70 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
and the constantλis usually called the cosmological constant. Today, a largenumber of
physical experiments manifest thatλ=0. However, the recent discovery of the acceleration
of our universe leads to some physicists to think thatλ 6 =0.
In Chapter 7 , based on the unified field theory developed by the authors, wepresent a
theory of dark matter and dark energy, which clearly explains the phenomena of dark matter
and dark energy, and shows that the constantλshould be zero, i.e.λ=0.
2.4 Gauge Invariance
2.4.1 U( 1 )gauge invariance of electromagnetism
Gauge symmetry is one of fundamental invariance principlesof physics, and determines the
Lagrange actions of the electromagnetic, the weak and the strong interactions. In order to
show the origin of gauge theory, we first introduce theU( 1 )gauge invariance of electromag-
netic fields.
Fermions under an electromagnetic field with potentialAμobey the Dirac equations:
iγμDμψ−mc
̄h(2.4.1) ψ= 0 ,
(2.4.2) Dμ= (∂μ+ieAμ),
where the electromagnetic potentialAμsatisfies the Maxwell equations (2.2.40):
∂νFμ ν=4 π
c(2.4.3) Jμ,
Fμ ν=gμ αgν β(
∂Aα
∂xβ−
∂Aβ
∂xα)
(2.4.4).
It is easy to see that the system of equations (2.4.1)-(2.4.4) is invariant under the following
transformation:
(2.4.5) ψ ̃=eiθψ, A ̃μ=Aμ−
1
e∂μθ.Sinceeiθ∈U( 1 ), (2.4.5) is called aU( 1 )-gauge transformation.
Now, we consider the gauge invariance from another point of view. Letψbe a Dirac
spinor describing a fermion:
ψ:M^4 →M^4 ⊗pC^4.
Experimentally, we cannot observe the phase angles ofψ. Namely, under a phase rotation
transformation
(2.4.6) ψ ̃=eiθψ, θ=θ(xμ),
we cannot distinguish the two statesψ ̃andψexperimentally. Mathematically speaking, the
phenomenon amounts to saying that the Dirac equations are covariant under the transforma-
tion (2.4.6). This covariance requires that the derivative be covariant:
(2.4.7) D ̃μψ ̃=eiθDμψ.