2.2. LORENTZ INVARIANCE 57
and by (2.2.45), we derive from Postulate2.23an relativistic equation, called the Klein-
Gordon (KG) equation:
(2.2.51)
[
1
c^2
∂^2
∂t^2
−∇^2 +
(mc
h ̄
) 2 ]
ψ= 0.
The KG equation describes a free boson with spinJ=0.
When an electromagnetic field presents, the operators in (2.2.45) and (2.2.46) are rewrit-
ten as^2
(2.2.52)
E=i ̄h
∂
∂t
−eA 0 , ~P=−ih ̄∇−
e
c
~A,
P 0 =i ̄h(~σ·~D), P 1 =−ih ̄(~α·~D),
whereAμ= (A 0 ,~A)is the electromagnetic potential, and
~D=∇+ie
hc ̄
~A.
Thus, as electromagnetic field presents, the Klein-Gordon equation (2.2.51) is expressed as
(2.2.53)
(
DμDμ−
(mc
̄h
) 2 )
ψ= 0 ,
where
(2.2.54) Dμ=∂μ+i
e
hc ̄
Aμ,
is a 4-dimensional vector operator. The expression (2.2.53) is clearly Lorentz invariant.
2.Weyl equations.Based on the de Broglie relation
E=h ̄ω, P= ̄h/λ, c=ω λ,
we obtain
(2.2.55) E=cP.
The relation (2.2.55) is valid to a massless free fermion, e.g. as neutrinos. InsertingEandP 0
in (2.2.45) and (2.2.46) into (2.2.55), we obtain that
(2.2.56)
∂ ψ
∂t
=c(~σ·∇)ψ,
which are called the Weyl equations describing the free massless neutrinos. Hereψ=
(ψ 1 ,ψ 2 )Tis a two component Weyl spinor.
3.Dirac equations.For a massive fermion, the de Broglie matter-wave duality relation
can be generalized in the form
E= ̄hω±mc^2 , P= ̄h/λ, c=ω λ.
(^2) Hereeis the electric charge of an electron, and is negative.