SUNSimplify[exp] simplifies color algebraic expressions
involving color matrices with implicit (SUNT) or explicit
fundamental indices (SUNTF) as well as structure constants
(SUND, SUNF) and Kronecker deltas
(SD, SDF).
If the option Explicit is set to True
(default is False), the structure constants will be
rewritten in terms of traces. However, since traces with 2 or 3 color
matrices are by default converted back into structure constants, you
must also set the option SUNTraceEvaluate to
False (default is Automatic) in order to have
unevaluated color traces in the output.
Many of the relations used in this routine (including derivations) can be found in arXiv:1912.13302
Overview, SUNTrace, SUNFierz, SUNT, SUNTF, SUNF, SUND, SUNTraceEvaluate.
SUNDelta[a, b] SUNDelta[b, c]
SUNSimplify[%]\delta ^{ab} \delta ^{bc}
\delta ^{ac}
SUNT[a] . SUNT[a]
SUNSimplify[%]T^a.T^a
C_F
SUNSimplify[SUNT[a] . SUNT[a], SUNNToCACF -> False]-\frac{1-N^2}{2 N}
SUNF[a, r, s] SUNF[b, r, s]
SUNSimplify[%]f^{ars} f^{brs}
C_A \delta ^{ab}
SUNF[a, b, c] SUNF[a, b, c]
SUNSimplify[%]\left(f^{abc}\right)^2
2 C_A^2 C_F
SUNF[a, b, c] SUNF[d, b, c]
SUNSimplify[%]f^{abc} f^{dbc}
C_A \delta ^{ad}
SUNF[a, b, c] SUND[d, b, c]
SUNSimplify[%, Explicit -> True]d^{bcd} f^{abc}
0
SUND[a, b, c] SUND[a, b, c]
SUNSimplify[%, SUNNToCACF -> False] // Factor2\left(d^{abc}\right)^2
\frac{\left(1-N^2\right) \left(4-N^2\right)}{N}
SUNSimplify[SUND[a, b, c] SUND[e, b, c], SUNNToCACF -> False] // Simplify\frac{\left(N^2-4\right) \delta ^{ae}}{N}
SUNSimplify[SUNF[a, b, c], Explicit -> True]f^{abc}
SUNSimplify[SUNF[a, b, c], Explicit -> True, SUNTraceEvaluate -> False]2 i \;\text{tr}\left(T^a.T^c.T^b\right)-2 i \;\text{tr}\left(T^a.T^b.T^c\right)
SUNSimplify[SUND[a, b, c], Explicit -> True]d^{abc}
SUNSimplify[SUND[a, b, c], Explicit -> True, SUNTraceEvaluate -> False]2 \;\text{tr}\left(T^a.T^b.T^c\right)+2 \;\text{tr}\left(T^a.T^c.T^b\right)
SUNF[a, b, c] SUNT[c, b, a]
SUNSimplify[%]f^{abc} T^c.T^b.T^a
-\frac{1}{2} i C_A C_F
SUNF[a, b, e] SUNF[c, d, e] + SUNF[a, b, z] SUNF[c, d, z]
SUNSimplify[%, SUNIndexNames -> {j}]f^{abe} f^{cde}+f^{abz} f^{cdz}
2 f^{abj} f^{cdj}
SUNSimplify[1 - SD[i, i]]2-C_A^2
SUNSimplify[SUNF[a, b, c] SUND[d, b, c]]0
SUNSimplify[SUNF[a, b, c] SUND[a, b, d]]0
SUNSimplify[SUNF[a, b, c] SUND[a, d, c]]0
SUNSimplify[SUND[a, b, c] SUND[d, b, c]]-\left(\left(4-C_A^2\right) \delta ^{ad} \left(C_A-2 C_F\right)\right)
SUNSimplify[SUNTrace[SUNT[i1, i2, i1, i2]], FCE -> True]-\frac{C_F}{2}
SUNSimplify can also deal with chains of color matrices
containing explicit fundamental indices (entered as
SUNTF)
SUNTF[{a}, i, j] SUNTF[{a}, k, l]
SUNSimplify[%]T_{ij}^a T_{kl}^a
\frac{1}{2} \delta _{il} \delta _{jk}-\frac{1}{2} \left(C_A-2 C_F\right) \delta _{ij} \delta _{kl}
SUNTF[{b, a, c}, i, j] SUNTF[{d, a, e}, k, l]
SUNSimplify[%]\left(T^bT^aT^c\right){}_{ij} \left(T^dT^aT^e\right){}_{kl}
\frac{1}{2} \left(T^bT^e\right){}_{il} \left(T^dT^c\right){}_{kj}-\frac{1}{2} \left(C_A-2 C_F\right) \left(T^bT^c\right){}_{ij} \left(T^dT^e\right){}_{kl}
```mathematica SUNTF[{a}, i, j] SUNTrace[SUNT[b, a, c]]
SUNSimplify[%]
```mathematica
T_{ij}^a \;\text{tr}\left(T^b.T^a.T^c\right)
\frac{1}{2} \left(T^cT^b\right){}_{ij}-\frac{1}{4} \left(C_A-2 C_F\right) \delta ^{bc} \delta _{ij}