4.1.3 Gauge Invariance#

Prompts

How do \(\psi\), \(\boldsymbol{A}\), and \(\Phi\) each transform under a gauge transformation? Why must they transform together for the Schrödinger equation to be invariant?
What quantities are gauge-invariant (physical and observable) and what are gauge-dependent (unphysical, convention)? Give examples of each.
Why is canonical momentum \(\hat{\boldsymbol{p}}\) gauge-dependent while kinetic momentum \(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\) is gauge-invariant? Which one corresponds to what you’d measure in an experiment?
“Gauge symmetry is not a symmetry of nature — it is a redundancy of description.” What does this mean? How is it different from a true physical symmetry like rotation invariance?

Lecture Notes#

Overview#

Physical quantities must be independent of the choice of gauge. Different physicists can choose different gauges—Coulomb, Lorenz, radiation—for convenience, but all must arrive at identical predictions for energy levels, transition probabilities, and observable properties. This section clarifies what is gauge-invariant (the real physics) and what is gauge-dependent (mere mathematical convention), and explains why gauge symmetry is a redundancy of description, not a physical symmetry.

Gauge Transformations#

A gauge transformation modifies the potentials \(\boldsymbol{A}\) and \(\Phi\) while simultaneously changing the wavefunction phase, leaving all physical observables unchanged.

Gauge Transformation Rules

Under a gauge transformation parameterized by an arbitrary scalar function \(\alpha(\boldsymbol{x}, t)\):

\[\psi \to \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi, \qquad \boldsymbol{A} \to \boldsymbol{A} + \nabla\alpha, \qquad \Phi \to \Phi - \partial_t\alpha\]

All three act in concert: the simultaneous change ensures that the Schrödinger equation \(\mathrm{i}\hbar\partial_t\psi = \hat{H}\psi\) with \(\hat{H} = \frac{1}{2m}(\hat{\boldsymbol{p}} - q\boldsymbol{A})^2 + q\Phi\) takes the same form in the new gauge.

Derivation: Gauge Invariance of the Schrödinger Equation

Set \(\psi' = \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\). The key identity is:

\[(\hat{\boldsymbol{p}} - q\boldsymbol{A}')\psi' = \mathrm{e}^{\mathrm{i}q\alpha/\hbar}(\hat{\boldsymbol{p}} - q\boldsymbol{A})\psi\]

This holds because applying \(\hat{\boldsymbol{p}} = -\mathrm{i}\hbar\nabla\) to \(\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\) generates a term \(q\nabla\alpha\) that cancels the shift \(\boldsymbol{A} \to \boldsymbol{A} + \nabla\alpha\). Squaring:

\[(\hat{\boldsymbol{p}} - q\boldsymbol{A}')^2\psi' = \mathrm{e}^{\mathrm{i}q\alpha/\hbar}(\hat{\boldsymbol{p}} - q\boldsymbol{A})^2\psi\]

For the time derivative:

\[\mathrm{i}\hbar\partial_t\psi' = -q(\partial_t\alpha)\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi + \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\mathrm{i}\hbar\partial_t\psi\]

The \(-q\partial_t\alpha\) term is compensated by \(\Phi \to \Phi - \partial_t\alpha\) in the Hamiltonian, giving \(\mathrm{i}\hbar\partial_t\psi' = \hat{H}'\psi'\). \(\checkmark\)

Gauge-Invariant vs Gauge-Dependent Quantities#

Physical vs Conventional Quantities

Quantity	Status	Reason
\(\boldsymbol{E}\), \(\boldsymbol{B}\) (fields)	Gauge-invariant	Derivatives of potentials; \(\nabla\alpha\) terms cancel
\(\vert\psi\vert^2\) (probability density)	Gauge-invariant	Phase shift cancels in \(\vert\psi\vert^2\)
\(\boldsymbol{j}\) (probability current)	Gauge-invariant	Built from kinetic momentum
Energy eigenvalues, transition probabilities	Gauge-invariant	Spectrum and overlaps are physical
\(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\) (kinetic momentum)	Gauge-invariant	Shift in \(\hat{\boldsymbol{p}}\) cancels shift in \(q\boldsymbol{A}\)
\(\boldsymbol{A}\), \(\Phi\) (potentials)	Gauge-dependent	Directly shifted by \(\alpha\)
\(\hat{\boldsymbol{p}} = -\mathrm{i}\hbar\nabla\) (canonical momentum)	Gauge-dependent	Depends on phase convention
Phase of \(\psi\)	Gauge-dependent	Transforms as \(\psi \to \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\)

Canonical vs Kinetic Momentum

In the presence of electromagnetic fields, the canonical momentum \(\hat{\boldsymbol{p}} = -\mathrm{i}\hbar\nabla\) is gauge-dependent and not directly observable. What experiments measure is the kinetic momentum:

\[\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\]

Different gauge choices give different values of \(\langle\hat{\boldsymbol{p}}\rangle\), but all agree on \(\langle\hat{\boldsymbol{\pi}}\rangle\). The kinetic momentum appears in the probability current, energy expressions, and equations of motion.

Discussion

When you expand a wavefunction in the plane-wave basis \(\mathrm{e}^{\mathrm{i}\boldsymbol{p}\cdot\boldsymbol{r}/\hbar}\), you are expanding in canonical momentum eigenstates. In the presence of a magnetic field, canonical and kinetic momenta differ. Is the momentum you measure in a time-of-flight experiment the canonical or kinetic momentum? How can two physicists in different gauges both report correct “momentum measurements”?

Gauge Symmetry as Redundancy#

Gauge = Redundancy of Description

Gauge symmetry is fundamentally different from physical symmetries like rotations or translations:

Physical symmetry (e.g., rotation): maps one state to a different state.
Gauge symmetry: maps one description to another description of the same state.

The set of gauge-equivalent descriptions forms an equivalence class. The physical state is the class itself, not any particular gauge choice. This redundancy is nonetheless powerful: the requirement that theories be gauge-invariant constrains what interactions are possible, leading directly to the minimal coupling prescription of §4.1.1.

Gauge Fixing#

In practice, we choose a gauge to perform calculations. The choice is arbitrary—all gauges give identical physics—but some simplify particular problems.

Common Gauge Choices

Coulomb gauge (\(\nabla \cdot \boldsymbol{A} = 0\)): Useful for nonrelativistic atomic and solid-state physics. \(\Phi\) becomes the instantaneous Coulomb potential.
Lorenz gauge (\(\partial_\mu A^\mu = 0\)): Manifestly Lorentz-covariant; preferred in relativistic QFT.
Temporal gauge (\(\Phi = 0\)): Used in lattice gauge theory and quantum simulation.

The gauge freedom is a computational advantage: choose whichever gauge simplifies your problem.

Discussion

The Aharonov-Bohm effect (§4.2.2) shows that the vector potential \(\boldsymbol{A}\) produces observable phase shifts in a region where \(\boldsymbol{B} = 0\). This seems to contradict the claim that \(\boldsymbol{A}\) is gauge-dependent and unphysical. How do you reconcile this? What quantity is actually gauge-invariant in the AB effect? (Hint: think about the line integral \(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\).)

Summary#

Gauge transformations act simultaneously on \(\psi\) (phase), \(\boldsymbol{A}\) (gradient), and \(\Phi\) (time derivative), preserving the Schrödinger equation.
Gauge-invariant: fields \(\boldsymbol{E}\), \(\boldsymbol{B}\), energy levels, transition probabilities, kinetic momentum \(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\).
Gauge-dependent: potentials \(\boldsymbol{A}\), \(\Phi\), canonical momentum \(\hat{\boldsymbol{p}}\), wavefunction phase.
Gauge symmetry is redundancy: multiple descriptions of the same physics, not a map between distinct states.
Gauge fixing is a computational choice; all gauges yield identical physical predictions.

Homework#

1. Verify that the Schrödinger equation with the minimal-coupling Hamiltonian \(\hat{H} = \frac{1}{2m}(\hat{\boldsymbol{p}} - q\boldsymbol{A})^2 + q\Phi\) is invariant under the simultaneous gauge transformation:

\[\psi \to \psi' = \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi, \quad \boldsymbol{A} \to \boldsymbol{A}' = \boldsymbol{A} + \nabla\alpha, \quad \Phi \to \Phi' = \Phi - \partial_t\alpha\]

Show that if \(\psi(\boldsymbol{x}, t)\) satisfies \(\mathrm{i}\hbar\partial_t\psi = \hat{H}\psi\) with potentials \((\boldsymbol{A}, \Phi)\), then \(\psi'\) satisfies the same equation with potentials \((\boldsymbol{A}', \Phi')\).

2. Show that the charge density \(\rho = q|\psi|^2\) and probability current \(\boldsymbol{j} = \frac{1}{m}\text{Im}(\psi^*\hat{\boldsymbol{\pi}}\psi)\) (where \(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\)) are both gauge-invariant:

(a) Show that \(|\psi'|^2 = |\psi|^2\) under a gauge transformation.

(b) Show that \(\psi'^*\hat{\boldsymbol{\pi}}'\psi' = \psi^*\hat{\boldsymbol{\pi}}\psi\) (the kinetic momentum is gauge-invariant).

3. In the Coulomb gauge \(\nabla \cdot \boldsymbol{A} = 0\), the gauge-fixing condition leaves a residual gauge freedom: \(\alpha(\boldsymbol{x}, t)\) can be any function satisfying \(\nabla^2\alpha = 0\) (harmonic functions).

(a) Show that if \(\boldsymbol{A}\) satisfies \(\nabla \cdot \boldsymbol{A} = 0\), then \(\boldsymbol{A}' = \boldsymbol{A} + \nabla\alpha\) also satisfies the Coulomb gauge if \(\nabla^2\alpha = 0\).

(b) Give an explicit example of a harmonic function \(\alpha(\boldsymbol{x})\) that generates a residual gauge transformation.

4. Two physicists compute the expectation value of canonical momentum \(\langle\hat{\boldsymbol{p}}\rangle\) for an electron in a uniform magnetic field \(\boldsymbol{B} = B\hat{\boldsymbol{z}}\). Physicist A uses the Coulomb gauge \(\boldsymbol{A} = \frac{B}{2}(-y, x, 0)\) (symmetric gauge). Physicist B uses the Landau gauge \(\boldsymbol{A} = (0, Bx, 0)\).

(a) Explain why they will compute different values for \(\langle\hat{\boldsymbol{p}}\rangle\) in their respective gauges.

(b) Which momentum is gauge-invariant: \(\hat{\boldsymbol{p}}\) or \(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\)?

(d) In an experiment, what quantity would actually be measured?

5. Show that energy eigenvalues are gauge-invariant: if \(\psi_n\) is an energy eigenstate in one gauge with eigenvalue \(E_n\):

\[\hat{H}[\boldsymbol{A}, \Phi]\psi_n = E_n\psi_n\]

then the gauge-transformed state \(\psi_n' = \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi_n\) is an eigenstate of the transformed Hamiltonian \(\hat{H}'[\boldsymbol{A}', \Phi']\) with the same eigenvalue \(E_n\).

6. A particle moves from point A to point B in an electromagnetic field. The phase acquired along a path \(\mathcal{C}\) is determined by the path integral of the kinetic momentum:

\[\phi_\mathcal{C} = \frac{q}{\hbar}\int_\mathcal{C} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{r}\]

Show that this phase difference (in interference patterns) is gauge-invariant even though the local vector potential \(\boldsymbol{A}\) is gauge-dependent. Specifically, show that if two paths enclose a region with magnetic flux \(\Phi_B\), the phase difference is:

\[\Delta\phi = \frac{q}{\hbar}\oint (\boldsymbol{A} + \nabla\alpha) \cdot \mathrm{d}\boldsymbol{r} = \frac{q}{\hbar}\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{r}\]

(Use Stokes’ theorem and the fact that \(\oint \nabla\alpha \cdot \mathrm{d}\boldsymbol{r} = 0\) for a closed loop.)

7. Explain conceptually why “gauge symmetry is a redundancy of description, not a physical symmetry of nature.” Compare and contrast gauge symmetry with a true physical symmetry, such as rotational invariance.

(a) How does rotational symmetry map physical states to different physical states?

(b) How does gauge symmetry map descriptions to different descriptions of the same state?

(d) Give an example of how two physicists using different gauges (Coulomb vs. Lorenz) would report different intermediate results but agree on all physical predictions.

8. The Aharonov-Bohm effect (discussed in 4.2.1) produces observable interference patterns due to the vector potential \(\boldsymbol{A}\) in a region where the magnetic field \(\boldsymbol{B} = 0\) everywhere on the particle’s path.

(a) Explain how this is consistent with the statement that \(\boldsymbol{A}\) is gauge-dependent and unphysical.

(b) What quantity in the Aharonov-Bohm effect is actually gauge-invariant: the local \(\boldsymbol{A}\), or the integrated phase \(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{r}\)?

(d) Discuss why the Aharonov-Bohm effect demonstrates that the potentials (\(\boldsymbol{A}\), \(\Phi\)) are more fundamental in quantum mechanics than in classical mechanics, even though they remain gauge-dependent quantities.