4.4. DUALITY AND DECOUPLING OF INTERACTION FIELDS 213
represents the force due to the nonlinear coupling ofgμ νand its dualφ. Formula (4.4.11) can
be approximatively written as
(4.4.13)
F=mMG(
−
1
r^2−
k 0
r+k 1 r)
,
k 0 = 4 × 10 −^18 km−^1 , k 1 = 10 −^57 km−^3.The formula (4.4.13) shows that a central gravitational field with massMhas an attract-
ing force−k 0 /rin addition to the Newtonian gravitational force. This explains the dark
matter phenomenon. Also there is a repelling forcek 1 r, which explains the dark energy phe-
nomenon; see (Ma and Wang,2014e) for details.
4.4.3 Modified QED model
For the electromagnetic interaction only, the decoupled QED field equations from (4.3.22)
and (4.3.25) are given by
∂ν(∂νAμ−∂μAν)−eJμ=(
∂μ+βe
hc ̄
Aμ)
(4.4.14) φe,
iγμ(
∂μ+i
e
hc ̄Aμ)
ψ−
mc
h ̄(4.4.15) ψ= 0 ,
whereβis a dimensionless constant, andJμ=ψ γμψis the current density satisfying
(4.4.16) ∂μJμ= 0.
Equations (4.4.14) and (4.4.15) are the modified QED model. Taking divergence on both
sides of (4.4.14), by (4.4.16) and
∂μ∂ν(∂νAμ−∂μAν) = 0 ,the equations (4.4.14)-(4.4.15) can be equivalently written as
∂ν(∂νAμ−∂μAν)−eJμ=(
∂μ+βe
hc ̄
Aμ)
(4.4.17) φe,
∂μ∂μφe+βe
̄hc(4.4.18) ∂μ(Aμφe) = 0 ,
iγμ(
∂μ+ieAμ)
ψ−
mc
h ̄(4.4.19) ψ= 0.
If we take(4.4.20) H=curl~A, E=−
1
c∂~A
∂t−∇φ,whereAμ= (φ,~A),~A= (A 1 ,A 2 ,A 3 ), then (4.4.17)-(4.4.18) and (4.4.20) are a modified ver-