Mathematical Principles of Theoretical Physics

28 CHAPTER 1. GENERAL INTRODUCTION

representation invariant (PRI), and 3) Lorentz invariant.The field equations are then derived
by using PID:

Gab

[

∂νGbν μ−gλcdbgα βGcα μGdβ

]

−gΨγμτaΨ=

[

∂μ−

k^2

4

xμ+gGμ+gG^0 μ

]

(1.11.5) φa,

iγμ

[

∂μ+igG^0 μ+igGaμτa

]

(1.11.6) Ψ−MΨ= 0 ,

for 1≤a≤N^2 −1, whereG^0 μis the interaction field of external systems, andGμ=αaGaμ. It

is important to note that coupling to the external fields is achieved by the terms involvingG^0 μ.

Unification and geometrization of matter fields

Also, we have established a unified field model coupling matter fields, which matches the
vision of Einstein and Nambu, as stated in Nambu’s Nobel lecture (Nambu, 2008 ):

Einstein used to express dissatisfaction with his famous equation of gravity

Gμ ν= 8 πTμ ν

His point was that, from an aesthetic point of view, the left hand side of the equa-

tion which describes the gravitational field is based on a beautiful geometrical

principle, whereas the right hand side, which describes everything else,...

looks arbitrary and ugly.

... [today] Since gauge fields are based on a beautiful geometrical principle, one

may shift them to the left hand side of Einsteins equation. What is left on the

right are the matter fields which act as the source for the gauge fields ... Can one

geometrize the matter fields and shift everything to the left?

Basically, one needs to geometrize the energy-momentum tensorTμ νappearing in the
Einstein field equations. For example, for multi-particle system under gravity and electro-
magnetism, using the basic postulates as outlined above, a unified field model can be natu-
rally derived so that the energy-momentum tensorTμ νis derived from first principles and is
geometrized.
The above vision of Einstein and Nambu is achieved by using the postulate for interact-
ing multi-particle systems, the PID, the PRI, as well as the principle of symmetry-breaking
(PSB):

1) fundamental symmetries of Nature dictate the actions for both the subsys-

tems as well as the global system,

2) the subsystems are coupled through PID, PRI and PSB.

We believe that this is the essence of physical modeling and aunique route for unification.

1.12 Weakton Model of Elementary Particles

The weakton model of elementary particles was first introduced by (Ma and Wang,2015b).
Hereafter we address the basic ingredients of this theory.