1.3. FIRST PRINCIPLES OF FOUR FUNDAMENTAL INTERACTIONS 9
Third, the PIDSU(N)gauge field equations (1.3.5) and (1.3.6) provide a first principle
based mechanism for the mass generation and spontaneous gauge symmetry-breaking: The
αbGbμon the right-hand side of (1.3.5) breaks theSU(N)gauge symmetry, and the mass
generation follows the Nambu-Jona Lasinio idea.
Fourth, one of the most challenging problems for strong interaction is the quark confinement–
no free quarks have been observed. One hopes to solve this mystery with the quantum chro-
modynamics (QCD) based on the classicalSU( 3 )gauge theory. Unfortunately, as we shall
see later that the classicalSU( 3 )Yang-Mills equations produces only repulsive force, and it
is the dual fields in the PID gauge field equations (1.3.5) that give rise to the needed attraction
for the binding quarks together forming hadrons.
Hence experimental evidence of quark confinement, as well asmany other properties
derived from the PID strong interaction model, clearly demonstrates the validity of PID for
strong interactions.
Fifth, from the mathematical point of view, the Einstein field equations are in general non
well-posed, as illustrated by a simple example in Section4.2.2. In addition, for the classical
Yang-Mills equations, the gauge-fixing problem will also pose issues on the well-posedness
of the Yang-Mills field equations; see Section4.3.5. The issue is caused by the fact that there
are more equations than the number of unknowns in the system.PID induced model brings
in additional unknowns to the equations, and resolves this problem.
3.5.5 Representation invariance of gauge theory.
Recently we have observed that there is a freedom to choose the set of generators for
representing elements inSU(N). In other words, basic logic dictates that theSU(N)gauge
theory should be invariant under the following representation transformations of the generator
bases:
(1.3.8) ̃τa=xbaτb,
whereX= (xba)are non-degenerate(N^2 − 1 )-th order matrices. Then we can define naturally
SU(N)tensors under the transformations (1.3.8). It is clear then thatθa,Gaμ, and the structure
constantsλabc are allSU(N)-tensors. In addition,Gab=^12 Tr(τaτb†)is a symmetric positive
definite 2nd-order covariantSU(N)-tensor, which can be regarded as a Riemannian metric on
SU(N). Consequently we have arrived at the following principle ofrepresentation invariance,
first discovered by the authors (Ma and Wang,2014h):
PRI (Ma and Wang,2014h). For the SU(N)gauge theory, under the represen-
tation transformations (1.3.8),1) the Yang-Mills action (1.3.2) of the gauge fields is invariant, and
2) the gauge field equations (1.3.5) and (1.3.6) are covariant.It is clear that PRI is a basic logic requirement for anSU(N)gauge theory, and has
profound physical implications.
First, as indicated in (Ma and Wang,2014h,2013a) and in Chapter 4 , the field model
based on PID appears to be the only model which obeys PRI. In fact, based on PRI, for the