5.2. QUARK MODEL 275
and their complex linear combination constituteCN:
CN=
{
N
∑
j= 1zkψk|zk∈C, 1 ≤k≤N}
,
which contains all physical states of theNparticles (5.2.25).
In addition, theNfundamental particles ofSU(N)are the antiparticles of (5.2.25) given
by
(5.2.26) ψ 1 ,···,ψN,
whereψkis the complex conjugate ofψk. The linear space
C
N
={
N
∑
j= 1ykψk|yk∈C, 1 ≤k≤N}
contains all physical states of theNantiparticles (5.2.26).
- Each matrixU∈SU(N)and eachU∈SU(N)represent the transformations of physical
states of particles (5.2.25) and antiparticles (5.2.26) as follows
(5.2.27)
N
∑
j= 1zkψk→N
∑
j= 1̃zkψk,N
∑
j= 1ykψk→N
∑
j= 1̃ykψk,where
(5.2.28)
̃z 1
..
.
̃zN
=U
z 1
..
.
zN
,
̃y 1
..
.
̃yN
=U
y 1
..
.
yN
.
- The tensor product of fundamental particles (5.2.25) and (5.2.26) are denoted by
(5.2.29) N︸⊗ ··· ⊗︷︷ N︸
k 1⊗N︸⊗ ··· ⊗︷︷ N︸
k 2={ψi 1 ···ψik 1 ψj 1 ···ψjk
2},
which stands for a new particle system where each particle
(5.2.30) ψi 1 ···ik 1 j 1 ···jk 2 =ψi 1 ···ψik 1 ψj 1 ···ψjk
2
is a composite particle made up ofψi 1 ,···,ψik 1 ,ψj 1 ,···,ψjk
2
.
For example forN=3, the tensor product3 ⊗ 3 =
ψ 1 ψ 1 ψ 1 ψ 2 ψ 1 ψ 3
ψ 2 ψ 1 ψ 2 ψ 2 ψ 2 ψ 3
ψ 3 ψ 1 ψ 3 ψ 2 ψ 3 ψ 3