Mathematical Principles of Theoretical Physics

188 CHAPTER 4. UNIFIED FIELD THEORY

5) SU( 3 )gauge transformation onM⊗p(C^4 )^3 :

(4.1.25)

Ω=eiθ

kτk

:(C^4 )^3 →(C^4 )^3 ∈SU( 3 ),

(

ψ ̃s,S ̃kμτk,m ̃s

)

=

(

Ωψs,SkμΩτkΩ−^1 +

i

gs

∂μΩΩ−^1 ,ΩmsΩ−^1

)

.

6) SU( 2 )representation transformation onTeSU( 2 ):

(4.1.26)

X= (xba):TeSU( 2 )→TeSU( 2 ), (yab)T=X−^1 ,

σ ̃s=xbaσb, (G ̃abw) =X(Gwab)XT,

W ̃μa=yabWμb.

7) SU( 3 )representation transformation onTeSU( 3 ):

(4.1.27)

X= (xlk):TeSU( 3 )→TeSU( 3 ), (ykl)T=X−^1 ,

τ ̃k=xlkτl, (G ̃kls) =X(Gskl)XT,

S ̃kμ=yklSlμ.

Remark 4.3.Here we adopt the linear transformations of the bundle spaces instead of the
coordinate transformations in the base manifoldM. In this case, the two transformations
(4.1.21) and (4.1.22) are compatible. Otherwise, we have to introduce the Vierbein tensors to
overcome the incompatibility between the Lorentz transformation and the general coordinate
transformation.

4.1.4 Gauge symmetry-breaking

In physics, symmetries are displayed in two levels in the laws of Nature:

(4.1.28) the invariance of Lagrangian actionsL,

(4.1.29) the covariance of variation equations ofL.

The following three symmetries:

(4.1.30)

the Einstein principle of general relativity (PGR),

the Lorentz invariance,

the principle of representation invariance (PRI),

represent the universality of physical laws— the validity of laws of Nature is independent of
the coordinate systems expressing them. Consequently, thesymmetries in (4.1.30) cannot be
broken at both levels of (4.1.28) and (4.1.29).
The physical implication of the gauge symmetry, however, isdifferent at the two levels:

(1) the gauge invariance of the Lagrangian action, (4.1.28), says that the energy contribu-

tions of particles in a physical system are indistinguishable; and