Mathematical Principles of Theoretical Physics

1.3 First Principles of Four Fundamental Interactions

Here theNwave functions:Ψ= (ψ 1 ,···,ψN)Trepresent theNparticle fields, theN^2 − 1
gauge fieldsGaμrepresent the interacting potentials between theseNparticles, and{τa|a=
1 ,···,N^2 − 1 }is the set of representation generators ofSU(N). The gauge symmetry is then
stated as follows:

Principle of Gauge Invariance.The electromagnetic, the weak, and the strong

interactions obey gauge invariance. Namely, the Dirac equations involved in the

three interactions are gauge covariant and the actions of the interaction fields

are gauge invariant under the gauge transformations (1.3.1).

The field equations involving the gauge fieldsGaμare determined by the corresponding
Yang-Mills action, is uniquely determined by both the gaugeinvariance and the Lorentz in-
variance, together with simplicity of laws of nature:

(1.3.2) LYM

(

Ψ,{Gaμ}

)

=

∫

M

[

−

1

4

Gabgμ αgν βGaμ νGbα β+Ψ

(

iγμDμ−

mc

̄h

)

Ψ

]

dx,

which is invariant under both the Lorentz and gauge transformations (1.2.3). HereΨ=Ψ†γ^0 ,
Ψ†= (Ψ∗)Tis the transpose conjugate ofΨ,Gab=^12 tr(τaτb†),λabc are the structure constants
of{τa|a= 1 ,···,N^2 − 1 },Dμis the covariant derivative andGaμ νstands for the curvature
tensor associated withDμ:

(1.3.3)

Dμ=∂μ+igGkμτk,

Gaμ ν=∂μGaν−∂νGaμ+gλbcaGbμGcν.

Principle of Interaction Dynamics (PID) (Ma and Wang,2014e,2015a)

With the Lagrangian action at our disposal, the physical lawof the underlying system
is then represented as the Euler-Lagrangian equations of the action, using the principle of
Lagrangian dynamics (PLD). For example, all laws of classical mechanics can be derived
using PLD. However, in Section1.1, we have demonstrated that the law of gravity obeys
the principle of interaction dynamics (PID), which takes the variation of the Einstein-Hilbert
action under energy-momentum conservation constraint (1.2.6).
We now state the general form of PID, and then illustrate the validity of PID for all
fundamental interactions.
PID was discovered in (Ma and Wang,2014e,2015a), and requires that for the four fun-
damental interactions, the variation be taken under the energy-momentum conservation con-
straints:

PID(Ma and Wang,2014e,2015a)

1) For the four fundamental interactions, the Lagrangian actions are given by

(1.3.4) L(g,A,ψ) =

∫

M

L(gμ ν,A,ψ)

√

−gdx,

where g={gμ ν}is the Riemannian metric representing the gravitational

potential, A is a set of vector fields representing the gauge potentials, and

ψare the wave functions of particles.