210 CHAPTER 4. UNIFIED FIELD THEORY
energy phenomena, the formulas of the weak and strong forces, the quark confinement, the
asymptotic freedom, and the strong interacting potentialsof nucleons. Also, this duality lay a
solid foundation for the weakton model of elementary particles and the energy level theory of
subatomic particles, and gives rise to a new mechanism for sub-atomic decay and scattering.
The unified field model can be easily decoupled to study each individual interaction when
other interactions are negligible. In other words, PID is certainly applicable to each individual
interaction. For gravity, for example, PID offers to a new gravitational field model, leading
to a unified model for dark energy and dark matter (Ma and Wang,2014e).
4.4.1 Duality
In the unified field equations (4.3.21)-(4.3.24), there exists a natural duality between the
interaction fields(gμ ν,Aμ,Wμa,Skμ)and their corresponding dual fields(φμg,φe,φaw,φks):
(4.4.1)
gμ ν ↔ φ
g
μ,
Aμ ↔ φe,
Wμa ↔ φwa for 1≤a≤ 3 ,
Skμ ↔ φsk for 1≤k≤ 8.Thanks to PRI, theSU( 2 )gauge fieldsWμa( 1 ≤a≤ 3 )and theSU( 3 )gauge fieldsSkμ( 1 ≤
k≤ 8 )are symmetric in their indicesa= 1 , 2 ,3 andk= 1 ,···,8 respectively. Therefore, the
corresponding relation (4.4.1) can be also considered as the following dual relation
(4.4.2)
gμ ν ↔ φμg,
Aμ ↔ φe,
{Wμa} ↔ {φwa},
{Skμ} ↔ {φsk}.The duality relation (4.4.1) can be regarded as the correspondence between field particles for
each interaction, and the relation (4.4.2) is the duality of interacting forces. We now address
these two different dualities.
1.Duality of field particles.In the duality relation (4.4.1), if the tensor fields on the left-
hand side are ofk-th order, then their dual tensor fields on the right-hand side are of(k− 1 )-th
order. Physically, this amounts to saying that if a mediatorfor an interaction has spin−k, then
the dual mediator for the dual field has spin−(k− 1 ). Hence, (4.4.1) leads to the following
important physical conclusion:
Duality of Interaction Mediators 4.13.Each interaction mediator possesses a dual field
particle, called the dual mediator, and if the mediator has spin-k, then its dual mediator has
spin-(k− 1 ).
The duality between interaction mediators is a direct consequence of PID used for deriv-
ing the unified field equations. Based on this duality, if there exist a graviton with spinJ=2,