Gauge invariance (and covariance) in quantum mechanicsmechanics

The Lorentz force law for a non-relativistic particle of charge q moving with velocity v under the influence of both electric and magnetic fields is

F = qE+qv ×B. (2.27)

It may be derived, via Hamilton’s equations, from the classical Hamiltonian² H = 1

(p −qA)²+qV. (2.28)

2m

The Schr¨odinger equation for such a particle in an electromagnetic field is

( )

1 ∂ψ(x, t)

(−i∇ −qA)²+qV ψ(x, t) = i (2.29)

2m ∂t

which is obtained from the classical Hamiltonian by the usual prescription, p → −i∇, for Schr¨odinger’s wave mechanics (ħ = 1). Note the appearance of the operator combinations

D ≡∇ −iqA

(2.30) D⁰ ≡∂/∂t+ iqV

in place of ∇ and ∂/∂t, in going from the free-particle Schr¨odinger equation to the electromagnetic field case.

The solution ψ(x, t) of the Schr¨odinger equation (2.29) describes com­

pletely the state of the particle moving under the influence of the potentials V, A. However, these potentials are not unique, as we have already seen:

they can be changed by a gauge transformation

A→A^′ = A+∇χ (2.31)

V →V ′ = V −∂χ/∂t (2.32) and the Maxwell equations for the fields E and B will remain the same.

This immediately raises a serious question: if we carry out such a change of potentials in equation (2.29), will the solution of the resulting equation describe the same physics as the solution of equation (2.29)? If it does, we shall be able to assume the validity of Maxwell’s theory for the quan­

tum world; if not, some modification will be necessary, since the gauge sym­

metry possessed by the Maxwell equations will be violated in the quantum theory.

2Wesetħ =c=1throughout(seeappendixB).

The answer to the question just posed is evidently negative, since it is clear that the same ‘ψ’ cannot possibly satisfy both (2.29) and the analogous equation with (V,A) replaced by (V ,′ A^′ ). Unlike Maxwell’s equations, the Schr¨odinger equation is not gauge invariant. But we must remember that the wavefunction ψ is not a directly observable quantity, as the electromagnetic fields E and B are. Perhaps ψ does not need to remain unchanged (invari­

ant) when the potentials are changed by a gauge transformation. In fact, in order to have any chance of ‘describing the same physics’ in terms of the gauge-transformed potentials, wewillhavetoallow ψ tochangeaswell. This is a crucial point: for quantum mechanics to be consistent with Maxwell’s equations it is necessary for the gauge transformations (2.31) and (2.32) of the Maxwell potentials to be accompanied also by a transformation of the quantum-mechanical wavefunction, ψ → ψ ^′ , where ψ ^′ satisfies the equation

( )

1 _′ ∂ψ ^′ (x, t)

(−i∇ − qA^′ )²+qV ψ ^′ (x, t) = i . (2.33)

2m ∂t

Note that the form of (2.33) is exactly the same as the form of (2.29) – it is this that will effectively ensure that both ‘describe the same physics’. Readers of appendix D will expect to be told that – if we can find such a ψ ^′ – we may then assert that (2.29) is gaugecovariant, meaning that it maintains the same form under a gauge transformation. (The transformations relevant to this use of ‘covariance’ are gauge transformations.)

Since we know the relations (2.31) and (2.32) between A, V and A^′ , V , ′

we can actually find what ψ ^′ (x, t) must be in order that equation (2.33) be consistent with (2.29). We shall state the answer and then verify it; then we shall discuss the physical interpretation. The required ψ ^′ (x, t) is

ψ ^′ (x, t) = exp[iqχ(x, t)]ψ(x, t) (2.34) where χ is the same space–time-dependent function as appears in equations (2.31) and (2.32). To verify this we consider

(−i∇ − qA^′ )ψ ^′ = [−i∇ −qA − q(∇χ)][exp(iqχ)ψ]

= q(∇χ) exp(iqχ)ψ + exp(iqχ) · (−i∇ψ)

+ exp(iqχ) · (−qAψ) − q(∇χ) exp(iqχ)ψ. (2.35) The first and the last terms cancel leaving the result:

(−i∇ − qA^′ )ψ ^′ = exp(iqχ) · (−i∇ − qA)ψ (2.36) which may be written using equation (2.30) as:

(−iD^′ ψ ^′ ) = exp(iqχ) ·(−iDψ). (2.37) Thus, although the space–time-dependent phase factor feels the action of the gradient operator ∇, it ‘passes through’ the combined operator D^′ and con­

verts it into D: in fact comparing the equations (2.34) and (2.37), we see that

51 2.4. Gaugeinvariance (andcovariance) inquantum mechanics

D^′ ψ ^′ bears to Dψ exactly the same relation as ψ ^′ bears to ψ. In just the same way we find (cf equation (2.30))

(iD^0′ ψ ^′ ) = exp(iqχ) · (iD⁰ψ) (2.38) where we have used equation (2.32) for V ^′ . Once again, D^0′ ψ ^′ is simply related to D⁰ψ. Repeating the operation which led to equation (2.37) we find

1 (−iD^′ )²ψ ^′ = exp(iqχ) · 1

(−iD)²ψ

2m 2m

= exp(iqχ) · iD⁰ψ (using equation (2.29))

= iD^0′ ψ ^′ (using equation (2.30)). (2.39) Equation (2.39) is just (2.33) written in the D notation of equation (2.30), so we have verified that (2.34) is the correct relationship between ψ ^′ and ψ to ensure consistency between equations (2.29) and (2.33). Precisely this consistency is summarized by the statement that (2.29) is gauge covariant.

Do ψ and ψ ^′ describe the same physics, in fact? The answer is yes, but it is not quite trivial. It is certainly obvious that the probability densities |ψ|² and |ψ ^′ |² are equal, since in fact ψ and ψ ^′ in equation (2.34) are related by a phase transformation. However, we can be interested in other observables involving the derivative operators ∇ or ∂/∂t– for example, the current, which is essentially ψ^∗(∇ψ) − (∇ψ)^∗ψ. It is easy to check that this current is not invariant under (2.34), because the phase χ(x, t) is x-dependent. But equations (2.37) and (2.38) show us what we must do to construct gauge-invariant currents: namely, we must replace ∇ by D (and in general also

∂/∂t by D⁰) since then:

ψ ^∗′ (D^′ ψ ^′ ) = ψ ^∗ exp(−iqχ)· exp(iqχ) ·(Dψ) = ψ ^∗ Dψ (2.40) for example. Thus the identity of the physics described by ψ and ψ ^′ is indeed ensured. Note, incidentally, that the equality between the first and last terms in (2.40) is indeed a statement of (gauge) invariance.

We summarize these important considerations by the statement that the gauge invariance of Maxwell equations re-emerges as a covariance in quantum mechanics provided we make the combined transformation

A → A^′= A+∇χ

V → V ′ = V −∂χ/∂t (2.41) ψ → ψ ^′ = exp(iqχ)ψ

on the potential and on the wavefunction.

The Schr¨odinger equation is non-relativistic, but the Maxwell equations are of course fully relativistic. One might therefore suspect that the prescriptions discovered here are actually true relativistically as well, and this is indeed

the case. We shall introduce the spin-0 and spin-₂¹ relativistic equations in chapter 3. For the present we note that (2.30) can be written in manifestly Lorentz covariant form as

D^μ ≡ ∂^μ + iqA^μ (2.42)

in terms of which (2.37) and (2.38) become

−iD ^′μψ ^′ = exp(iqχ) · (−iD^μψ). (2.43) It follows that any equation involving the operator ∂^μ can be made gauge invariant under the combined transformation

A^μ → A ^′μ = A^μ −∂^μχ ψ → ψ ^′ = exp(iqχ)ψ

if ∂^μ is replaced by D^μ. In fact, we seem to have a very simple prescription for obtaining the wave equation for a particle in the presence of an electro­

magnetic field from the corresponding free particle wave equation: make the replacement

∂^μ → D^μ ≡ ∂^μ + iqA^μ . (2.44) In the following section this will be seen to be the basis of the so-called ‘gauge principle’ whereby, in accordance with the idea advanced in the previous sec­

tions, the form of the interaction is determined by the insistence on (local) gauge invariance.

One final remark: this new kind of derivative

D^μ ≡ ∂^μ + iqA^μ (2.45)

turns out to be of fundamental importance – it will be the operator which generalizes from the (Abelian) phase symmetry of QED (see comment (iii) of section 2.6) to the (non-Abelian) phase symmetry of our weak and strong interaction theories. It is called the ‘gauge covariant derivative’, the term being usually shortened to ‘covariant derivative’ in the present context. The geometrical significance of this term will be explained in volume 2.