4.1. PRINCIPLES OF UNIFIED FIELD THEORY 191
We are now in position to introduce the principle of representation invariance (PRI). We
end this section by recalling the principle of representation invariance (PRI) first postulated
in (Ma and Wang,2014h). We proceed with theSU(N)representation. In a neighborhood
U⊂SU(N)of the unit matrix, a matrixΩ∈Ucan be written as
Ω=eiθ
aτa
,where(4.1.36) τa={τ 1 ,···,τK} ⊂TeSU(N), K=N^2 − 1 ,is a basis of generators of the tangent spaceTeSU(N); see Section3.5for the mathematical
theory. AnSU(N)representation transformation is a linear transformationof the basis in
(4.1.36) as(4.1.37) ̃τa=xbaτb,whereX= (xba)is a nondegenerate complex matrix.
Mathematical logic dictates that a physically sound gauge theory should be invariant un-
der theSU(N)representation transformation (4.1.37). Consequently, the following princi-
ple of representation invariance (PRI) must be universallyvalid and was first postulated in
(Ma and Wang,2014h).Principle 4.6(Principle of Representation Invariance).All SU(N)gauge theories are invari-
ant under the transformation (4.1.37). Namely, the actions of the gauge fields are invariant
and the corresponding gauge field equations as given by (4.1.33)-(4.1.35) are covariant un-
der the transformation (4.1.37).Both PID and PRI are very important. As far as we know, it appears that the only unified
field model, which obeys not only PRI but also the principle ofgauge symmetry breaking,
Principle4.4, is the unified field model based on PID introduced in this chapter. From this
model, we can derive not only the same physical conclusions as those from the standard
model, but also many new results and predictions, leading tothe solution of a number of
longstanding open questions in physics, including the 10 problems mentioned in Section
4.1.2.
A few further remarks on PID and PRI are now in order.
First, there are strong theoretical, experimental and observational evidence for PID; see
the next section for details.
Second, PID is based on variations with divA-free constraint defined by (4.1.31). Phys-
ically, the divA-free condition: divAX=0 stands for the energy-momentum conservation
constraints.
Third, PRI means that the gauge theory is universally valid,and therefore should be in-
dependent of the choice of the generatorsτaofSU(N). In other words, PRI is basic a logic
requirement.
Fourth, the electroweak interactions is aU( 1 )×SU( 2 )gauge theory coupled with the
Higgs mechanism. An unavoidable feature for the Higgs mechanism is that the gauge fields