Mathematical Principles of Theoretical Physics

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.