Mathematical Principles of Theoretical Physics

5.2. QUARK MODEL 277

can also be decomposed in the form

(5.2.36) H(U) =H 1 (U)⊕ ··· ⊕HK(U) ∀U∈SU(N),

andHj(U) ( 1 ≤j≤K)are as in (5.2.23), such that

(5.2.37) Hj(U):Ej→Ej, dimEj=mj ( 1 ≤j≤K).

In other words, the subspaceEjof (5.2.35) is invariant for the linear transformation (5.2.36)-
(5.2.37).

7.Sakata’s explanation of irreducible representation of SU(N).Now, we can deduce the
following physical conclusions from the discussions in above Steps 1-6.

Physical Explanation 5.5.Let (5.2.29) be a family of composite particles as given by (5.2.30).
The irreducible representation (5.2.36), which usually is expressed as

N︸⊗ ··· ⊗︷︷ N︸

k 1

⊗N︸⊗ ··· ⊗︷︷ N︸

k 2

=m 1 ⊕ ··· ⊕mK,

means that

• the composite particle system (5.2.29) can be classified into K groups of particles:

Gj={Ψ 1 j,···,Ψmjj}, 1 ≤j≤K;

• each group Gjhas mjparticles, such that under the state transformation (5.2.27)-
  (5.2.28) of fundamental particles:

U:CN→CN, U:C

N

→C

N

(U∈SU(N)),

the particles in Gjonly transform between themselves as in (5.2.37).

We now examine the Physical Explanation5.5from the mathematical viewpoint. The

Nk(k=k 1 +k 2 )elements of (5.2.29)-(5.2.30) form a basis ofCN
k 1
⊗C
Nk^2
.We denote these
elements as

(5.2.38) E={Ψ 1 ,···,ΨNk},

with eachΨjas in (5.2.30). The irreducible representation (5.2.23):

AH(U)A−^1 =H 1 (U)⊕ ··· ⊕HK(U), ∀U∈SU(N)

implies that if we take the basis transformation for (5.2.38)

(5.2.39)







Ψ ̃ 1

..

.

Ψ ̃Nk





=A







Ψ 1

..

.

ΨNk





,