WOLFRAM|DEMONSTRATIONS PROJECT

Everything about Gell-Mann Matrices (Part 2): Binary Operations

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first λ
λ
1
λ
2
λ
3
λ
4
λ
5
λ
6
λ
7
λ
8
second λ
λ
1
λ
2
λ
3
λ
4
λ
5
λ
6
λ
7
λ
8
operation
product
commutator
anticommutator
λ
1
=
0
1
0
1
0
0
0
0
0
λ
2
=
0
-
0

0
0
0
0
0
λ
3
=
1
0
0
0
-1
0
0
0
0
λ
4
=
0
0
1
0
0
0
1
0
0
λ
5
=
0
0
-
0
0
0

0
0
λ
6
=
0
0
0
0
0
1
0
1
0
λ
7
=
0
0
0
0
0
-
0

0
λ
8
=
1
3
0
0
0
1
3
0
0
0
-
2
3
I
=
1
0
0
0
1
0
0
0
1
λ
1
λ
2
= 
λ
3
Gell–Mann
λ
-matrices are a complete set of Hermitian 3×3 noncommuting trace-orthogonal matrices,
Tr
λ
i
λ
j
=2
δ
ij
,
i=1,…,8
. They are at the heart of Quantum Chromodynamics (QCD), an integral part of the Standard Model. They are also used in quantum information theory to represent qutrits. Gell–Mann matrices are to
SU(3)
what the Pauli matrices are to
SU(2)
.
This Demonstration shows binary operations involving
λ
-matrices.