Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

164 CHAPTER 3. MATHEMATICAL FOUNDATIONS


The structure constants are


λabc = 2 fabc, 1 ≤a,b,c≤ 8 ,

andfabcare anti-symmetric, given by


f 123 = 1 , f 147 =f 246 =f 257 =f 345 =

1


2


,


f 156 =f 367 =−

1


2


, f 458 =f 678 =


(3.5.39) 3 / 2 ,


fabc= 0 for others.

By (3.5.39) we can deduce that


λadcλcbd=

{


0 ifa 6 =b,
12 ifa=b.

Hence we get


gab=

1


12


λadcλcbd=δab.

Namely, under the Gell-Mann representation (3.5.38),(gab) =I.
We are now in position to consider generalSU(N). In fact, for allN≥2, there exists
a generator basis{τa| 1 ≤a≤N^2 − 1 }ofSU(N)such that(gab) = 41 N(λadcλcbd) =I. The
generatorsτaare given by the following traceless Hermitian matrices:
(3.5.40)


τ
( 1 )
1 =

(


σ 1 0
0 0

)


, τ
( 1 )
2 =

(


σ 2 0
0 0

)


, τ
( 1 )
3 =

(


σ 3 0
0 0

)


,


τ
( 2 )
1 =

(


λ 4 0
0 0

)


, τ
( 2 )
2 =

(


λ 5 0
0 0

)


, τ
( 2 )
2 =

(


λ 6 0
0 0

)


,


τ
( 2 )
4 =

(


λ 7 0
0 0

)


, τ
( 2 )
5 =

(


λ 8 0
0 0

)


,


..


.


τ 1 (N−^1 )=




0 ··· 1


..


.


..


.


1 ··· 0




, τ
(N− 1 )
2 =




0 ··· −i
..
.

..


.


i ··· 0



,


τ
(N− 1 )
3 =






0 0 ··· 0


0 0 ··· 1


..


.


..


.


..


.


0 1 ··· 0







, τ
(N− 1 )
4 =






0 0 ··· 0


0 0 ··· −i
..
.

..


.


..


.


0 i ··· 0






,


..


.


τ 2 (NN−−^11 )=


2



N(N− 1 )





1 0


..


. 1


0 −(N− 1 )




,

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