55
2.5. The argument reversed: the gauge principle
then the wavefunction ψ
′ , given by the local phase transformation, will not,
since both ∇ and ∂/∂t now act on α(x, t) in the phase factor. Thus local phase
invariance is not an invariance of the free-particle wave equation. If we wish
to satisfy the demands of local phase invariance, we are obliged to modify the
free-particle Schr¨ odinger equation into something for which there is a local
phase invariance – or rather, more accurately, a corresponding covariance.
But this modified equation will no longer describe a free particle: in other
words, the freedom to alter the phase of a charged particle’s wavefunction
locally is only possible if some kind of force field is introduced in which the
particle moves. In more physical terms, the covariance will now be manifested
in the inability to distinguish observationally between the effect of making a
local change in phase convention and the effect of some new field in which the
particle moves.
What kind of field will this be? In fact, we know immediately what the
answer is, since the local phase transformation
ψ → ψ
′ = exp[iα(x, t)]ψ
(2.56)
with α = qχ is just the phase transformation associated with electromagnetic
gauge invariance! Thus we must modify the Schr¨ odinger equation
1 (−i∇)
2 ψ = i∂/∂t
(2.57)
2m
to
1 (−i∇ − qA)
2 ψ = (i∂/∂t − qV )ψ
(2.58)
2m
and satisfy the local phase invariance
ψ → ψ
′ = exp[iα(x, t)]ψ
(2.59)
by demanding that A and V transform by
A → A
′ = A + q
−1
∇α
(2.60)
′
V → V = V − q
−1 ∂α/∂t
when ψ → ψ
′ . The modified wave equation is of course precisely the Schr¨ odinger
equation describing the interaction of the charged particle with the electromagnetic field described by A and V .
In a Lorentz covariant treatment, A and V will be regarded as parts of a
4-vector A
μ , just as −∇ and ∂/∂t are parts of ∂
μ (see problem 2.1). Thus the
presence of the vector field A
μ , interacting in a ‘universal’ prescribed way with
any particle of charge q, is dictated by local phase invariance. A vector field
such as A
μ , introduced to guarantee local phase invariance, is called a ‘gauge
field’. The principle that the interaction should be so dictated by the phase
(or gauge) invariance is called the gauge principle: it allows us to write down
the wave equation for the interaction directly from the free particle equation
2.5. The argument reversed: the gauge principle
then the wavefunction ψ
′ , given by the local phase transformation, will not,
since both ∇ and ∂/∂t now act on α(x, t) in the phase factor. Thus local phase
invariance is not an invariance of the free-particle wave equation. If we wish
to satisfy the demands of local phase invariance, we are obliged to modify the
free-particle Schr¨ odinger equation into something for which there is a local
phase invariance – or rather, more accurately, a corresponding covariance.
But this modified equation will no longer describe a free particle: in other
words, the freedom to alter the phase of a charged particle’s wavefunction
locally is only possible if some kind of force field is introduced in which the
particle moves. In more physical terms, the covariance will now be manifested
in the inability to distinguish observationally between the effect of making a
local change in phase convention and the effect of some new field in which the
particle moves.
What kind of field will this be? In fact, we know immediately what the
answer is, since the local phase transformation
ψ → ψ
′ = exp[iα(x, t)]ψ
(2.56)
with α = qχ is just the phase transformation associated with electromagnetic
gauge invariance! Thus we must modify the Schr¨ odinger equation
1 (−i∇)
2 ψ = i∂/∂t
(2.57)
2m
to
1 (−i∇ − qA)
2 ψ = (i∂/∂t − qV )ψ
(2.58)
2m
and satisfy the local phase invariance
ψ → ψ
′ = exp[iα(x, t)]ψ
(2.59)
by demanding that A and V transform by
A → A
′ = A + q
−1
∇α
(2.60)
′
V → V = V − q
−1 ∂α/∂t
when ψ → ψ
′ . The modified wave equation is of course precisely the Schr¨ odinger
equation describing the interaction of the charged particle with the electromagnetic field described by A and V .
In a Lorentz covariant treatment, A and V will be regarded as parts of a
4-vector A
μ , just as −∇ and ∂/∂t are parts of ∂
μ (see problem 2.1). Thus the
presence of the vector field A
μ , interacting in a ‘universal’ prescribed way with
any particle of charge q, is dictated by local phase invariance. A vector field
such as A
μ , introduced to guarantee local phase invariance, is called a ‘gauge
field’. The principle that the interaction should be so dictated by the phase
(or gauge) invariance is called the gauge principle: it allows us to write down
the wave equation for the interaction directly from the free particle equation