334 CHAPTER 6. QUANTUM PHYSICS
In addition to these Hermitians in (6.2.2) and (6.2.5), the following 5 types of particle
current operators are very important in quantum field theory:
(6.2.6)
scalar current operator: I(identity),
pseudo-scalar current operator: γ^5 ,
vector current operator: γμ,
axial vector current operator: γμγ^5 ,tensor current operator: σμ ν=i
2[γμ,γν],and the corresponding currents are
(6.2.7)
scalar current: ρ=ψ†ψ,
pseudo-scalar current: P=ψ†γ^5 ψ,
vector current: Vμ=ψ γμψ,
axial vector current : Aμ=ψ γμγ^5 ψ,
tensor current: Tμ ν=ψ σμ νψ,whereψ=ψ†γ^0 , andγμ( 0 ≤μ≤ 3 )andγ^5 are the Dirac matrices, which are expressed in
the forms:
(6.2.8)
γ^0 =(
I 0
0 −I
)
, γk=(
0 σk
−σk 0)
for 1≤k≤ 3 ,γ^5 =iγ^0 γ^1 γ^2 γ^3 =(
0 I
I 0
)
.
Remark 6.6.The particle currents defined in (6.2.7) are very important in the transition
theory of particle decays and scatterings. In fact, the general form of particle currents is
written as
JAB=ψAγ ψB+h·c (Hermitian conjugate),whereγis a current operator in (6.2.7),ψAandψBare wave functions of particlesAandB.
For example, for theβ-decay
n→p+e−+νe,by the Fermi theory, the transition amplitude is
M=
Gf
√
2(ψeγμψνe)(ψpγμψn)+h·c,whereGfis the Fermi constant, and
γμ=gμ νγν= (−γ^0 ,γ^1 ,γ^2 ,γ^3 ).