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 16. 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 64.
Notice that this routine does not perform any canonicalization. Use
FCCanonicalizeDummyIndices for that.
Overview, ComplexConjugate, FCCanonicalizeDummyIndices.
FVD[q, mu] FVD[p, mu] + FVD[q, nu] FVD[p, nu] + FVD[q, si] FVD[r, si]
FCRenameDummyIndices[%] // Factor2p^{\text{mu}} q^{\text{mu}}+p^{\text{nu}} q^{\text{nu}}+q^{\text{si}} r^{\text{si}}
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^{\text{\$AL}(\text{\$22})} p^{\text{\$AL}(\text{\$23})} q^{\text{\$AL}(\text{\$22})} q^{\text{\$AL}(\text{\$23})}
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-\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]
% // FCRenameDummyIndicesp^i q^i+p^j r^j
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]
% // FCRenameDummyIndicesT^a.T^b.T^a+T^c.T^b.T^c
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\left(\bar{\gamma }^{\text{mu}}\right){}_{ij} \left(\bar{\gamma }^{\text{nu}}\right){}_{jk}
\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\left(\overline{\sigma }^a\right){}_{ij} \left(\overline{\sigma }^b\right){}_{jk}
\left(\overline{\sigma }^a\right){}_{i\text{\$AL}(\text{\$32})} \left(\overline{\sigma }^b\right){}_{\text{\$AL}(\text{\$32})k}