5.2. QUARK MODEL 273
Thus, each matrixU∈SU(N)is a linear transformation:
(5.2.14) U:CN→CN,
and eachU∈SU(N)gives
(5.2.15) U∗:C
N
→C
N
.
Remark 5.4.In particle physics, a complex orthogonal basis{e 1 ,···,eN}ofCNstands forN
different particles, and its conjugate basis{e 1 ,···,eN}stands for theNantiparticles. Hence
we have
(5.2.16)
CN=the space of all states of particlese 1 ,···,eN,
C
N
=the space of all states of antiparticlese 1 ,···,eN.Thus, the mappingsU∈SU(N)in (5.2.14) stand for state transformations of particlese 1 ,···,eN,
andU∗∈SU(N)for state transformations of antiparticlese 1 ,···,eN.
4.Tensor product of matrices.In quantum physics we often see tensor products of matri-
ces. Here we give their definition. LetA,Bbe two matrices given by
A=
a 11 ··· a 1 n
..
...
.
an 1 ··· ann
, B=
b 11 ··· b 1 m
..
...
.
bm 1 ··· bmm
.
Then the tensor productA⊗Bis defined by
(5.2.17) A⊗B=
a 11 B ··· a 1 nB
..
...
.
an 1 B ··· annB
,
whereaijBare the block matrices
aijB=
aijb 11 ··· aijb 1 m
..
...
.
aijbm 1 ··· aijbmm
.
HenceA⊗Bis an(n×m)-th order matrix.
5.Irreducible representation of SU(N).In the quark model, we shall meet the notations:(5.2.18)
meson= 3 ⊗ 3 ,
baryon= 3 ⊗ 3 ⊗ 3 ,representing the tensor products:
(5.2.19)
meson=SU( 3 )⊗SU( 3 ),
baryon=SU( 3 )⊗SU( 3 )⊗SU( 3 ).