Subatomic Physics

12.1. Introduction 385
D0 ≡ 1 ∂ iqA0
+
c ∂t �c ,
D ≡ ∇ − iqA
�c .
(12.5)
If these derivatives replace the normal ones, (1/c)∂/∂t and ∇, it follows with
Eqs. (7.1) and (12.2) that
D ′ 0ψ ′ q = D ′ 0UQψq = UQD0ψq,
D ′ ψ ′ q = D ′ UQψq = UQDψq, (12.6)
where D ′ 0 and D ′ have A ′ 0 and A ′ as dependent variables. It is important to note
that if UQ stands to the left of D0 and D, it is a simple phase factor, since the
derivatives only act on quantities to their right. With the introduction of the gauge
covariant derivatives, D0 and D transform under local gauge transformations just
like 1
c
∂
∂t
and ∇ do under a global gauge transformation (ɛ = constant). The vec-
tor nature of the compensating field which appears in the covariant derivative is
determined by the vector property of the momentum p for the free Hamiltonian
and the time dependence of Eq. (7.1). When the covariant derivative is introduced
in the Schrödinger equation, including the compensating field, the resulting particle
Hamiltonian has the form Eq. (12.1). Thus, the requirement of local gauge
invariance generates the qA0 and j · A interaction of a charged particle with the
electromagnetic field. We note, in addition, that space and time transformations
are tied together.
So far, we have neglected the equation of motion for the vector field (A0, A). In
the case of the electromagnetic field, it is given by Maxwell’s equations (i = x, y, z)
1
c2 1
c2 if we use the Lorentz condition
∂2A0 ∂t2 − ∇2A0 = ρ = ψ ∗ qψ,
∂2Ai ∂t2 − ∇2Ai = ji
c
= ψ∗ qvi
c ψ,
(12.7)
1 ∂A0
+ ∇·A =0. (12.8)
c ∂t
The equations (12.7) are invariant under the gauge transformations, Eq. (12.4), if
we impose the condition
1
c2 • In four-vector notation Eq. (12.7) becomes
�
�Aµ ≡ gα ν∇α∇νAµ = jµ
c .
�
∂ 2 ɛ(x,t)
∂t 2 − ∇2 ɛ(x,t)=0. (12.9)