53
2.5. The argument reversed: the gauge principle
We then checked its gauge invariance under the combined transformation
A → A
′ = A + ∇χ
′
V → V
= V − ∂χ/∂t
(2.47)
ψ → ψ
′ = exp(iqχ)ψ.
We now want to reverse the argument: we shall start by demanding that our
theory is invariant under the space–time-dependent phase transformation
ψ(x, t) → ψ
′ (x, t) = exp[iqχ(x, t)]ψ(x, t).
(2.48)
We shall demonstrate that such a phase invariance is not possible for a free
theory, but rather requires an interacting theory involving a (4-vector) field
whose interactions with the charged particle are precisely determined, and
which undergoes the transformation
A → A
′ = A + ∇χ
(2.49)
′
V → V
= V − ∂χ/∂t
(2.50)
when ψ → ψ
′ . The demand of this type of phase invariance will have then
dictated the form of the interaction – this is the basis of the gauge principle.
Before proceeding we note that the resulting equation – which will of course
turn out to be (2.29) – will not strictly speaking be invariant under (2.48),
but rather covariant (in the gauge sense), as we saw in the preceding section.
Nevertheless, we shall in this section sometimes continue (slightly loosely) to
speak of ‘local phase invariance’. When we come to implement these ideas
in quantum field theory in chapter 7 (section 7.4), using the Lagrangian formalism, we shall see that the relevant Lagrangians are indeed invariant under
(2.48).
We therefore focus attention on the phase of the wavefunction. The absolute phase of a wavefunction in quantum mechanics cannot be measured; only
relative phases are measurable, via some sort of interference experiment. A
simple example is provided by the diffraction of particles by a two-slit system.
Downstream from the slits, the wavefunction is a coherent superposition of
two components, one originating from each slit: symbolically,
ψ = ψ 1 + ψ 2 .
(2.51)
The probability distribution |ψ|
2 will then involve, in addition to the separate
intensities |ψ 1 |
2 and |ψ 2 |
2 , the interference term
2 Re(ψ 1
∗ ψ 2 ) = 2|ψ 1 ||ψ 2 | cos δ
(2.52)
where δ (= δ 1 −δ 2 ) is the phase difference between components ψ 1 and ψ 2 . The
familiar pattern of alternating intensity maxima and minima is then attributed
to variation in the phase difference δ. Where the components are in phase,
the interference is constructive and |ψ|
2 has a maximum; where they are out
2.5. The argument reversed: the gauge principle
We then checked its gauge invariance under the combined transformation
A → A
′ = A + ∇χ
′
V → V
= V − ∂χ/∂t
(2.47)
ψ → ψ
′ = exp(iqχ)ψ.
We now want to reverse the argument: we shall start by demanding that our
theory is invariant under the space–time-dependent phase transformation
ψ(x, t) → ψ
′ (x, t) = exp[iqχ(x, t)]ψ(x, t).
(2.48)
We shall demonstrate that such a phase invariance is not possible for a free
theory, but rather requires an interacting theory involving a (4-vector) field
whose interactions with the charged particle are precisely determined, and
which undergoes the transformation
A → A
′ = A + ∇χ
(2.49)
′
V → V
= V − ∂χ/∂t
(2.50)
when ψ → ψ
′ . The demand of this type of phase invariance will have then
dictated the form of the interaction – this is the basis of the gauge principle.
Before proceeding we note that the resulting equation – which will of course
turn out to be (2.29) – will not strictly speaking be invariant under (2.48),
but rather covariant (in the gauge sense), as we saw in the preceding section.
Nevertheless, we shall in this section sometimes continue (slightly loosely) to
speak of ‘local phase invariance’. When we come to implement these ideas
in quantum field theory in chapter 7 (section 7.4), using the Lagrangian formalism, we shall see that the relevant Lagrangians are indeed invariant under
(2.48).
We therefore focus attention on the phase of the wavefunction. The absolute phase of a wavefunction in quantum mechanics cannot be measured; only
relative phases are measurable, via some sort of interference experiment. A
simple example is provided by the diffraction of particles by a two-slit system.
Downstream from the slits, the wavefunction is a coherent superposition of
two components, one originating from each slit: symbolically,
ψ = ψ 1 + ψ 2 .
(2.51)
The probability distribution |ψ|
2 will then involve, in addition to the separate
intensities |ψ 1 |
2 and |ψ 2 |
2 , the interference term
2 Re(ψ 1
∗ ψ 2 ) = 2|ψ 1 ||ψ 2 | cos δ
(2.52)
where δ (= δ 1 −δ 2 ) is the phase difference between components ψ 1 and ψ 2 . The
familiar pattern of alternating intensity maxima and minima is then attributed
to variation in the phase difference δ. Where the components are in phase,
the interference is constructive and |ψ|
2 has a maximum; where they are out