Mixing
Trace of 4 Gell-Mann matrices
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Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices?
This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty common, but I could not find a proper reference.
In general is there any reference for trace of arbitrary number of Gell Mann matrices?
homework-and-exercises quantum-chromodynamics lie-algebra trace
edited Aug 9 at 7:54
Qmechanic♦
96.5k121631019
asked Aug 9 at 6:11
Angela
634
add a comment |Â
up vote
2
down vote
favorite
Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices?
This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty common, but I could not find a proper reference.
In general is there any reference for trace of arbitrary number of Gell Mann matrices?
homework-and-exercises quantum-chromodynamics lie-algebra trace
edited Aug 9 at 7:54
Qmechanic♦
96.5k121631019
asked Aug 9 at 6:11
Angela
634
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices?
This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty common, but I could not find a proper reference.
In general is there any reference for trace of arbitrary number of Gell Mann matrices?
homework-and-exercises quantum-chromodynamics lie-algebra trace
edited Aug 9 at 7:54
Qmechanic♦
96.5k121631019
asked Aug 9 at 6:11
Angela
634
Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices?
This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty common, but I could not find a proper reference.
In general is there any reference for trace of arbitrary number of Gell Mann matrices?
homework-and-exercises quantum-chromodynamics lie-algebra trace
edited Aug 9 at 7:54
Qmechanic♦
96.5k121631019
asked Aug 9 at 6:11
Angela
634
edited Aug 9 at 7:54
Qmechanic♦
96.5k121631019
edited Aug 9 at 7:54
Qmechanic♦
96.5k121631019
edited Aug 9 at 7:54
Qmechanic♦
96.5k121631019
96.5k121631019
asked Aug 9 at 6:11
Angela
634
asked Aug 9 at 6:11
Angela
634
asked Aug 9 at 6:11
Angela
634
634
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
6
down vote
accepted
I take the SU(N) generators in the fundamental representation normalized such that
$$
textTrleft[T^a T^bright] = frac12delta^ab
$$
The commutator of two generators define the structure constants $f^abc$
$$
left[T^a,T^bright] = if^abcT^c
$$
The anticommutator of two generators is
$$
leftT^a,T^bright = frac1Ndelta^ab1 +d^abcT^c
$$
where by $1$ I mean the identity matrix and $d^abc$ are the "d-symbol" defined as
$$
d^abc = 2textTrleft[ leftT^a,T^brightT^c right]
$$
Then, there is a useful identity
$$
textTrleft[T^aT^bT^cT^dright] = frac14Ndelta^abdelta^cd + frac18left(d^abed^cde - f^abef^cde+if^abed^cde+if^cded^aberight)
$$
I suggest you this reference http://scipp.ucsc.edu/~haber/ph218/sunid17.pdf where different trace identitites are collected. For your case, look at Equation 75 in Appendix B, page 9.
Check the normalization of the generators before to use this identity.
answered Aug 9 at 7:36
apt45
944514
As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone.
– StephenG
Aug 9 at 7:49
@StephenG thanks for the advise
– apt45
Aug 9 at 8:04
@apt45. Thanks.
– Angela
Aug 9 at 14:26
1
@Angela if this answers your question, you should mark it as answered.
– apt45
Aug 9 at 14:27
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
I take the SU(N) generators in the fundamental representation normalized such that
$$
textTrleft[T^a T^bright] = frac12delta^ab
$$
The commutator of two generators define the structure constants $f^abc$
$$
left[T^a,T^bright] = if^abcT^c
$$
The anticommutator of two generators is
$$
leftT^a,T^bright = frac1Ndelta^ab1 +d^abcT^c
$$
where by $1$ I mean the identity matrix and $d^abc$ are the "d-symbol" defined as
$$
d^abc = 2textTrleft[ leftT^a,T^brightT^c right]
$$
Then, there is a useful identity
$$
textTrleft[T^aT^bT^cT^dright] = frac14Ndelta^abdelta^cd + frac18left(d^abed^cde - f^abef^cde+if^abed^cde+if^cded^aberight)
$$
I suggest you this reference http://scipp.ucsc.edu/~haber/ph218/sunid17.pdf where different trace identitites are collected. For your case, look at Equation 75 in Appendix B, page 9.
Check the normalization of the generators before to use this identity.
answered Aug 9 at 7:36
apt45
944514
As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone.
– StephenG
Aug 9 at 7:49
@StephenG thanks for the advise
– apt45
Aug 9 at 8:04
@apt45. Thanks.
– Angela
Aug 9 at 14:26
1
@Angela if this answers your question, you should mark it as answered.
– apt45
Aug 9 at 14:27
add a comment |Â
up vote
6
down vote
accepted
I take the SU(N) generators in the fundamental representation normalized such that
$$
textTrleft[T^a T^bright] = frac12delta^ab
$$
The commutator of two generators define the structure constants $f^abc$
$$
left[T^a,T^bright] = if^abcT^c
$$
The anticommutator of two generators is
$$
leftT^a,T^bright = frac1Ndelta^ab1 +d^abcT^c
$$
where by $1$ I mean the identity matrix and $d^abc$ are the "d-symbol" defined as
$$
d^abc = 2textTrleft[ leftT^a,T^brightT^c right]
$$
Then, there is a useful identity
$$
textTrleft[T^aT^bT^cT^dright] = frac14Ndelta^abdelta^cd + frac18left(d^abed^cde - f^abef^cde+if^abed^cde+if^cded^aberight)
$$
I suggest you this reference http://scipp.ucsc.edu/~haber/ph218/sunid17.pdf where different trace identitites are collected. For your case, look at Equation 75 in Appendix B, page 9.
Check the normalization of the generators before to use this identity.
answered Aug 9 at 7:36
apt45
944514
As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone.
– StephenG
Aug 9 at 7:49
@StephenG thanks for the advise
– apt45
Aug 9 at 8:04
@apt45. Thanks.
– Angela
Aug 9 at 14:26
1
@Angela if this answers your question, you should mark it as answered.
– apt45
Aug 9 at 14:27
add a comment |Â
up vote
6
down vote
accepted
up vote
6
down vote
accepted
I take the SU(N) generators in the fundamental representation normalized such that
$$
textTrleft[T^a T^bright] = frac12delta^ab
$$
The commutator of two generators define the structure constants $f^abc$
$$
left[T^a,T^bright] = if^abcT^c
$$
The anticommutator of two generators is
$$
leftT^a,T^bright = frac1Ndelta^ab1 +d^abcT^c
$$
where by $1$ I mean the identity matrix and $d^abc$ are the "d-symbol" defined as
$$
d^abc = 2textTrleft[ leftT^a,T^brightT^c right]
$$
Then, there is a useful identity
$$
textTrleft[T^aT^bT^cT^dright] = frac14Ndelta^abdelta^cd + frac18left(d^abed^cde - f^abef^cde+if^abed^cde+if^cded^aberight)
$$
I suggest you this reference http://scipp.ucsc.edu/~haber/ph218/sunid17.pdf where different trace identitites are collected. For your case, look at Equation 75 in Appendix B, page 9.
Check the normalization of the generators before to use this identity.
answered Aug 9 at 7:36
apt45
944514
I take the SU(N) generators in the fundamental representation normalized such that
$$
textTrleft[T^a T^bright] = frac12delta^ab
$$
The commutator of two generators define the structure constants $f^abc$
$$
left[T^a,T^bright] = if^abcT^c
$$
The anticommutator of two generators is
$$
leftT^a,T^bright = frac1Ndelta^ab1 +d^abcT^c
$$
where by $1$ I mean the identity matrix and $d^abc$ are the "d-symbol" defined as
$$
d^abc = 2textTrleft[ leftT^a,T^brightT^c right]
$$
Then, there is a useful identity
$$
textTrleft[T^aT^bT^cT^dright] = frac14Ndelta^abdelta^cd + frac18left(d^abed^cde - f^abef^cde+if^abed^cde+if^cded^aberight)
$$
I suggest you this reference http://scipp.ucsc.edu/~haber/ph218/sunid17.pdf where different trace identitites are collected. For your case, look at Equation 75 in Appendix B, page 9.
Check the normalization of the generators before to use this identity.
answered Aug 9 at 7:36
apt45
944514
edited Aug 9 at 8:04
answered Aug 9 at 7:36
apt45
944514
answered Aug 9 at 7:36
apt45
944514
answered Aug 9 at 7:36
apt45
944514
944514
As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone.
– StephenG
Aug 9 at 7:49
@StephenG thanks for the advise
– apt45
Aug 9 at 8:04
@apt45. Thanks.
– Angela
Aug 9 at 14:26
1
@Angela if this answers your question, you should mark it as answered.
– apt45
Aug 9 at 14:27
add a comment |Â
As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone.
– StephenG
Aug 9 at 7:49
@StephenG thanks for the advise
– apt45
Aug 9 at 8:04
@apt45. Thanks.
– Angela
Aug 9 at 14:26
1
@Angela if this answers your question, you should mark it as answered.
– apt45
Aug 9 at 14:27
As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone.
– StephenG
Aug 9 at 7:49
As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone.
– StephenG
Aug 9 at 7:49
@StephenG thanks for the advise
– apt45
Aug 9 at 8:04
@StephenG thanks for the advise
– apt45
Aug 9 at 8:04
@apt45. Thanks.
– Angela
Aug 9 at 14:26
@apt45. Thanks.
– Angela
Aug 9 at 14:26
1
1
@Angela if this answers your question, you should mark it as answered.
– apt45
Aug 9 at 14:27
@Angela if this answers your question, you should mark it as answered.
– apt45
Aug 9 at 14:27
add a comment |Â
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