4.3. UNIFIED FIELD MODEL BASED ON PID AND PRI 207
4.3.4 Potentials of the weak and strong forces
It is known that theU( 1 )gauge fields
(4.3.36) Aμ= (A 0 ,A 1 ,A 2 ,A 3 )
represent the electromagnetic potentials, with
(4.3.37)
A 0 =the Coulomb potential,
~A=magnetic potential, ~A= (A 1 ,A 2 ,A 3 ),and the electric chargeeis
(4.3.38) e=theU( 1 )gauge coupling constant.
The electromagnetic forces are given by
(4.3.39)
Fe=−e∇A 0 the Coulomb force,
Fm=e
c~v×curl~A the Lorentz force.Now, we consider theSU( 2 )andSU( 3 )gauge fields:(4.3.40)
SU( 2 )gauge fields: Wμa= (W 0 a,W 1 a,W 2 a,W 3 a), 1 ≤a≤ 3 ,
SU( 3 )gauge fields: Skμ= (Sk 0 ,Sk 1 ,Sk 2 ,Sk 3 ), 1 ≤k≤ 8.They areSU(N)tensors withN= 2 ,3, and haveN^2 −1 components. These components will
change under the transformation ofSU(N)generators. Thanks to PRI, theN^2 − 1 (N= 2 , 3 )
gauge fields in (4.3.40) can be combined into two vector fields as in (4.3.30):
(4.3.41)
Wμ=ωaWμa= (W 0 ,W 1 ,W 2 ,W 3 ),
Sμ=ρkSkμ= (S 0 ,S 1 ,S 2 ,S 3 ),which have the same role as (4.3.36)-(4.3.39) for the electro-magneticU( 1 )gauge fields.
In the same spirit as the electromagnetic fields, for the two fields given by (4.3.41), we
have
(4.3.42)
W 0 =the weak force potential,
W~ =the weak magnetic potential, W~ = (W 1 ,W 2 ,W 3 ),and
(4.3.43)
S 0 =the strong force potential,
~S=the strong magnetic potential, ~S= (S 1 ,S 2 ,S 3 ).In addition, the weak and strong chargesgwandgsare
(4.3.44)
weak chargegw=SU( 2 )gauge coupling constant,
strong chargegs=SU( 3 )gauge coupling constant.