Subatomic Physics

410 The Electroweak Theory of the Standard Model
These currents can now be introduced in the equation of motion. As we learned
in chapter 12, the form of this equation is dictated by gauge invariance and the
interaction is generated thereby. The leptons e and ν are light, so that a nonrelativistic
equation of motion cannot be used except at very low energies for the
electron. The Schrödinger equation must be modified, since it has first-order time
derivatives but second-order space derivatives; it is, consequently, not relativistically
invariant. The problem was solved by Dirac, who invented an equation that
is first order in both space and time. We will not introduce the Dirac equation, but
use the fact that the electroweak theory is most important at high energies where
the lepton masses can be neglected. The generalization of the Schrödinger equation
for a particle of zero mass was found by Weyl, and it can be written down easily for
a massless electron by noting that the only observables are spin and momentum.
The simplest equation consequently is
i�∂ψ
∂t
= σ · pψ = −i�σ · ∇ψ.
This equation has the right form but is not general enough for the electroweak
theory. We generalize it by introducing the vector Vµ =(V0, V ) as in Eq. (13.13)
and writing (9)
∂ψ
i�V0 = V · pcψ = −i�cV · ∇ψ. (13.20)
∂t
The vector Vµ is related to the spin of the fermion. In the presence of the electromagnetic
field, gauge invariance dictates that the derivatives ∂/∂t and ∇ be
replaced by D0 and D, so that Eq. (13.20) becomes
i�cV0D0ψ = i�V0
� �
∂ ieA0
+ ψ
∂t �
= −i�cV · Dψ = −i�cV ·
�
∇ − ieA
�
ψ. (13.21)
�c
This equation applies to particles of charge e. In the electroweak theory, we need
gauge invariance with respect to both the isoscalar B field and the isovector W
fields. The latter are non-commuting and thus non-Abelian. We also have both
the neutrino and electron to consider. The left-handed components of the electron
and the neutrino couple to both the isovector weak, W , and isoscalar, B, fields,
whereas the right-handed component of the electron couples only to the isosinglet
field B, since it does not participate in the weak interaction. Consider the equation
of motion for eR first. For a free electron, Eq. (13.20) can be used. In the presence
of the B field, we have from Eq. (13.21)
�
∂ g′
i�V0 − i
∂t � B0
�
�
Y
ψeR = −i�cV · ∇ + i
2
g′
�
Y
B ψeR, (13.22)
�c 2
9Merzbacher, Ch. 24.