4.4. DUALITY AND DECOUPLING OF INTERACTION FIELDS 217
- Strong force. The strong interaction forces are governed by the field equations
(4.3.31)-(4.3.35). The decoupled field equations are given by
∂νSkν μ−gs
̄hcfijkgα βSiα μSβj−gsQkμ=[
∂μ−1
4
k^2 sxμ+gsδ
̄hcSμ]
(4.4.39) φsk,
∂μ∂μφsk−k^2 φsk+1
4
k^2 sxμ∂μφsk+gsδ
hc ̄
(4.4.40) ∂μ(Sμφsk)
=−gs∂μQkμ−gs
̄hcfijkgα β∂μ(Siα μSβj),iγμ[
∂μ+i
gs
̄hcSlμτl]
ψ−
mc
̄h(4.4.41) ψ= 0 ,
for 1≤k≤8, whereδis a parameter,andSμis as in (4.3.41).
In the next section, we shall deduce the layeredformulas of strong interaction potentials
from the equations (4.4.39)-(4.4.41) with gauge equations (4.4.38).
Remark 4.16.We need to explain the physical significance of the parametersksandδ.
Usually,ksandδare regarded as masses of the field particles. However, since(4.4.39)-
(4.4.41) are the field equations for the interaction forces, the parametersksandδare no
longer viewed as masses. In fact,k−^1 represents the range of attracting force for the strong
interaction, and
(
gsφs^0
̄hcδ)− 1
is the range of the repelling force, whereφs^0 is a ground state ofφs.
4.4.5 Weak interaction field equations
The unified field model (4.3.21)-(4.3.25) can be decoupled to study the weak interaction only,
leading to the following weak interaction field equations:
∂νWν μa −gw
̄hcεbcagα βWα μbWβc−gwJμa=[
∂μ−1
4
(m
Hc
h ̄) 2
xμ+gw
hc ̄
γbwWμb]
(4.4.42) φwa,
iγμ[
∂μ+i
gw
hc ̄Wμaσa]
ψ−
mc
h ̄(4.4.43) ψ= 0 ,
wheremHrepresents the mass of the Higgs particle,σa=σa( 1 ≤a≤ 3 )are the Pauli
matrices as in (3.5.36), and
(4.4.44)
Wμ νa =∂μWνa−∂νWμa+gw
hc ̄εabcWμbWνc,Jaμ=ψ γμσaψ, γμ=gμ νγν.
Taking divergence on both sides of (4.4.42) we get∂μ∂μφwa−(mHc
̄h) 2
φwa+gw
hc ̄
γbw∂μ(Wμbφwa)−1
4
(
mHc
h ̄
(4.4.45) )^2 xμ∂μφwa
=−gw
hc ̄εbcagα β∂μ(Wα μbWβc)−gw∂μJaμ.Also, we need to supplement (4.4.42)-(4.4.43) with three additional 3 gauge equations to
compensate the induced dual fieldsφwa:
(4.4.46) Fwa(Wμ,φw,ψ) = 0 for 1≤a≤ 3.