The Gell-Mann matrices , named after Murray Gell-Mann , are a possible representation of the infinitesimal generators of the special unitary group SU (3) .
This group has eight Hermitian generators that can be written with . They satisfy the commutator relation (see: Lie algebra )
T
j
{\ displaystyle T_ {j}}
j
=
1
,
...
,
8th
{\ displaystyle j = 1, \ dotsc, 8}
[
T
a
,
T
b
]
=
i
f
a
b
c
T
c
{\ displaystyle \ left [T_ {a}, T_ {b} \ right] = {\ mathrm {i}} \, f ^ {abc} \, T_ {c}}
(using Einstein's summation convention ). The are called structure constants called and are fully-antisymmetric with respect to interchange of the indices. For the SU (3) they have the values:
f
a
b
c
{\ displaystyle f ^ {abc}}
f
123
=
1
,
f
147
=
f
246
=
f
257
=
f
345
=
1
2
,
f
156
=
f
367
=
-
1
2
,
f
458
=
f
678
=
3
2
{\ displaystyle f ^ {123} = 1, ~ f ^ {147} = f ^ {246} = f ^ {257} = f ^ {345} = {\ frac {1} {2}}, ~ f ^ {156} = f ^ {367} = - {\ frac {1} {2}}, ~ f ^ {458} = f ^ {678} = {\ frac {\ sqrt {3}} {2}}}
Any set of matrices that satisfy the commutator relation can be used as generators of the group.
The Gell-Mann matrices are a standard set of such matrices. They are linked to the above generators (analogous to the Pauli matrices ) by:
T
a
=
1
2
λ
a
{\ displaystyle T_ {a} = {\ frac {1} {2}} \ lambda _ {a}}
They are chosen as 3 × 3 matrices and have the form:
λ
1
=
(
0
1
0
1
0
0
0
0
0
)
{\ displaystyle \ lambda _ {1} = {\ begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \ end {pmatrix}}}
λ
2
=
(
0
-
i
0
i
0
0
0
0
0
)
{\ displaystyle \ lambda _ {2} = {\ begin {pmatrix} 0 & - \ mathrm {i} & 0 \\\ mathrm {i} & 0 & 0 \\ 0 & 0 & 0 \ end {pmatrix}}}
λ
3
=
(
1
0
0
0
-
1
0
0
0
0
)
{\ displaystyle \ lambda _ {3} = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \ end {pmatrix}}}
λ
4th
=
(
0
0
1
0
0
0
1
0
0
)
{\ displaystyle \ lambda _ {4} = {\ begin {pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \ end {pmatrix}}}
λ
5
=
(
0
0
-
i
0
0
0
i
0
0
)
{\ displaystyle \ lambda _ {5} = {\ begin {pmatrix} 0 & 0 & - \ mathrm {i} \\ 0 & 0 & 0 \\\ mathrm {i} & 0 & 0 \ end {pmatrix}}}
λ
6th
=
(
0
0
0
0
0
1
0
1
0
)
{\ displaystyle \ lambda _ {6} = {\ begin {pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \ end {pmatrix}}}
λ
7th
=
(
0
0
0
0
0
-
i
0
i
0
)
{\ displaystyle \ lambda _ {7} = {\ begin {pmatrix} 0 & 0 & 0 \\ 0 & 0 & - \ mathrm {i} \\ 0 & \ mathrm {i} & 0 \ end {pmatrix}}}
λ
8th
=
1
3
(
1
0
0
0
1
0
0
0
-
2
)
.
{\ displaystyle \ lambda _ {8} = {\ frac {1} {\ sqrt {3}}} {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \ end {pmatrix}}.}
With the SU (2) one has the three Pauli matrices instead of the eight matrices .
λ
{\ displaystyle \ lambda}
The matrices have the following properties:
λ
{\ displaystyle \ lambda}
They are Hermitian , so they only have real eigenvalues .
You are without a trace , that is .
tr
(
λ
i
)
=
0
{\ displaystyle \ operatorname {tr} (\ lambda _ {i}) = 0}
They are orthogonal with respect to the Frobenius scalar product , that is .
tr
(
λ
i
λ
j
)
=
2
δ
i
j
{\ displaystyle \ operatorname {tr} (\ lambda _ {i} \ lambda _ {j}) = 2 \ delta _ {ij}}
They are used e.g. B. in calculations in quantum chromodynamics , which is described by a SU (3) theory. From this one can understand the choice as 3 × 3 matrices, since the matrices are intended to act on color charge triplets.
See also literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![\ left [T_ {a}, T_ {b} \ right] = {{\ mathrm i}} \, f ^ {{abc}} \, T_ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/178fbbadbd583006914477dd81c459b232f1ba49)
