8 CHAPTER 1. GENERAL INTRODUCTION
2) The action (1.3.4) satisfy the invariance of general relativity, Lorentz in-
variance, gauge invariance and the gauge representation invariance.
3) The states(g,A,ψ)are the extremum points of (1.3.4) with the divA-free
constraint.We now illustrate that PID is also valid for both the weak and strong interactions, replac-
ing the classical PLD.
First, PID applied to the Yang-Mills action (1.3.2) takes the following form:Gab[
∂νGbν μ−gλcdbgα βGcα μGdβ]
(1.3.5) −gΨγμτaΨ= (∂μ+αbGbμ)φa,
(1.3.6) (iγμDμ−m)Ψ= 0 ,
where the Dirac equations (1.3.6) are the gauge covariant, and are the dynamic equations for
fermions participating the interaction. The right hand side of (1.3.5) is due to PID, with the
operator
∇A=∂μ+αbGbμ
taken in such a way thatαbGbμis PRI-covariant under the representation transformations
(1.3.8) below.
One important consequence of the PIDSU(N)theory is the natural introduction of the
scalar fieldsφa, reminiscent of the Higgs field in the standard model in particle physics.
Second, for the Yang-Mills action for anSU(N)gauge theory, the resulting Yang-Mills
equations are the Euler-Lagrange equations of the Yang-Mills action:
Gab[
∂νGbν μ−gλcdbgα βGcα μGdβ]
(1.3.7) −gΨγμτaΨ= 0 ,
supplemented with the Dirac equations (1.3.6).
Historically, the most important difficulty encountered bythe gauge theory and the field
equations (1.3.7) is that the gauge vectorial field particles{Gaμ}for the weak interaction are
massless, in disagreement with experimental observations.
A great deal of efforts have been made toward to the modification of the gauge theory by
introducing proper mass generation mechanism. A historically important breakthrough is the
discovery of the mechanism of spontaneous symmetry breaking in subatomic physics. Al-
though the phenomenon was discovered in superconductivityby Ginzburg-Landau in 1951,
the mechanism of spontaneous symmetry breaking in particlephysics was first proposed by
Y. Nambu in 1960; see (Nambu, 1960 ;Nambu and Jona-Lasinio,1961a,b). The Higgs mech-
anism, introduced in (Higgs, 1964 ;Englert and Brout, 1964 ;Guralnik, Hagen and Kibble,
1964 ), is an ad hoc method based on the Nambu-Jona-Lasinio spontaneous symmetry break-
ing, leading to the mass generation of the vector bosons for the weak interaction.
In all these efforts associated with Higgs fields, the modification of the Yang-Mills action
are artificial as in the case for modifying the Einstein-Hilbert action for gravity.
As we indicated before, the Yang-Mills action is uniquely determined by the gauge sym-
metry, together with the simplicity principle of laws of Nature. All artificial modification of
the action will only lead to certain approximations of the underlying physical laws.