4.1.3 Gauge Invariance#

Prompts

How do \(\psi\), \(\boldsymbol{A}\), and \(\phi\) transform under a gauge transformation? Why must all three transform together?
What are the three types of momentum (canonical, kinetic, potential) and which are gauge-invariant? Make the same distinction for energy.
Which physical quantities are gauge-invariant and which are gauge-dependent? How would you organize them?
Explain the statement “gauge invariance is not a symmetry—it is a redundancy.” How does this differ from a physical symmetry like rotation?
Why is \(\oint\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{l}\) gauge-invariant, even though \(\boldsymbol{A}\) itself is not? What does this tell us about which features of \(\boldsymbol{A}\) can be physical?

Lecture Notes#

Overview#

In §4.1.1–4.1.2, we built the gauge-invariant Hamiltonian and derived the electromagnetic fields and forces. This section asks: which quantities in this framework are physical (gauge-invariant) and which are conventional (gauge-dependent)? The answer reveals that gauge invariance is not a symmetry of nature—it is a redundancy in our description.

Gauge Transformations#

A gauge transformation modifies the potentials and the wavefunction simultaneously, leaving all physics unchanged:

Gauge Transformation

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

All three transform in concert: 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 every gauge — both sides pick up the same overall phase factor \(\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\) by the homogeneous transformation of \(\boldsymbol{D}\psi\) and \(D_{t}\psi\) established in §4.1.1.

Gauge Invariance: a Redundancy, not a Symmetry

A physical symmetry (rotation, translation) maps one physical state to a different physical state that happens to obey the same laws.
A gauge transformation maps one description to another description of the same physical state. The physics has not changed—only the bookkeeping.

Gauge invariance is therefore a redundancy: multiple mathematical descriptions (\(\psi\), \(\boldsymbol{A}\), \(\phi\) in different gauges) correspond to one physical reality. Requiring gauge invariance is not imposing a new law of nature—it is projecting out the redundant degrees of freedom to identify what is physical.

This perspective explains why gauge invariance is so powerful: it constrains the theory not because it adds structure, but because it removes unphysical freedom. The minimal coupling prescription, the form of the Hamiltonian, the Lorentz force—all follow from demanding that the redundant description be self-consistent.

Momentum and Energy: Total = Kinetic + Potential#

The gauge-invariant Hamiltonian \(\hat{H} = \frac{1}{2m}(\hat{\boldsymbol{p}} - q\boldsymbol{A})^2 + q\phi\) decomposes both momentum and energy into three parts with the same structure:

Total = Kinetic + Potential

	Total (Canonical)	Kinetic	Potential
Momentum	\(\hat{\boldsymbol{p}} = -\mathrm{i}\hbar\nabla\)	\(\hat{\boldsymbol{\pi}} = -\mathrm{i}\hbar\boldsymbol{D} = m\hat{\boldsymbol{v}}\)	\(q\boldsymbol{A}\)
Energy	\(\hat{H} = \mathrm{i}\hbar\partial_t\)	\(\frac{\hat{\boldsymbol{\pi}}^2}{2m} = \mathrm{i}\hbar D_t\)	\(q\phi\)
Gauge	Dependent	Invariant	Dependent
Conserved	Yes	No	No

\[\begin{split} \begin{split} \hat{\boldsymbol{p}} &= \hat{\boldsymbol{\pi}} + q\boldsymbol{A} \\ \hat{H} &= \frac{\hat{\boldsymbol{\pi}}^2}{2m} + q\phi \end{split} \end{split}\]

Total (canonical): \(\hat{\boldsymbol{p}}\) and \(\hat{H}\) appear in conservation laws and determine the action accumulated in spacetime.
Kinetic: \(\hat{\boldsymbol{\pi}} = m\hat{\boldsymbol{v}}\) and \(\hat{\boldsymbol{\pi}}^2/2m\) are directly linked to the particle’s motion.
Potential: \(q\boldsymbol{A}\) and \(q\phi\) represent the interaction of the charged particle with the background electromagnetic field, present even when the particle is at rest.

Gauge-Invariant vs Gauge-Dependent Quantities#

Classification of Physical Quantities

Gauge-invariant (physical):

Quantity	Symbol
Electric field	\(\boldsymbol{E}\)
Magnetic field	\(\boldsymbol{B}\)
Field strength tensor	\(F_{\mu\nu}\) (with \(c=1\) in 4-tensor notation; cf. §4.1.2)
Kinetic momentum	\(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\)
Kinetic energy	\(\hat{\boldsymbol{\pi}}^2/2m\)
Probability density	\(\rho=\vert\psi\vert^2\)
Probability current	\(\boldsymbol{j} = \frac{1}{m}\mathrm{Re}(\psi^*\hat{\boldsymbol{\pi}}\psi)\)
Closed-loop phase	\(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\)

Gauge-dependent (conventional):

Quantity	Symbol
Vector potential	\(\boldsymbol{A}\)
Scalar potential	\(\phi\)
Canonical momentum	\(\hat{\boldsymbol{p}} = -\mathrm{i}\hbar\nabla\)
Canonical energy	\(\hat{H}\)
Wavefunction phase	\(\arg(\psi)\)
Open-path phase	\(\int_{\mathcal{C}} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\)

Open path vs closed loop

The phase factor \((q/\hbar)\int_{\mathcal{C}} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\) along an open path \(\mathcal{C}\) from \(\boldsymbol{r}_{\text{start}}\) to \(\boldsymbol{r}_{\text{end}}\) is not gauge-invariant: under \(\boldsymbol{A} \to \boldsymbol{A} + \nabla\alpha\) it shifts by the boundary values of \(\alpha\),

\[ \frac{q}{\hbar}\int_{\mathcal{C}} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l} \;\to\; \frac{q}{\hbar}\int_{\mathcal{C}} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l} + \frac{q}{\hbar}\bigl[\alpha(\boldsymbol{r}_{\text{end}}) - \alpha(\boldsymbol{r}_{\text{start}})\bigr], \]

exactly cancelling the boundary contributions \(\mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol{r})/\hbar}\) picked up by \(\psi(\boldsymbol{r})\) at the two endpoints. The combination \(\mathrm{e}^{\mathrm{i}q\int_{\mathcal{C}}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{l}/\hbar}\,\psi(\boldsymbol{r}_{\text{end}})\) that compares the wavefunction at the two endpoints through parallel transport along \(\mathcal{C}\) is gauge-invariant — but \(\int_{\mathcal{C}}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{l}\) alone is not a physical observable.

When the path closes, \(\boldsymbol{r}_{\text{end}} = \boldsymbol{r}_{\text{start}}\), the boundary terms cancel and \(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\) becomes gauge-invariant — this is the closed-loop holonomy, a genuine physical observable that drives the Aharonov-Bohm effect (§4.2.2) and serves as the prototype for the Berry phase (§4.2.1).

Gauge Fixing#

In practice, we choose a gauge to simplify calculations. Common choices:

Common Gauge Choices

Coulomb gauge (\(\nabla \cdot \boldsymbol{A} = 0\)): Standard for nonrelativistic problems. \(\phi\) becomes the instantaneous Coulomb potential.
Lorenz gauge (\(\partial_\mu A^\mu = 0\), with \(c = 1\) in 4-vector notation; cf. §4.1.2): Manifestly Lorentz-covariant; preferred in relativistic quantum field theory.
Temporal gauge (\(\phi = 0\)): Used in lattice gauge theory.

All gauges give identical physics; the choice is a computational convenience.

Discussion: \(\boldsymbol{A}\) is observable through phases

The Aharonov-Bohm effect shows that \(\boldsymbol{A}\) produces observable phase shifts in a region where \(\boldsymbol{B} = 0\). This seems to contradict the claim that \(\boldsymbol{A}\) is gauge-dependent. How do you reconcile this? (Hint: the observable is \(\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\), which is gauge-invariant.)

Poll: Observable quantities under gauge freedom

Which of the following is NOT gauge-invariant?

(A) The electric field \(\boldsymbol{E} = -\nabla\phi - \partial_t\boldsymbol{A}\).

(B) The probability density \(\vert\psi(\boldsymbol{r},t)\vert^2\).

(D) The magnetic field \(\boldsymbol{B} = \nabla \times \boldsymbol{A}\).

Summary#

Gauge transformations act on \(\psi\) (phase), \(\boldsymbol{A}\) (gradient), and \(\phi\) (time derivative) simultaneously, preserving all physics.
Total = Kinetic + Potential applies to both momentum (\(\hat{\boldsymbol{p}} = \hat{\boldsymbol{\pi}} + q\boldsymbol{A}\)) and energy (\(\hat{H} = \hat{\boldsymbol{\pi}}^2/2m + q\phi\)). Total quantities are conserved; kinetic quantities are gauge-invariant; potential quantities encode the coupling to the electromagnetic field.
Gauge invariance is redundancy, not symmetry: multiple descriptions of the same state, projected to physical reality by requiring consistency.

Homework#

1. Gauge-transforming an observable. Consider the expectation value \(\langle\hat{O}\rangle = \langle\psi\vert\hat{O}\vert\psi\rangle\).

(a) Under the gauge transformation \(\vert\psi\rangle \to \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\vert\psi\rangle\), show that \(\langle\hat{O}\rangle\) is invariant if and only if \([\hat{O}, \mathrm{e}^{\mathrm{i}q\alpha/\hbar}] = 0\). Which familiar operators satisfy this, and which do not?

(b) One claims that \(\langle\hat{\boldsymbol{p}}\rangle\) is gauge-invariant because “momentum is physical.” Find the flaw by computing \(\langle\hat{\boldsymbol{p}}\rangle\) in two gauges for a particle in a uniform field \(\boldsymbol{B} = B\hat{z}\).

2. Probability current. The probability current is \(\boldsymbol{j} = \frac{1}{m}\mathrm{Re}(\psi^*\hat{\boldsymbol{\pi}}\psi)\) with \(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - q\boldsymbol{A}\).

(a) Show that \(\boldsymbol{j}\) is gauge-invariant.

(b) Show that the continuity equation \(\partial_t\vert\psi\vert^2 + \nabla \cdot \boldsymbol{j} = 0\) holds in every gauge.

3. Canonical momentum shifts. Two physicists compute \(\langle\hat{\boldsymbol{p}}\rangle\) for the same electron in a uniform magnetic field \(\boldsymbol{B} = B\hat{z}\). Physicist A uses the symmetric gauge \(\boldsymbol{A} = \frac{B}{2}(-y, x, 0)\); physicist B uses the Landau gauge \(\boldsymbol{A} = (0, Bx, 0)\).

(a) Explain why they get different values of \(\langle\hat{\boldsymbol{p}}\rangle\).

(b) Show that both agree on \(\langle\hat{\boldsymbol{\pi}}\rangle\).

4. Energy under gauge transformation. One might argue: “The eigenvalue equation \(\hat{H}\psi_n = E_n\psi_n\) contains \(\hat{H}\), which depends on \(\boldsymbol{A}\) and \(\phi\). So changing gauge must change the energy eigenvalues.”

(a) Refute this by showing explicitly that if \(\hat{H}\psi_n = E_n\psi_n\), then \(\hat{H}'\psi_n' = E_n\psi_n'\) with the transformed \(\hat{H}'\) and \(\psi_n' = \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi_n\).

(b) The kinetic energy \(\frac{1}{2m}\langle\hat{\boldsymbol{\pi}}^2\rangle\) is gauge-invariant, but \(q\langle\phi\rangle\) is not. How can \(E_n\) be gauge-invariant if one of its terms is not? Resolve the apparent paradox.

(c) An experimentalist measures the hydrogen atom spectrum. In what sense is the result “gauge-invariant” even though the calculation was done in a specific gauge?

5. Coulomb gauge residual freedom. In the Coulomb gauge \(\nabla \cdot \boldsymbol{A} = 0\), show that the gauge is not fully fixed: \(\boldsymbol{A}' = \boldsymbol{A} + \nabla\alpha\) still satisfies \(\nabla \cdot \boldsymbol{A}' = 0\) provided \(\nabla^2\alpha = 0\). Give an explicit example of such a residual transformation.

6. Gauge-invariant classification. Classify each of the following quantities as gauge-invariant or gauge-dependent. For each gauge-dependent quantity, construct a related gauge-invariant one.

(a) \(\vert\psi(\boldsymbol{r})\vert^2\), \(\arg(\psi(\boldsymbol{r}))\), \(\langle\hat{\boldsymbol{p}}\rangle\)

(b) \(\boldsymbol{B} = \nabla \times \boldsymbol{A}\), \(\boldsymbol{A}(\boldsymbol{r})\), \(\oint \boldsymbol{A}\cdot\mathrm{d}\boldsymbol{l}\) (closed loop)

(c) The energy eigenvalue \(E_n\), the canonical momentum eigenvalue \(\hbar k\), the Berry phase \(\Phi_{\mathrm{Berry}} = \oint \mathcal{A}\cdot\mathrm{d}\boldsymbol{R}\)

7. Aharonov-Bohm phase. A particle travels along two paths enclosing a region with magnetic flux \(\Phi = \oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\). Show that the phase difference \(\Delta\Phi_{\mathrm{AB}} = \frac{q}{\hbar}\oint \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}\) is gauge-invariant by verifying \(\oint \nabla\alpha \cdot \mathrm{d}\boldsymbol{l} = 0\) for any single-valued \(\alpha\). Explain how \(\boldsymbol{A}\) can produce observable effects in a region where \(\boldsymbol{B} = 0\).