Mathematical Principles of Theoretical Physics

274 CHAPTER 5. ELEMENTARY PARTICLES

To understand the implications of (5.2.18) and (5.2.19), we have to know the irreducible
representation of the tensor products

(5.2.20) SU(N)⊗ ··· ⊗SU(N)
︸ ︷︷ ︸
k 1

⊗SU(N)⊗ ···SU(N)

︸ ︷︷ ︸

k 2

,

and its physical significance, which will be discussed in more detail in the next two subsec-
tions. Here we just give a brief introduction to the irreducible representation of (5.2.20).
A representation of (5.2.20) is a mapping defined as

(5.2.21)

H:SU(N)→SU(N)⊗ ··· ⊗SU(N)

︸ ︷︷ ︸

k 1

⊗SU(N)⊗ ··· ⊗SU(N)

︸ ︷︷ ︸

k 2

,

H(U) =U︸⊗ ··· ⊗︷︷ U︸

k 1

⊗U︸⊗ ··· ⊗︷︷ U︸

k 2

forU∈SU(N).

We can show that mapping (5.2.21) satisfies (5.2.6), i.e.

H(U 1 ·U 2 ) =H(U 1 )H(U 2 ) ∀U 1 ,U 2 ∈SU(N).

By Definition (5.2.17) for tensor products of matrices, each representationH(U)in (5.2.21)
is anNk-th order matrix, and is also a linear transformation of the complex space as

(5.2.22) H(U):X→X, X=C︸N⊗ ··· ⊗︷︷ CN︸

k 1

⊗C

N

⊗ ··· ⊗C

N

︸ ︷︷ ︸

k 2

.

It is (5.2.16) that bestows the physical implication of the representation (5.2.21)-(5.2.22) of
SU(N), which will be explained in the next subsection.
Based on the irreducible representation theory ofSU(N), ifk=k 1 +k 2 ≥2 andN≥2, the
representation (5.2.21) must be reducible, i.e.H(U)can be split into smaller block diagonal
form:

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

Usually, (5.2.23) is simply denoted as

(5.2.24) N︸⊗ ··· ⊗︷︷ N︸

k 1

⊗N︸⊗ ··· ⊗︷︷ N︸

k 2

=m 1 ⊕ ··· ⊕mK,

wheremjis the order ofHj(U). In Subsection5.2.4we shall give the computational method
of (5.2.23) (or (5.2.24)) by the Young tableau.

5.2.3 Physical explanation of irreducible representations

The physical implications of irreducible representationsofSU(N)were revealed first by
Sakata in 1950’s. In the following we give the Sakata explanation in a few steps.

1. The dimensionNofSU(N)representsNparticles:

(5.2.25) ψ 1 ,···,ψN,