FeynCalc manual (development version)

ComplexConjugate

ComplexConjugate[exp] returns the complex conjugate of exp, where the input expression must be a proper matrix element. All Dirac matrices are assumed to be inside closed Dirac spinor chains. If this is not the case, the result will be inconsistent. Denominators may not contain explicit ii’s.

See also

Overview, FCRenameDummyIndices, FermionSpinSum, DiracGamma.

Examples

ComplexConjugate is meant to be applied to amplitudes, i.e. given a matrix element M\mathcal{M}, it will return M∗\mathcal{M}^\ast.

amp = (Spinor[Momentum[k1], SMP["m_e"], 1] . GA[\[Mu]] . Spinor[Momentum[p2], SMP["m_e"], 1]*
     Spinor[Momentum[k2], SMP["m_e"], 1] . GA[\[Nu]] . Spinor[Momentum[p1], SMP["m_e"], 1]*
     FAD[k1 - p2, Dimension -> 4]*SMP["e"]^2 - Spinor[Momentum[k1], SMP["m_e"], 
       1] . GA[\[Mu]] . Spinor[Momentum[p1], SMP["m_e"], 1]*Spinor[Momentum[k2], 
       SMP["m_e"], 1] . GA[\[Nu]] . Spinor[Momentum[p2], SMP["m_e"], 1]*FAD[k2 - p2, 
      Dimension -> 4]*SMP["e"]^2)

e2(φ(k1‾,me)).γˉμ.(φ(p2‾,me))(φ(k2‾,me)).γˉν.(φ(p1‾,me))(k1‾−p2‾)2−e2(φ(k1‾,me)).γˉμ.(φ(p1‾,me))(φ(k2‾,me)).γˉν.(φ(p2‾,me))(k2‾−p2‾)2\frac{\text{e}^2 \left(\varphi (\overline{\text{k1}},m_e)\right).\bar{\gamma }^{\mu }.\left(\varphi (\overline{\text{p2}},m_e)\right) \left(\varphi (\overline{\text{k2}},m_e)\right).\bar{\gamma }^{\nu }.\left(\varphi (\overline{\text{p1}},m_e)\right)}{(\overline{\text{k1}}-\overline{\text{p2}})^2}-\frac{\text{e}^2 \left(\varphi (\overline{\text{k1}},m_e)\right).\bar{\gamma }^{\mu }.\left(\varphi (\overline{\text{p1}},m_e)\right) \left(\varphi (\overline{\text{k2}},m_e)\right).\bar{\gamma }^{\nu }.\left(\varphi (\overline{\text{p2}},m_e)\right)}{(\overline{\text{k2}}-\overline{\text{p2}})^2}

ComplexConjugate[amp]

e2(φ(p2‾,me)).γˉμ.(φ(k1‾,me))(φ(p1‾,me)).γˉν.(φ(k2‾,me))(k1‾−p2‾)2−e2(φ(p1‾,me)).γˉμ.(φ(k1‾,me))(φ(p2‾,me)).γˉν.(φ(k2‾,me))(k2‾−p2‾)2\frac{\text{e}^2 \left(\varphi (\overline{\text{p2}},m_e)\right).\bar{\gamma }^{\mu }.\left(\varphi (\overline{\text{k1}},m_e)\right) \left(\varphi (\overline{\text{p1}},m_e)\right).\bar{\gamma }^{\nu }.\left(\varphi (\overline{\text{k2}},m_e)\right)}{(\overline{\text{k1}}-\overline{\text{p2}})^2}-\frac{\text{e}^2 \left(\varphi (\overline{\text{p1}},m_e)\right).\bar{\gamma }^{\mu }.\left(\varphi (\overline{\text{k1}},m_e)\right) \left(\varphi (\overline{\text{p2}},m_e)\right).\bar{\gamma }^{\nu }.\left(\varphi (\overline{\text{k2}},m_e)\right)}{(\overline{\text{k2}}-\overline{\text{p2}})^2}

Although one can also apply the function to standalone Dirac matrices, it should be understood that the result is not equivalent to the complex conjugation of such matrices.

GA[\[Mu]] 
 
ComplexConjugate[%]

γˉμ\bar{\gamma }^{\mu }

γˉμ\bar{\gamma }^{\mu }

GA[5] 
 
ComplexConjugate[%]

γˉ5\bar{\gamma }^5

−γˉ5-\bar{\gamma }^5

(GS[Polarization[k1, -I, Transversality -> True]] . (GS[k1 - p2] + SMP["m_e"]) . 
    GS[Polarization[k2, -I, Transversality -> True]]) 
 
ComplexConjugate[%]

(γˉ⋅εˉ∗(k1)).(γˉ⋅(k1‾−p2‾)+me).(γˉ⋅εˉ∗(k2))\left(\bar{\gamma }\cdot \bar{\varepsilon }^*(\text{k1})\right).\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)+m_e\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*(\text{k2})\right)

(γˉ⋅εˉ(k2)).(γˉ⋅(k1‾−p2‾)+me).(γˉ⋅εˉ(k1))\left(\bar{\gamma }\cdot \bar{\varepsilon }(\text{k2})\right).\left(\bar{\gamma }\cdot \left(\overline{\text{k1}}-\overline{\text{p2}}\right)+m_e\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }(\text{k1})\right)

SUNTrace[SUNT[a, b, c]] 
 
ComplexConjugate[%]

tr(Ta.Tb.Tc)\text{tr}(T^a.T^b.T^c)

tr(Tc.Tb.Ta)\text{tr}(T^c.T^b.T^a)

Since FeynCalc 9.3 ComplexConjugate will automatically rename dummy indices.

PolarizationVector[p1, \[Mu]] PolarizationVector[p2, \[Nu]] MT[\[Mu], \[Nu]] 
 
ComplexConjugate[%]

gˉμνεˉμ(p1)εˉν(p2)\bar{g}^{\mu \nu } \bar{\varepsilon }^{\mu }(\text{p1}) \bar{\varepsilon }^{\nu }(\text{p2})

gˉ$AL($19)$AL($20)εˉ∗$AL($19)(p1)εˉ∗$AL($20)(p2)\bar{g}^{\text{\$AL}(\text{\$19})\text{\$AL}(\text{\$20})} \bar{\varepsilon }^{*\text{\$AL}(\text{\$19})}(\text{p1}) \bar{\varepsilon }^{*\text{\$AL}(\text{\$20})}(\text{p2})

GA[\[Mu], \[Nu]] LC[\[Mu], \[Nu]][p1, p2] 
 
ComplexConjugate[%]

γˉμ.γˉνϵˉμνp1‾  p2‾\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu } \bar{\epsilon }^{\mu \nu \overline{\text{p1}}\;\overline{\text{p2}}}

γˉ$AL($21).γˉ$AL($22)ϵˉ$AL($22)$AL($21)p1‾  p2‾\bar{\gamma }^{\text{\$AL}(\text{\$21})}.\bar{\gamma }^{\text{\$AL}(\text{\$22})} \bar{\epsilon }^{\text{\$AL}(\text{\$22})\text{\$AL}(\text{\$21})\overline{\text{p1}}\;\overline{\text{p2}}}

This behavior can be disabled by setting the option FCRenameDummyIndices to False.

ComplexConjugate[GA[\[Mu], \[Nu]] LC[\[Mu], \[Nu]][p1, p2], FCRenameDummyIndices -> False]

γˉν.γˉμϵˉμνp1‾  p2‾\bar{\gamma }^{\nu }.\bar{\gamma }^{\mu } \bar{\epsilon }^{\mu \nu \overline{\text{p1}}\;\overline{\text{p2}}}

If particular variables must be replaced with their conjugate values, use the option Conjugate.

GA[\[Mu]] . (c1 GA[6] + c2 GA[7]) . GA[\[Nu]] 
 
ComplexConjugate[%]

γˉμ.(c1γˉ6+c2γˉ7).γˉν\bar{\gamma }^{\mu }.\left(\text{c1} \bar{\gamma }^6+\text{c2} \bar{\gamma }^7\right).\bar{\gamma }^{\nu }

γˉν.(c1γˉ7+c2γˉ6).γˉμ\bar{\gamma }^{\nu }.\left(\text{c1} \bar{\gamma }^7+\text{c2} \bar{\gamma }^6\right).\bar{\gamma }^{\mu }

ex = ComplexConjugate[GA[\[Mu]] . (c1 GA[6] + c2 GA[7]) . GA[\[Nu]], Conjugate -> {c1, c2}]

γˉν.(γˉ7  c1∗+γˉ6  c2∗).γˉμ\bar{\gamma }^{\nu }.\left(\bar{\gamma }^7 \;\text{c1}^*+\bar{\gamma }^6 \;\text{c2}^*\right).\bar{\gamma }^{\mu }

ex // StandardForm

(*DiracGamma[LorentzIndex[\[Nu]]] . (Conjugate[c2] DiracGamma[6] + Conjugate[c1] DiracGamma[7]) . DiracGamma[LorentzIndex[\[Mu]]]*)

It may happen that one needs to deal with amplitudes with amputated spinors, i.e. with open Dirac or Pauli indices. If the amplitude contains only a single chain of Dirac/Pauli matrices, everything remains unambiguous and the missing spinors are understood

GA[\[Mu], \[Nu], \[Rho], 5] CSI[i, j] 
 
ComplexConjugate[%]

σ‾i.σ‾jγˉμ.γˉν.γˉρ.γˉ5\overline{\sigma }^i.\overline{\sigma }^j \bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^5

−σ‾j.σ‾iγˉ5.γˉρ.γˉν.γˉμ-\overline{\sigma }^j.\overline{\sigma }^i \bar{\gamma }^5.\bar{\gamma }^{\rho }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\mu }

However, when there are at least two spinor chains of the same type involved, such expressions do not make sense anymore. In these cases one should introduce explicit spinor indices to avoid ambiguities

DCHN[GA[\[Mu], \[Nu], \[Rho], 5], i, j] DCHN[GA[\[Mu], \[Nu], \[Rho], 5], k, l] 
 
ComplexConjugate[%]

(γˉμ.γˉν.γˉρ.γˉ5)ij(γˉμ.γˉν.γˉρ.γˉ5)kl\left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^5\right){}_{ij} \left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^5\right){}_{kl}

(γˉ5.γˉ$AL($23).γˉ$AL($24).γˉ$AL($25))ji(γˉ5.γˉ$AL($23).γˉ$AL($24).γˉ$AL($25))lk\left(\bar{\gamma }^5.\bar{\gamma }^{\text{\$AL}(\text{\$23})}.\bar{\gamma }^{\text{\$AL}(\text{\$24})}.\bar{\gamma }^{\text{\$AL}(\text{\$25})}\right){}_{ji} \left(\bar{\gamma }^5.\bar{\gamma }^{\text{\$AL}(\text{\$23})}.\bar{\gamma }^{\text{\$AL}(\text{\$24})}.\bar{\gamma }^{\text{\$AL}(\text{\$25})}\right){}_{lk}

PCHN[CSI[i, j, k], a, b] PCHN[CSI[i, j, k], c, d] 
 
ComplexConjugate[%]

(σ‾i.σ‾j.σ‾k)ab(σ‾i.σ‾j.σ‾k)cd\left(\overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k\right){}_{ab} \left(\overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k\right){}_{cd}

(σ‾$AL($26).σ‾$AL($27).σ‾$AL($28))ba(σ‾$AL($26).σ‾$AL($27).σ‾$AL($28))dc\left(\overline{\sigma }^{\text{\$AL}(\text{\$26})}.\overline{\sigma }^{\text{\$AL}(\text{\$27})}.\overline{\sigma }^{\text{\$AL}(\text{\$28})}\right){}_{ba} \left(\overline{\sigma }^{\text{\$AL}(\text{\$26})}.\overline{\sigma }^{\text{\$AL}(\text{\$27})}.\overline{\sigma }^{\text{\$AL}(\text{\$28})}\right){}_{dc}

The function does not apply Conjugate to symbols that do not depend on I and are unrelated to Dirac/Pauli/Color matrices. One can specify symbols that need to be explicitly conjugated using the Conjugate option

cc SpinorU[p1] . GA[mu] . SpinorV[p2] 
 
ComplexConjugate[%]

ccu(p1).γˉmu.v(p2)\text{cc} u(\text{p1}).\bar{\gamma }^{\text{mu}}.v(\text{p2})

cc(φ(−p2‾)).γˉmu.(φ(p1‾))\text{cc} \left(\varphi (-\overline{\text{p2}})\right).\bar{\gamma }^{\text{mu}}.\left(\varphi (\overline{\text{p1}})\right)

cc SpinorU[p1] . GA[mu] . SpinorV[p2] 
 
ComplexConjugate[%, Conjugate -> {cc}]

ccu(p1).γˉmu.v(p2)\text{cc} u(\text{p1}).\bar{\gamma }^{\text{mu}}.v(\text{p2})

cc∗(φ(−p2‾)).γˉmu.(φ(p1‾))\text{cc}^* \left(\varphi (-\overline{\text{p2}})\right).\bar{\gamma }^{\text{mu}}.\left(\varphi (\overline{\text{p1}})\right)