208 CHAPTER 4. UNIFIED FIELD THEORY
The weak and strong forces are given by
(4.3.45)
weak force: Fw=−gw∇W 0 ,weak magnetic force: Fwm=gw
c~v×curlW~,and
(4.3.46)
strong force : Fs=−gs∇S 0 ,
strong magnetic force : Fsm=gs
c~v×curl~S.Remark 4.12.It is the PRI that provides a physical approach to combine theN^2 −1 com-
ponents of theSU(N)gauge fields into the forms (4.3.41)-(4.3.46) for the interacting forces.
With this physical interpretation, the electromagnetic, weak and strong interactions can be re-
garded as a unified force, separated by three different interaction charges: the electric charge
e, weak chargegw, strong chargegs.
4.3.5 Gauge-fixing problem
For a gauge theory, one has to supplement gauge-fixing equations to ensure a unique physical
solution. In theU( 1 )gauge theory the gauge-fixing problem is well-posed. However for the
SU(N)gauge theory, the problem is generally not well-posed underthe Principle of Lagrange
Dynamics (PLD).
We first recall the classicalU( 1 )gauge theory describing electromagnetism. The field
equations by PLD are given by
(4.3.47) ∂ν(∂νAμ−∂μAν) =eψγμψ (γμ=gμ νγν),
(4.3.48) iγμ(∂μ+ieAμ)ψ−mψ= 0 ,
which are invariant under theU( 1 )gauge transformation
ψ ̃=eiθψ, A ̃μ=Aμ−1
e∂μθ.It implies thatA ̃μ,ψ ̃are also solutions. Hence, the equations (4.3.47)-(4.3.48) have infinitely
many solutions. However, in these solutions only one is physical. To find the physical solu-
tion, one has to provide a supplementary equation, called gauge-fixing equation,
(4.3.49) F(Aμ) = 0 ,
such that the system (4.3.47)-(4.3.49) has a unique physical solution. Observe that
∂μ∂ν(∂νAμ−∂μAν) = 0 ,
∂μψ γμψ= 0 (by( 4. 3. 48 )).Only three equations in (4.3.47) are independent. Therefore the number of independent equa-
tions in (4.3.47)-(4.3.49) isNEQ=8, the same as the number of unknowns. Hence, (4.3.47)-
(4.3.49) are well-posed. Usually physical gauge-fixing equation (4.3.47) takes one of the