units\; of\; |}\mathrm{g\_\_e}\mathrm{|\; for\; 1)\; the\; electron,\; muon\; and\; tauon,\; 2)\; the\; up,\; charm\; and\; top,\; and\; 3)\; the\; down,\; strange\; and\; bottom:}\mathrm{q\_\_e}=\mathrm{-1},\mathrm{q\_\_u}=\frac{2}{3},\mathrm{q\_\_d}=-\frac{1}{3}$

$\mathrm{Defined\; as\; the\; weak\; isospin\; for\; 1)\; the\; electron,\; muon\; and\; tauon,\; 2)\; the\; up,\; charm\; and\; top,\; 3)\; the\; down,\; strange\; and\; bottom,\; and\; 4)\; all\; the\; neutrinos:}\mathrm{I\_\_e}=-\frac{1}{2},\mathrm{I\_\_u}=\frac{1}{2},\mathrm{I\_\_d}=-\frac{1}{2},\mathrm{I\_\_n}=\frac{1}{2}$

$\mathrm{You\; can\; use\; the\; active\; form\; without\; the\; \%\; prefix,\; or\; the\; \text{'}value\text{'}\; command\; to\; give\; the\; corresponding\; value\; to\; any\; of\; the\; inert\; representations}\mathrm{q\_\_e},\mathrm{q\_\_u},\mathrm{q\_\_d},\mathrm{I\_\_e},\mathrm{I\_\_u},\mathrm{I\_\_d},\mathrm{I\_\_n}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{Minkowski\; spacetime\; with\; signatre}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{I\_\_d}\,\mathrm{I\_\_e}\,\mathrm{I\_\_n}\,\mathrm{I\_\_u}\,\mathrm{q\_\_d}\,\mathrm{q\_\_e}\,\mathrm{q\_\_u}\,\mathrm{BField}\,\mathrm{BFieldStrength}\,\mathrm{Bottom}\,\mathrm{CKM}\,\mathrm{Charm}\,\mathrm{Down}\,\mathrm{ElectromagneticField}\,\mathrm{ElectromagneticFieldStrength}\,\mathrm{Electron}\,\mathrm{ElectronNeutrino}\,\mathrm{FSU3}\,\mathrm{Glambda}\,\mathrm{GluonField}\,\mathrm{GluonFieldStrength}\,\mathrm{HiggsBoson}\,\mathrm{Lagrangian}\,\mathrm{Muon}\,\mathrm{MuonNeutrino}\,\mathrm{Strange}\,\mathrm{Tauon}\,\mathrm{TauonNeutrino}\,\mathrm{Top}\,\mathrm{Up}\,\mathrm{WField}\,\mathrm{WFieldStrength}\,\mathrm{WMinusField}\,\mathrm{WMinusFieldStrength}\,\mathrm{WPlusField}\,\mathrm{WPlusFieldStrength}\,\mathrm{WeinbergAngle}\,\mathrm{ZField}\,\mathrm{ZFieldStrength}\,\mathrm{g\_\_e}\,\mathrm{g\_\_s}\,\mathrm{g\_\_w}\right]$

$\mathrm{CompactDisplay}\left(\mathrm{Electron}\left(X\right)\right)$

$\mathbf{e}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathbf{e}$

$\mathrm{Lagrangian}\left(\mathrm{QED}\right)$

${\overline{\mathbf{e}}}_i\left(\mathrm{I}{\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{i,j}{▿}_{\mathrm{\mu }}-m_{\mathbf{e}}{\mathrm{\delta }}_{i,j}\right){\mathbf{e}}_j-\frac{{\mathbf{𝔽}}_{\mathrm{\mu },\mathrm{\nu }}{\mathbf{𝔽}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}}{4}$

$\mathrm{DiracConjugate}\left(\right)$

$\overline{{\mathbf{e}}_j}\left(-\mathrm{I}{\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,i}\stackrel{\&conjugate0;}{{▿}_{\mathrm{\mu }}}-\overline{{\mathrm{\delta }}_{i,j}}{m}_{\mathbf{e}}\right){\mathbf{e}}_i-\frac{{\mathbf{𝔽}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{\mathbf{𝔽}}_{\mathrm{\mu },\mathrm{\nu }}}{4}$

$\mathrm{am}≔\mathrm{Annihilation}\left(A\,1\right)$

$\mathrm{am}≔a^{-}$

$\mathrm{ap}≔\mathrm{Creation}\left(A\,1\right)$

$\mathrm{ap}≔a^{+}$

$\mathrm{DiracConjugate}\left(\mathrm{am}\right)$

$a^{+}$

$\mathrm{DiracConjugate}\left(\mathrm{ap}\right)$

$a^{-}$

$\mathrm{Setup}\left(\mathrm{noncommutativeprefix}=Z\right)$

$\left[\mathrm{noncommutativeprefix}=\left\{Z\right\}\right]$

$\mathrm{Commutator}\left(Z\left[1\right]\,Z\left[2\right]\right)$

${\left[Z_1\,Z_2\right]}_{-}$

$=\mathrm{expand}\left(\right)$

${\left[Z_1\,Z_2\right]}_{-}=Z_1{Z}_2-Z_2{Z}_1$

$\mathrm{DiracConjugate}\left(\right)$

${\left[{Z_2}^{†}\,{Z_1}^{†}\right]}_{-}={Z_2}^{†}{Z_1}^{†}-{Z_1}^{†}{Z_2}^{†}$

$\mathrm{AntiCommutator}\left(Z\left[1\right]\,Z\left[2\right]\right)$

${\left[Z_1\,Z_2\right]}_{+}$

$=\mathrm{expand}\left(\right)$

${\left[Z_1\,Z_2\right]}_{+}=Z_1{Z}_2+Z_2{Z}_1$

$\mathrm{DiracConjugate}\left(\right)\mathbf{assuming}\mathrm{Hermitian}$

${\left[Z_1\,Z_2\right]}_{+}=Z_1{Z}_2+Z_2{Z}_1$

$\mathrm{DiracConjugate}\left(\right)$

${\left[{Z_1}^{†}\,{Z_2}^{†}\right]}_{+}={Z_1}^{†}{Z_2}^{†}+{Z_2}^{†}{Z_1}^{†}$

$\mathrm{Setup}\left(\mathrm{diff}=X\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{differentiationvariables}=\left[X\right]\right]$

$\mathrm{d\_}\left[\mathrm{\mu }\right]\left(Z\left[1\right]\left(X\right)\right)\mathrm{d\_}\left[\mathrm{\nu }\right]\left(Z\left[2\right]\left(X\right)\right)+\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]\mathrm{dAlembertian}\left(F\left(X\right)\right)$

${\partial }_{\mathrm{\mu }}\left(Z_1\left(X\right)\right){\partial }_{\mathrm{\nu }}\left(Z_2\left(X\right)\right)+g_{\mathrm{\mu },\mathrm{\nu }}\mathrm{\square }\left(F\left(X\right)\right)$

$\mathrm{DiracConjugate}\left(\right)$

${\partial }_{\mathrm{\nu }}\left({Z_2\left(X\right)}^{†}\right){\partial }_{\mathrm{\mu }}\left({Z_1\left(X\right)}^{†}\right)+\mathrm{\square }\left(\stackrel{\&conjugate0;}{F\left(X\right)}\right)g_{\mathrm{\mu },\mathrm{\nu }}$