Mathematical Principles of Theoretical Physics

78 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

whereλabc are the structure constants ofSU(N).
In the classicalSU(N)gauge theory, theSU(N)generatorτk( 1 ≤k≤K)are taken to be
Hermitian and traceless, and satisfy

1

2

tr(τaτb†) =δab.

In this case, the Lagrange densityFin (2.4.48) becomes

F=Fμ νaFμ νa.

The Lagrange action of anSU(N)gauge theory is usually taken in the following form,
called the Yang-Mills action:

(2.4.50) LYM=

∫

M^4

[

−

1

4

Fμ νaFμ νa+Ψ

(

iγμDμ−

cm

̄h

)

Ψ

]

dx,

whereΨ=Ψ†γ^0 ,

(2.4.51)

Fμ νa =∂μGνa−∂νGaμ+gλbcaGbμGcν,

Dμ=∂μ+igGaμτa.

The second term on the right-hand side of (2.4.50) is the action for the Dirac equations
(2.4.30)-(2.4.32).

2.4.4 Principle of gauge invariance

In Sections2.4.2-2.4.3, we introduced the mathematical framework ofSU(N)gauge fields,
leading to the following principle of gauge invariance.

Principle 2.32(Gauge Invariance).The electromagnetic, the weak, and the strong interac-
tions obey gauge invariance. Namely, the motion equations involved in the three interactions
are gauge covariant and the actions of the interaction fieldsare gauge invariant.

A few remarks are now in order.

Remark 2.33.The Standard Model in particle physics is currently a prevailing theory de-
scribing all, except the gravity, fundamental interactions. It consists of the Glashow-Weinberg-
Salam (GWS) electroweak theory, the transition theory of weak interaction decay, the quark
model, and the Quantum Chromodynamics (QCD). Based on the Standard Model, the strong
interaction is described by anSU( 3 )gauge theory, and the electromagnetic and weak in-
teractions are unified in an action ofU( 1 )×SU( 2 )gauge fields, combing with the Higgs
mechanism and the Yukawa coupling.

Remark 2.34.All up-to-date experiments illustrate that the electromagnetic, weak, strong
(EWS) interactions obey Principle of Gauge Invariance2.32.