250 CHAPTER 4. UNIFIED FIELD THEORY
- Violation of PRI.In the classical electroweak theory, a key ingredient is thelinear
combinations ofWμ^3 andBμas given by (4.6.70). By PRI,
Wμ^3 is the third component of aSU( 2 )tensor{Wμa},
Bμ is theU( 1 )gauge field.Hence, for the combinations of two different types of tensors:
Zμ=cosθwWμ^3 +sinθwBμ,
Aμ=−sinθwWμ^3 +cosθwBμ,their field equations (4.6.69) must vary under generalSU( 2 )generator transformations as
follows
(4.6.74) σ ̃a=xbaσb.
In other words, such linear combinations violates PRI.
3.Decoupling obstacle.The classical electroweak theory has a difficulty for decoupling
the electromagnetic and the weak interactions. In reality,electromagnetism and weak interac-
tion often are independent to each other. Hence, as a unified theory for both interactions, one
should be able to decouple the model to study individual interactions. However, the classical
electroweak theory manifests a radical decoupling obstacle.
In fact, it is natural to require that under the condition
(4.6.75) Wμ±= 0 , Zμ= 0 ,
the WS field equations (4.6.69) should return to theU( 1 )gauge invariant Maxwell equations.
But we see that
Aμ=cosθwBμ−sinθwWμ^3 ,
whereBμis aU( 1 )gauge field, andWμ^3 is a component ofSU( 2 )gauge field. Therefore,Aμis
not independent ofSU( 2 )gauge transformation. In particular, the condition (4.6.75) means
(4.6.76) Wμ^1 = 0 , Wμ^2 = 0 , Wμ^3 =−tgθwBμ.
Hence, as we take the transformation (4.6.74),Wμabecomes
W ̃μ^1
W ̃μ^2
W ̃μ^3
=
y^13 Wμ^3
y^23 Wμ^3
y^33 Wμ^3
, (yba)T= (xba)−^1.It implies that under a transformation (4.6.74), a nonzero weak interaction can be generated
from a zero weak interaction field of (4.6.75)-(4.6.76):
W ̃μ± 6 = 0 , Z ̃μ 6 =0 asya 36 = 0 ( 1 ≤a≤ 3 ),and the nonzero electromagnetic fieldAμ 6 =0 will become zero:
A ̃μ=0 as y^33 =cotθw.