Mathematical Principles of Theoretical Physics

220 CHAPTER 4. UNIFIED FIELD THEORY

where

φs=ρkφsk, Qμ=ρkQkμ, λij=ρkλijk, Sμ ν=∂μSν−∂νSμ+

gs

hc ̄

λijSiμSνj.

Based on the superposition property of the strong potentialfor strong charges,Φs=S 0
andφsobey a linear relationship. Namely, the time-componentμ=0 equation of (4.5.6)
and the equation (4.5.7) should be linear. In other words, we have to take eight gaugefixing
equations as in (4.4.38) such that they contain the following two equations:

(4.5.8)

λij

[

∂ν(SνiS 0 j)−gα βSiαoSβj

]

+δS 0 φs= 0 ,

∂μ

[

λijgα βSiα βSβj−δSμφs

]

= 0 ,

Also, in the eight supplement equations we take

(4.5.9) xμ∂μφs= 0 , ∂μSμ= 0 ,

together with the following static assumption:

(4.5.10)

∂S 0

∂t

= 0 ,

∂ φs

∂t

= 0.

With the gauge fixing equations (4.5.8)-(4.5.9) and the static assumption (4.5.10), we
derive from (4.5.6) and (4.5.7) that

−∆Φs=gsQ−

1

4

(4.5.11) k^20 cτ φs,

(4.5.12) −∆φs+k^20 φs=gs∂μQμ,

wherecτis the wave length ofφs,Q=−Q 0.
In the following, we deduce the solutionΦsandφsof (4.5.11)-(4.5.12) in a few steps.
Step 1. Solution of (4.5.12).By definition ofQμ, we have

∂μQμ=ρk∂μψ γμτkψ+ρkψ γμτk∂μψ.

In view of the Dirac equation (4.4.41),

∂μψγμτkψ=i

gs

̄hc

Sμjψ γμτjτkψ+i

mc

h ̄

ψ τkψ,

ψ γμτk∂μψ=−i

gs

̄hc

Sjμψ γμτkτjψ−i

mc

̄h

ψ τkψ.

Hence we arrive at

(4.5.13) ∂μQμ=

igs

hc ̄

ρkSμjψ γμ[τj,τk]ψ=−

2 gs

hc ̄

ρkSμjλjkiQμi

SinceQμi=Qiμis the current density, we have

ρkλijkQiμ=θjμδ(r),