Mathematical Principles of Theoretical Physics

276 CHAPTER 5. ELEMENTARY PARTICLES

contains 9 new particles
ψij=ψiψj, 1 ≤i,j≤ 3 ,

composed of a fundamental particleψiand an anti-particleψj.

4. The state space of composite particle system (5.2.29) is the tensor product ofk 1 com-

plex spacesCNandk 2 complex conjugate spacesC
N
, expressed as

C︸N⊗ ··· ⊗︷︷ CN︸

k 1

⊗C

N

⊗ ··· ⊗C

N

︸ ︷︷ ︸

k 2

(5.2.31)

=

{ N

∑

i 1 = 1

···

N

∑

ik 1 = 1

N

∑

j 1 = 1

···

N

∑

jk 2 = 1

zi 1 ···ik 1 j 1 ···jk 2 ψi 1 ···ik 1 j 1 ···jk 2 |

zi 1 ···ik 1 j 1 ···jk 2 ∈C,ψi 1 ···ik 1 j 1 ···jk 2 are as in (5.2.30)

}

.

5. Denote the space (5.2.31) as

CN

k 1

⊗C

Nk^2

=C︸N⊗ ··· ⊗︷︷ CN︸

k 1

⊗C

N

⊗ ··· ⊗C

N

︸ ︷︷ ︸

k 2

.

Then a representationH(U)of (5.2.21) forU∈SU(N)is a linear transformation of state
space of composite particle system:

(5.2.32) H(U):CN
k 1
⊗C
Nk 2
→CN
k 1
⊗C
Nk 2
,

which represents state transformation of composite particles, similar to the state transforma-
tion (5.2.27)-(5.2.28) for a single particle system.

6. We recall the irreducible representation ofSU(N)given by (5.2.23) and (5.2.24). The
   irreducible representation implies that there is a decomposition in theNkcomposite particles
   (5.2.29), which are classified intoKgroups as

(5.2.33) G 1 ={Ψ^11 ,···,Ψ^1 m 1 },···,GK={ΨK 1 ,···,ΨKmK},

where each groupGjcontainmjcomposite particlesΨ 1 j,···,Ψmjj, as in (5.2.30), such that
the space (5.2.31) can be decomposed into the direct sum of theKsubspaces of spanGjas

(5.2.34) CN

k 1

⊗C

Nk^2

=E 1 ⊕ ··· ⊕EK,

where

(5.2.35) Ej=spanGj=span{Ψ 1 j,···,Ψmjj}, 1 ≤j≤K,

and then, the linear transformationsH(U)in (5.2.32) are also decomposed into the direct
sum for allU∈SU(N)as in (5.2.23). Namely, under the decomposition (5.2.33) of basis of

CN

k 1

⊗C

Nk^2

,the representations

H(U)∈SU(N)⊗ ··· ⊗SU(N)

︸ ︷︷ ︸

k 1

⊗SU(N)⊗ ··· ⊗SU(N)

︸ ︷︷ ︸

k 2