Mathematical Principles of Theoretical Physics

182 CHAPTER 4. UNIFIED FIELD THEORY

which also stands for the scalar curvature of the spinor bundle:M⊗p(C^4 )^2. Here

Wμ νa =∂μWνa−∂νWμa+gwλbcaWμbWνc for 1≤a≤ 3.

4.Strong interaction. The strong fields are theSU( 3 )gauge fields

Skμ= (Sk 0 ,Sk 1 ,Sk 2 ,Sk 3 ) for 1≤k≤ 8 ,

and the action is

(4.1.8) LS=−

1

4

GklsSkμ νSμ νl,

which corresponds to the scalar curvature ofM⊗p(C^4 )^3. Here

Skμ ν=∂μSkν−∂νSkμ+gsΛkrlSrμSlν for 1≤k≤ 8.

The Yukawa Interaction Mechanism, briefly mentioned in Section2.1.6and restated be-
low, is the main reason why the weak interaction is describedby anSU( 2 )gauge theory and
the strong interaction is described by anSU( 3 )gauge theory.
One great vision of Albert Einstein is his principle of equivalence, which says that gravity
is manifested as the curved effect of the space-time manifold{M,gμ ν}. Based on the recent
work by the authors (Ma and Wang,2015a,2014h,d), the Einstein vision leads us to postulate
the Geometric Interaction Mechanism2.13, which is restated here for convenience:

Geometric Interaction Mechanism 4.1.The gravitational force is the curved effect of the
time-space, and the electromagnetic, weak, strong interactions are the twisted effects of the
underlying complex vector bundlesM⊗pCn.

Yukawa’s viewpoint, entirely different from Einstein’s, is that the other three fundamen-
tal forces—the electromagnetism, the weak and the strong interactions–take place through
exchanging intermediate bosons:

Yukawa Interaction Mechanism 4.2.The four fundamental interactions of Nature are me-
diated by exchanging interaction field particles, called the mediators. The gravitational force
is mediated by the graviton, the electromagnetic force is mediated by the photon, the strong
interaction is mediated by the gluons, and the weak interaction is mediated by the intermedi-
ate vector bosons W±and Z.

It is the Yukawa mechanism that leads to theSU( 2 )andSU( 3 )gauge theories respectively
for the weak and the strong interactions. In fact, the three mediatorsW±andZfor the weak
interaction are regarded as theSU( 2 )gauge fieldsWμa( 1 ≤a≤ 3 ), and the eight gluons for

the strong interaction are considered as theSU( 3 )gauge fieldsSkμ( 1 ≤k≤ 8 ). Of course,
the three color quantum numbers for the quarks are an important evidence for choosing the
SU( 3 )gauge theory to describe the strong interaction.
The two interaction mechanisms lead to two entirely different directions to develop the
unified field theory. The need for quantization for all current theories for the four interactions
is based on the Yukawa Interaction Mechanism. The new unifiedfield theory in this article is