Mathematical Principles of Theoretical Physics

6 CHAPTER 1. GENERAL INTRODUCTION

1.3 First Principles of Four Fundamental Interactions

The four fundamental interactions of Nature are the gravitational interaction, the electromag-
netism, the weak and the strong interactions. Seeking laws of the four fundamental interac-
tions is the most important human endeavor. In this section,we demonstrate that laws for the
four fundamental interactions are determined by the following principles:

1) the principle of general relativity, the principle of gaugeinvariance, and

the principle of Lorentz invariance, together with the simplicity principle

of laws of Nature, dictates the Lagrangian actions of the four interactions,

and

2) the principle of interaction dynamics and the principle of representation

invariance determines the field equations.

Symmetry principles

We have shown that for gravity, the basic symmetry principleis the Einstein PGR, which
dictates the Einstein-Hilbert action, and induces the gravitational field equations (1.2.7) using
PID.
Quantum mechanics provides a mathematical description about the Nature in the molecule,
the atomic and subatomic levels. Modern theory and experimental evidence have suggested
that the electromagnetic, the weak and the strong interactions obey the gauge symmetry. In
fact, these symmetries and the Lorentz symmetry, together with the simplicity of laws of
Nature dictate the Lagrangian actions for the electromagnetic, the weak and the strong inter-
actions:

Symmetry Dictates Actions of fundamental interactions:

(a) The principle of general relativity dictates the actionfor gravity, the Einstein-

Hilbert action.

(b) The principle of Lorentz invariance and the principle ofgauge invariance,

together with the simplicity principle of laws of Nature, dictate the La-

grangian actions for the electromagnetic, the weak and the strong interac-

tions.

This represents clearly the intrinsic beauty of Nature.
The abelianU( 1 )gauge theory describes quantum electrodynamics (QED). Thenon-
abelianSU(N)gauge theory was originated from the early work of (Weyl, 1919 ;Klein, 1938 ;
Yang and Mills, 1954 ). Physically, gauge invariance refers to the conservationof certain
quantum property of the underlying interaction. Such quantum property of theNparticles
with wave functions cannot be distinguished for the interaction, and consequently, the energy
contribution of theseNparticles associated with the interaction is invariant under the general
SU(N)phase (gauge) transformations:

(1.3.1) (Ψ ̃, G ̃aμτa) =

(

ΩΨ,GaμΩτaΩ−^1 +

i

g

(∂μΩ)Ω−^1

)

∀Ω=eiθ

k(x)τk

∈SU(N).