Mathematical Principles of Theoretical Physics

4.4. DUALITY AND DECOUPLING OF INTERACTION FIELDS 211

then there must exist a dual graviton with spinJ=1. In fact, for all interaction mediators, we
have the following duality correspondence:

(4.4.3)

graviton(J= 2 ) ↔ dual vector graviton(J= 1 ),

photon(J= 1 ) ↔ dual scalar photon(J= 0 ),

W±bosons(J= 1 ) ↔ charged HiggsH±(J= 0 ),

Zboson(J= 1 ) ↔ neutral HiggsH^0 (J= 0 ),

gluonsgk(J= 1 ) ↔ dual scalar gluonsφgk(J= 0 ).

The neutral HiggsH^0 (the adjoint particle ofZ) had been discovered experimentally. We
remark that the duality (4.4.3) can also be derived using the weakton model (Ma and Wang,
2015b), which is also presented in the next chapter.

2. Duality of interacting forces. The correspondence (4.4.2) provides a dual relation
   between the attracting and repelling forces. In fact, from the interaction potentials we find
   that the even-spin fields yield attracting forces, and the odd-spin fields yield repelling forces.

Duality of Interaction Forces 4.14.Each interaction generates both attracting and repelling
forces. Moreover, for each pair of dual fields, the even-spinfield generates an attracting force,
and the odd-spin field generates a repelling force.

This duality of interaction forces is illustrated as follows:

(4.4.4)

Gravitation force = attraction due togμ ν+repelling due toφμg,

Electromagnetism = attraction due toφe+repelling due toAμ,

Weak force = attraction due toφw+repelling due toWμ,

Strong force = attraction due toφs+repelling due toSμ.

Here, we remark that the electromagnetic force in (4.4.4) is between the charged particles
with the same sign, and the force generated byφeis attractive.

4.4.2 Gravitational field equations derived by PID

If we only consider the gravitational interaction, then thegravitational field equations can be
decoupled from the unified field model (4.3.21)-(4.3.25), and are given by

(4.4.5) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν+

[

∇μ+

e

hc ̄

Aμ

]

Φν,

where the term ̄hceAμΦνrepresents the coupling between the gravitation and the cosmic mi-
crowave background (CMB) radiation.
By the Bianchi identity (4.2.2), taking divergence on both sides of (4.4.5) yields

(4.4.6) ∇μ∇μΦν+

e

̄hc

∇μ(AμΦν) =

8 πG

c^4

∇μTμ ν.

The duality of gravity is based on the field equations (4.4.5) and (4.4.6).