FeynCalc manual (development version)

FCRenameDummyIndices

FCRenameDummyIndices[expr] identifies dummy indices and changes their names pairwise to random symbols. This can be useful if you have an expression that contains dummy indices and want to compute the square of it. For example, the square of GA[a, l, a] equals 1616. However, if you forget to rename the dummy indices and compute GA[a, l, a, a, l, a] instead of GA[a, l, a, b, l, b], you will get 6464.

Notice that this routine does not perform any canonicalization. Use FCCanonicalizeDummyIndices for that.

See also

Overview, ComplexConjugate, FCCanonicalizeDummyIndices.

Examples

FVD[q, mu] FVD[p, mu] + FVD[q, nu] FVD[p, nu] + FVD[q, si] FVD[r, si] 
 
FCRenameDummyIndices[%] // Factor2

pmuqmu+pnuqnu+qsirsip^{\text{mu}} q^{\text{mu}}+p^{\text{nu}} q^{\text{nu}}+q^{\text{si}} r^{\text{si}}

p$AL($19)q$AL($19)+p$AL($20)q$AL($20)+q$AL($21)r$AL($21)p^{\text{\$AL}(\text{\$19})} q^{\text{\$AL}(\text{\$19})}+p^{\text{\$AL}(\text{\$20})} q^{\text{\$AL}(\text{\$20})}+q^{\text{\$AL}(\text{\$21})} r^{\text{\$AL}(\text{\$21})}

Uncontract[SPD[q, p]^2, q, p, Pair -> All] 
 
FCRenameDummyIndices[%]

p$AL($22)p$AL($23)q$AL($22)q$AL($23)p^{\text{\$AL}(\text{\$22})} p^{\text{\$AL}(\text{\$23})} q^{\text{\$AL}(\text{\$22})} q^{\text{\$AL}(\text{\$23})}

p$AL($24)p$AL($25)q$AL($24)q$AL($25)p^{\text{\$AL}(\text{\$24})} p^{\text{\$AL}(\text{\$25})} q^{\text{\$AL}(\text{\$24})} q^{\text{\$AL}(\text{\$25})}

amp = -(Spinor[Momentum[k1], SMP["m_mu"], 1] . GA[Lor1] . Spinor[-Momentum[k2], 
         SMP["m_mu"], 1]*Spinor[-Momentum[p2], SMP["m_e"], 1] . GA[Lor1] . Spinor[Momentum[p1], 
         SMP["m_e"], 1]*FAD[k1 + k2, Dimension -> 4]*SMP["e"]^2); 
 
amp // FCRenameDummyIndices

−e2(φ(−p2‾,me)).γˉ$AL($26).(φ(p1‾,me))(φ(k1‾,mμ)).γˉ$AL($26).(φ(−k2‾,mμ))(k1‾+k2‾)2-\frac{\text{e}^2 \left(\varphi (-\overline{\text{p2}},m_e)\right).\bar{\gamma }^{\text{\$AL}(\text{\$26})}.\left(\varphi (\overline{\text{p1}},m_e)\right) \left(\varphi (\overline{\text{k1}},m_{\mu })\right).\bar{\gamma }^{\text{\$AL}(\text{\$26})}.\left(\varphi (-\overline{\text{k2}},m_{\mu })\right)}{(\overline{\text{k1}}+\overline{\text{k2}})^2}

CVD[p, i] CVD[q, i] + CVD[p, j] CVD[r, j] 
 
% // FCRenameDummyIndices

piqi+pjrjp^i q^i+p^j r^j

p$AL($27)q$AL($27)+p$AL($28)r$AL($28)p^{\text{\$AL}(\text{\$27})} q^{\text{\$AL}(\text{\$27})}+p^{\text{\$AL}(\text{\$28})} r^{\text{\$AL}(\text{\$28})}

SUNT[a, b, a] + SUNT[c, b, c] 
 
% // FCRenameDummyIndices

Ta.Tb.Ta+Tc.Tb.TcT^a.T^b.T^a+T^c.T^b.T^c

T$AL($29).Tb.T$AL($29)+T$AL($30).Tb.T$AL($30)T^{\text{\$AL}(\text{\$29})}.T^b.T^{\text{\$AL}(\text{\$29})}+T^{\text{\$AL}(\text{\$30})}.T^b.T^{\text{\$AL}(\text{\$30})}

DCHN[GA[mu], i, j] DCHN[GA[nu], j, k] 
 
% // FCRenameDummyIndices

(γˉmu)ij(γˉnu)jk\left(\bar{\gamma }^{\text{mu}}\right){}_{ij} \left(\bar{\gamma }^{\text{nu}}\right){}_{jk}

(γˉmu)i$AL($31)(γˉnu)$AL($31)k\left(\bar{\gamma }^{\text{mu}}\right){}_{i\text{\$AL}(\text{\$31})} \left(\bar{\gamma }^{\text{nu}}\right){}_{\text{\$AL}(\text{\$31})k}

PCHN[CSI[a], i, j] PCHN[CSI[b], j, k] 
 
% // FCRenameDummyIndices

(σ‾a)ij(σ‾b)jk\left(\overline{\sigma }^a\right){}_{ij} \left(\overline{\sigma }^b\right){}_{jk}

(σ‾a)i$AL($32)(σ‾b)$AL($32)k\left(\overline{\sigma }^a\right){}_{i\text{\$AL}(\text{\$32})} \left(\overline{\sigma }^b\right){}_{\text{\$AL}(\text{\$32})k}