Mathematical Principles of Theoretical Physics

4.4 Duality and decoupling of Interaction Fields.

forms:

Coulomb gauge :

∂A 1

∂x^1

+

∂A 2

∂x^2

+

∂A 3

∂x^3

= 0 ,

Lorentz gauge : ∂μAμ= 0 ,

Axial gauge : A 3 = 0 ,

Temporal gauge : A 0 = 0.

However, for theSU(N)gauge theory by PLD, the gauge-fixing problem is in general not
well-posed. In fact, theSU(N)gauge field equations are given by

(4.3.50) ∂νFν μa =gψγμτaψ, for 1≤a≤N^2 − 1 ,

(4.3.51) iγμ(∂μ+igGaμτa)ψ−mψ= 0 ,

where
Fν μa =∂νGμa−∂μGaν+gλbcaGbμGcν.

The equations (4.3.50)-(4.3.51) are invariant under theSU(N)gauge transformations

(4.3.52) ψ ̃=eiθ

a

τaψ, G ̃aμτa=Gaμeiθ

bτb

τae−iθ

bτb

−

1

g

∂μθaτa.

Hence, if there is a solution for (4.3.50)-(4.3.51), then there are infinitely many solutions.
Thus, we have to supplementN^2 −1 gauge-fixing equations in order to get a unique physical
solution:

(4.3.53) Fa(Gμ) = 0 for 1≤a≤N^2 − 1.

The reason why takeN^2 −1 equations in (4.3.53) are that there areN^2 −1 free functionsθa
in (4.3.52).
Now, the gauge-fixing problem (4.3.50)-(4.3.51) with (4.3.53) is not well-posed either,
because the number of independent equations of (4.3.50) are 4(N^2 − 1 )due to

∂μ(ψ γμτaψ) 6 = 0.

Namely, the number of independent equations in the gauge-fixing problem (4.3.50)-(4.3.51)
with (4.3.53) isNEQ= 5 (N^2 − 1 )+ 4 N, larger than the number of unknownsNUF= 4 (N^2 −
1 )+ 4 N.
The non well-posedness ofSU(N)gauge-fixing problem implies the the PLD is not ap-
plicable for theSU(N)gauge field theory. However, based on PID, theSU(N)gauge-fixing
problem is well-posed.

4.4 Duality and decoupling of Interaction Fields

The natural duality of four fundamental interactions to be addressed in this section is a di-
rect consequence of PID. It is with this duality, together with the PRI invariant potentialsSμ
andWμgiven by (4.5.1) and (4.6.1), that we establish a clear explanation for many long-
standing challenging problems in physics, including for example the dark matter and dark