Mathematical Principles of Theoretical Physics

72 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

whereFμ νis given by (2.4.4). The left-hand side of (2.4.13) is covariant under the gauge
transformation (2.4.5), and so doesFμ ν. Consequently the contraction

(2.4.14) F=Fμ νFμ ν,

is invariant under both the gauge and the Lorentz transformations.
Thus, the invariantFin (2.4.14) is the main part of the action of theU( 1 )gauge fieldAμ,
which is given by

(2.4.15) LEM=

∫

M^4

[

−

1

4

F+

4 π

c

AμJμ

]

dxdt.

This is the action of the Maxwell field.
The form of (2.4.15) is similar to that of the Einstein-Hilbert action (2.3.35). In mathemat-
ics, the gauge fieldAμin (2.4.2) is the connection of the complex vector bundleM^4 ⊗pC^4 ,
with which the Dirac spinorψis defined:

ψ:M^4 →M^4 ⊗pC^4.

The tensorFμ νin (2.4.13) is the curvature tensor of the bundleM^4 ⊗pC^4 , andF=Fμ νFμ νin
(2.4.15) is the scalar curvature. In addition, the current energyAμJμin (2.4.15) corresponds
to the energy-momentumgμ νSμ νin (2.3.35).
In the same fashion as the electromagnetism, we can derive the gauge theories for both
the weak and the strong interactions from theSU( 2 )andSU( 3 )gauge invariances. In the
next subsections we shall introduce the mathematical theory ofSU(N)gauge fields and the
principle of gauge invariance in physics.

2.4.2 Generator representations ofSU(N)

Both the weak and strong interactions are described by theSU(N)gauge theory, which is a
generalization of theU( 1 )gauge theory introduced in the last subsection.
In anSU(N)gauge theory, there areNwave functions, representingNfermions:

Ψ= (ψ^1 ,···,ψN)T,

where eachψk( 1 ≤k≤N)is a 4-component Dirac spinor. We seek for a set of gauge fields
Gkμsuch that the Dirac equations

(2.4.16)

[

iγμDμ−

cm

h ̄

]

Ψ= 0

are invariant under theSU(N)gauge transformation

(2.4.17) Ψ ̃=ΩΨ ∀Ω=eiθ

k(x)τk

∈SU(N),

wheremis the mass matrix, andθkare real parameters,

(2.4.18) Dμ=∂μ+igGkμτk,