2.2. LORENTZ INVARIANCE 59
the left-hand side of (2.2.60) should be
(2.2.62)
(
iγμ ̃∂μ−mc
h ̄)
ψ ̃=R(
iγμ∂μ−mc
h ̄)
ψ,where ̃∂μ=∂/∂x ̃μ, and
(2.2.63) ψ ̃=Rψ.
HereRis a 4×4 matrix depending on the Lorentz matrixLin (2.2.61). To determine the
matrixR, we note that
(2.2.64)
∂
∂ ̃xμ
=Lνμ∂
∂xν
, (Lνμ) =L.Inserting (2.2.63) and (2.2.64) into (2.2.62), we deduce that
(
iR−^1 γμLνμR∂
∂xν+
mc
̄h)
ψ=(
iγν∂
∂xν+
mc
h ̄)
ψ.It follows that
R−^1 γμLνμR=γν,
which is equivalent to
(2.2.65) R−^1 γμR=Lνμγν forμ= 0 , 1 , 2 , 3.
Hence the covariance of (2.2.62) is equivalent to the transformation matrixRin (2.2.63)
obeying equations (2.2.65).
To derive an explicit form ofRin (2.2.65), we need to write the Lorentz matrixLin the
form
L
μ
ν=
coshθ −sinhθ 0 0
−sinhθ coshθ 0 0
0 0 1 0
0 0 0 1
,
where coshθand sinhθare the hyperbolic functions andθsatisfies
coshθ=1
√
1 −v^2 /c^2, sinhθ=v/c
√
1 −v^2 /c^2.
In this form, the equations (2.2.65) can be written as
(2.2.66)
R−^1 γ^0 R=coshθ γ^0 −sinhθ γ^1 ,
R−^1 γ^1 R=−sinhθ γ^0 +coshθ γ^1 ,
R−^1 γ^2 R=γ^2 ,
R−^1 γ^3 R=γ^3.