248 CHAPTER 4. UNIFIED FIELD THEORY
which requires the following transformation ofSU( 2 )generators from the Pauli matricesσa:
σ ̃ 1
σ ̃ 2
σ ̃ 3
=√^1
2
1 i 0
0 −i 0
0 0√
2
σ 1
σ 2
σ 3
.
In addition, we also need to obtain the massmZofZboson by diagonalizing the massive
matrix (4.6.64). It is easy to see that
(4.6.66)
UMU†=
c^2
h ̄^2
m^2 W 0 0 0
0 mW^2 0 0
0 0 m^2 Z 0
0 0 0 0
,
U=
√^1
2
√i
2 0 0
√^1
2 −
√i
2 0 0
0 0 α β
0 0 −β α
, α=g 1
|g|
, β=g 2
|g|,
where|g|=
√
g^21 +g^22 , and(4.6.67)
c^2 mW^2
̄h^2=
a^2
2g^21 ,c^2 mZ
̄h^2=
a^2
2|g|^2.- The field equations governingW±andZbosons are obtained from the equations
(4.6.63) under the following transformation
(4.6.68)
Wμ+
Wμ−
Zμ
Aμ
=U
Wμ^1
Wμ^2
Wμ^3
Bμ
forU as in (4.6.66).In this case, the equations (4.6.63) become
(4.6.69)
∂νWν μ+−(mWc
h ̄) 2
Wμ+
∂νWν μ−−(mWc
h ̄) 2
Wμ−
∂νZν μ−(mZc
h ̄) 2
Zμ
∂νAν μ
=
√g^1
2 J+
μ
√g^1
2 J−
μ
|g|Jμ^0
−eJemμ
+higher order terms.wheree=g 1 g 2 /|g|, and
J±μ=1
2
(Jμ^1 ±iJμ^2 ),J^0 μ=1
2 |g|^2(g^21 J^3 μ−g^22 JμL− 2 g^22 JμR),Jemμ =1
2
(Jμ^3 +JμL+ 2 JRμ).