Mathematical Principles of Theoretical Physics

10 CHAPTER 1. GENERAL INTRODUCTION

gauge interacting fieldsAμand{Wμa}^3 a= 1 , corresponding to two different gauge groupsU( 1 )
for electromagnetism andSU( 2 )for the weak interaction, the following combination

(1.3.9) αAμ+βWμ^3

is prohibited. The reason is thatAμis anU( 1 )-tensor with tensor, andWμ^3 is simply the
third component of anSU( 2 )-tensor. The above combination violates PRI. This point of view
clearly shows that the classical electroweak theory violates PRI, so does the standard model.
The difficulty comes from the artificial way of introducing the Higgs field. The PID based
approach for introducing Higgs fields by the authors appearsto be the only model obeying
PRI.

Another important consequence of PRI is that for the termαbGbμin the right-hand side of

the PID gauge field equations (1.3.5), both{αb|b= 1 ,···,N^2 − 1 }and{Gbμ|b= 1 ,···,N^2 −
1 }areSU(N)tensors under the representation transformations (1.3.8).
The coefficientsαbrepresent the portions distributed to the gauge potentialsby the charge,
represented by the coupling constantg. Then it is clear that

In the field equations (1.3.5) and (1.3.6) of the SU(N)gauge theory for an fun-

damental interaction,

(a) the coupling constant g represents the interaction charge, playing the same

role as the electric charge e in the U( 1 )abelian gauge theory for quantum

electrodynamics (QED);

(b) the potential

(1.3.10) Gμ

def

=αbGbμ

represents the total interacting potential, where the SU(N)covectorαb

represents the portions of each interacting potential Gbμcontributed to the

total interacting potential; and

(c) the temporal component G 0 and the spatial components~G= (G 1 ,G 2 ,G 3 )

represent, respectively the interaction potential and interaction magnetic

potential. The force and the magnetic force generated by theinteraction

are given by:

F=−g∇G 0 , Fm=

g

c

~v×curl~G,

where∇and curl are the spatial gradient and curl operators.

2.1.6 Geometric interaction mechanism

A simple yet the most challenging problem throughout the history of physics is the mech-
anism or nature of a force.
One great vision of Albert Einstein is his principle of equivalence, which, in the mathe-
matical terms, says that the space-time is a 4-dimensional (4D) Riemannian manifold{M,gμ ν}