1.4 Symmetry and Symmetry-Breaking
with the metricgμ νrepresenting the gravitational potential. In other words,gravity is mani-
fested as the curved effect of the space-time manifold{M,gμ ν}. In essence, gravity is mani-
fested by the gravitational fields{gμ ν,∇μφ}, determined by the gravitational field equations
(1.2.7) and (1.2.8) together with the matter distribution{Tμ ν}.
The gauge theory provides a field theory for describing the electromagnetic, the weak and
the strong interactions. The geometry of theSU(N)gauge theory is determined by
1) the complex bundle^2 M⊗p(C^4 )Nfor the wave functionsΨ= (ψ 1 ,···,ψN)T, repre-
senting theNparticles,2) the gauge interacting fields{Gaμ|a= 1 ,···,N^2 − 1 }, and their dual fields{φa|a=
1 ,···,N^2 − 1 }, and3) the gauge field equations (1.3.5) coupled with the Dirac equations (1.3.6).In other words, the geometry of the complex bundleM⊗p(C^4 )N, dictated by the gauge
field equations (1.3.5) together with the matter equations (1.3.6), manifests the underlying
interaction.
Consequently, it is natural for us to postulate the Geometric Interaction Mechanism4.1
for all four fundamental interactions:
Geometric Interaction Mechanism (Ma and Wang,2014d)1) (Einstein, 1915 ) The gravitational force is the curved effect of the time-
space; and
2) the electromagnetic, weak, strong interactions are the twisted effects of the
underlying complex vector bundlesM⊗pCn.We note that Yukawa’s viewpoint, entirely different from Einstein’s, is that the other three
fundamental forces—the electromagnetism, the weak and thestrong interactions–take place
through exchanging intermediate bosons such as photons forthe electromagnetic interaction,
the W±and Z intermediate vector bosons for the weak interaction, and the gluons for the
strong interaction.
It is worth mentioning that the Yukawa Mechanism is orientedtoward to computing the
transition probability for particle decays and scattering, and the above Geometric Interaction
Mechanism is oriented toward to establishing fundamental laws, such as interaction poten-
tials, of the four interactions.
1.4 Symmetry and Symmetry-Breaking
As we have discussed so far, symmetry plays a fundamental role in understanding Nature.
In mathematical terms, each symmetry, associated with particular physical laws, consists
(^2) Throughout this book, we use the notation⊗pto denote ”gluing a vector space to each point of a manifold” to
form a vector bundle. For example,
M⊗pCn=
⋃
p∈M
{p} ×Cn
is a vector bundle with base manifoldMand fiber complex vector spaceCn.