Mathematical Principles of Theoretical Physics

270 CHAPTER 5. ELEMENTARY PARTICLES

K^0 K+

S= 1

S= 0

S=− 1

π

−

π

0

π+

η

I 3

K−

K ̄^0

Q=− 1 Q= 0 Q= 1

Y = 1

Y= 0

Y =− 1

-1 −^1

2 0

1

2 1

I 3

Figure 5.3

1.Irreducible representation of Lie groups. We begin with the definition of represen-
tation of abstract groups. LetGandMbe two groups, andMconsist ofN-th order real or
complex matrices satisfying certain properties. LetHbe a mapping fromGtoM:

(5.2.5) H:G→M,

preserving the multiplication:

(5.2.6) H(g 1 ·g 2 ) =H(g 1 )H(g 2 ) ∀g 1 ,g 2 ∈G.

Then, the imageH(G)⊂Mof groupGis called anN-dimensional representation ofG.
BecauseMis a group of matrices, for anyg∈G,H(g)∈Mis anN-th order matrix,
written as

H(g) =







a 11 ··· a 1 N

..

.

..

.

aN 1 ··· aNN





.

If there exists a matrixA(not necessary inM) such thatA−^1 H(g)A∈Mfor allg∈G, and

(5.2.7) AH(g)A−^1 =







H 1 (g) 0

..

.

0 Hn(g)





 ∀g∈G,

whereHk(g) ( 1 ≤k≤n)aremk-th order matrices with

n

∑

k= 1

mk=N, then the representation

H(G)is reducible, and each block matrixHk(g)in (5.2.7) is a smaller representation ofG. In
mathematics, (5.2.7) can be equivalently expressed as

(5.2.8) H(G) =H 1 (G)⊕ ··· ⊕Hn(G).