4.1.1 Gauge Principle#

Prompts

What does it mean to call the phase of \(\psi\) local internal data attached to each spacetime point? Why does locality forbid a single globally synchronized phase convention for \(\psi\) across spacetime?
If the phase convention at each spacetime point is independent, why does computing \(\partial_{i}\psi\) require a connection field? How does locality force one?
Define the covariant derivatives \(\boldsymbol{D}\) and \(D_{t}\). Why must \(\phi\) accompany \(\boldsymbol{A}\) once \(\alpha\) depends on time?
How does the minimal-coupling substitution \(\partial_t \to D_t,\ \nabla \to \boldsymbol{D}\) turn the free Schrödinger equation into a gauge-invariant one? Why does this make electromagnetism the geometric price of a local phase redundancy?
How do emergent gauge fields arise in frustrated magnets and quantum spin liquids? What plays the role of the local phase, and of \((\phi,\boldsymbol{A})\)?

Lecture Notes#

Overview#

The wavefunction’s phase is local internal data attached to each spacetime point, and locality forbids any global synchronization of those phases. A local phase rotation is therefore a redundancy in our description, not a symmetry. This section shows that the redundancy alone forces the existence of the electromagnetic gauge field \((\phi,\boldsymbol{A})\) — the connection required to compare \(\psi\) at distinct points — and that the same logic produces emergent gauge fields in frustrated magnets and quantum spin liquids.

Phase as Local Internal Data#

Quantum mechanics models a probability density by a complex amplitude:

\[ p(\boldsymbol{r}) \;=\; \vert\psi(\boldsymbol{r})\vert^{2}. \]

The amplitude \(\psi\) carries one piece of information that \(p\) does not — its phase. Two phase rotations of \(\psi\) leave \(p\) unchanged:

Global phase rotation. \(\psi \to \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\) with \(\alpha\) constant leaves every expectation value invariant. Quantum states therefore live in projective Hilbert space — the global phase is unobservable (§1.1.1).
Local phase rotation. Going further, \(\psi(\boldsymbol{r}) \to \mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol{r})/\hbar}\psi(\boldsymbol{r})\) with a spatially varying \(\alpha\) leaves local densities unchanged but makes ordinary derivatives gauge-dependent, which motivates introducing a connection and covariant derivatives. (Conceptual interpretation is developed in §4.1.3.)

Locality and Gauge Connection#

Schrödinger dynamics depends on derivatives of \(\psi\), which compare \(\psi\) at neighboring points \(\boldsymbol{r}\) and \(\boldsymbol{r}+\delta\boldsymbol{r}\). But if the phase convention at each point is independent, the difference \(\psi(\boldsymbol{r}+\delta\boldsymbol{r}) - \psi(\boldsymbol{r})\) has no gauge-invariant meaning — different choices of phase at the two points give different differences without any physical change.

We can see this concretely. Apply the local phase rotation

(144)#\[ \psi(\boldsymbol{r}, t) \;\to\; \psi'(\boldsymbol{r}, t) = \mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol{r},t)/\hbar}\,\psi(\boldsymbol{r}, t), \]

and compute the action of the bare derivatives:

\[\begin{split} \begin{split} \partial_{i}\psi' &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\bigl(\partial_{i} + \mathrm{i}(q/\hbar)\,\partial_{i}\alpha\bigr)\psi,\\ \partial_{t}\psi' &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\bigl(\partial_{t} + \mathrm{i}(q/\hbar)\,\partial_{t}\alpha\bigr)\psi. \end{split} \end{split}\]

The extra \(\partial_{i}\alpha\) and \(\partial_{t}\alpha\) pieces are exactly the artifacts of independent phase conventions at neighboring points. They show that the free Schrödinger equation \(\mathrm{i}\hbar\partial_{t}\psi = -(\hbar^{2}/2m)\nabla^{2}\psi\) does not retain its form when \(\alpha\) varies in spacetime.

Derivation: bare derivatives pick up gauge artifacts

Apply the product rule to differentiate \(\psi'=\mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol{r},t)/\hbar}\psi\). Each derivative acts on both the local phase factor and on \(\psi\), picking up an extra term proportional to the corresponding derivative of \(\alpha\).

For the spatial derivative, the product rule gives

\[\begin{split} \begin{split} \partial_{i}\psi' &= \partial_{i}\bigl(\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\bigr)\\ &= \bigl(\partial_{i}\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\bigr)\psi + \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,\partial_{i}\psi\\ &= \mathrm{i}(q/\hbar)\,(\partial_{i}\alpha)\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi + \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,\partial_{i}\psi\\ &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\bigl(\partial_{i}+\mathrm{i}(q/\hbar)\,\partial_{i}\alpha\bigr)\psi. \end{split} \end{split}\]

The time derivative is identical in form:

\[\begin{split} \begin{split} \partial_{t}\psi' &= \partial_{t}\bigl(\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\bigr)\\ &= \bigl(\partial_{t}\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\bigr)\psi + \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,\partial_{t}\psi\\ &= \mathrm{i}(q/\hbar)\,(\partial_{t}\alpha)\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi + \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,\partial_{t}\psi\\ &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\bigl(\partial_{t}+\mathrm{i}(q/\hbar)\,\partial_{t}\alpha\bigr)\psi. \end{split} \end{split}\]

Equivalently, commuting bare derivatives through the local phase factor displaces them by an extra piece:

\[\begin{split} \begin{split} \mathrm{e}^{-\mathrm{i}q\alpha/\hbar}\,\partial_{i}\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar} &= \partial_{i}+\mathrm{i}(q/\hbar)\,\partial_{i}\alpha,\\ \mathrm{e}^{-\mathrm{i}q\alpha/\hbar}\,\partial_{t}\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar} &= \partial_{t}+\mathrm{i}(q/\hbar)\,\partial_{t}\alpha. \end{split} \end{split}\]

Bare derivatives therefore do not commute with the local phase rotation: they get dressed by extra pieces \(\mathrm{i}(q/\hbar)\,\partial_{i}\alpha\) and \(\mathrm{i}(q/\hbar)\,\partial_{t}\alpha\) that depend on how \(\alpha\) varies in space and time. The next section asks what kind of object can absorb this dressing while keeping the dynamics gauge-invariant.

Gauge field \(=\) connection on a U(1) bundle

The vector potential \(\boldsymbol{A}\) and scalar potential \(\phi\) are the geometric infrastructure required to compare phases of \(\psi\) across spacetime. Mathematically, \((\phi, \boldsymbol{A})\) is a connection on a \(U(1)\) fiber bundle whose fibers are phase circles attached to each spacetime point.

Matter \(=\) stuff with local internal degrees of freedom (here, the phase of \(\psi\)).
Gauge field \(=\) the connection that tells us how to compare those degrees of freedom across spacetime.

Once we grant the redundancy, the connection field is forced; it is not added by hand.

Covariant Derivatives and Gauge Transformation#

Demanding that the phase rotation \(\psi \to \mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol{r},t)/\hbar}\psi\) leave the dynamics invariant fixes how the connection \((\phi,\boldsymbol{A})\) must transform together with \(\psi\):

Gauge transformation and covariant derivatives

Under the gauge transformation

(145)#\[\begin{split} \begin{split} \psi &\to \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi,\\ \boldsymbol{A} &\to \boldsymbol{A} + \nabla\alpha,\\ \phi &\to \phi - \partial_{t}\alpha, \end{split} \end{split}\]

the covariant derivatives

(146)#\[\begin{split} \begin{split} D_{i} &= \partial_{i} - \mathrm{i}\frac{q}{\hbar}A_{i},\\ D_{t} &= \partial_{t} + \mathrm{i}\frac{q}{\hbar}\phi, \end{split} \end{split}\]

(with vector spatial form \(\boldsymbol{D} = \nabla - \mathrm{i}(q/\hbar)\boldsymbol{A}\)) transform homogeneously — they pick up exactly the same phase factor as \(\psi\):

\[\begin{split} \begin{split} D_{i}\psi &\to \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,D_{i}\psi,\\ D_{t}\psi &\to \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,D_{t}\psi. \end{split} \end{split}\]

The opposite signs in \(D_{t}\) vs \(\boldsymbol{D}\) — and equivalently in the gauge shifts of \(\phi\) vs \(\boldsymbol{A}\) — are not arbitrary. They reflect the fact that time and space appear with opposite signature in the underlying spacetime structure, and they are exactly what makes the unwanted \(\partial_{t}\alpha\) and \(\partial_{i}\alpha\) terms cancel against the connection shifts.

Derivation: covariance of \(\boldsymbol{D}\) and \(D_{t}\)

First check the spatial covariant derivative. In components, with \(A_i\to A_i+\partial_i\alpha\),

\[\begin{split} \begin{split} D'_i\psi' &=\left(\partial_i-\mathrm{i}\frac{q}{\hbar}(A_i+\partial_i\alpha)\right)\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\\ &=\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\partial_i\psi +\mathrm{i}\frac{q}{\hbar}(\partial_i\alpha)\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi -\mathrm{i}\frac{q}{\hbar}A_i\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi -\mathrm{i}\frac{q}{\hbar}(\partial_i\alpha)\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\\ &=\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\left(\partial_i-\mathrm{i}\frac{q}{\hbar}A_i\right)\psi =\mathrm{e}^{\mathrm{i}q\alpha/\hbar}D_i\psi. \end{split} \end{split}\]

The two \(\partial_i\alpha\) terms cancel: one comes from differentiating the local phase factor, and the other comes from the shift of \(A_i\).

Now check the temporal covariant derivative, using \(\phi \to \phi - \partial_t\alpha\):

\[\begin{split} \begin{split} D'_{t}\psi' &= \bigl(\partial_{t} + \mathrm{i}(q/\hbar)(\phi - \partial_{t}\alpha)\bigr)\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\\ &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,\partial_{t}\psi \;+\; \mathrm{i}(q/\hbar)(\partial_{t}\alpha)\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\\ &\quad +\; \mathrm{i}(q/\hbar)\phi\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi \;-\; \mathrm{i}(q/\hbar)(\partial_{t}\alpha)\,\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi\\ &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\bigl(\partial_{t} + \mathrm{i}(q/\hbar)\phi\bigr)\psi \;=\; \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\,D_{t}\psi. \end{split} \end{split}\]

Again the \(\partial_t\alpha\) terms cancel. The opposite signs in \(D_i=\partial_i-\mathrm{i}(q/\hbar)A_i\) and \(D_t=\partial_t+\mathrm{i}(q/\hbar)\phi\) are exactly what match the opposite gauge shifts \(\boldsymbol{A}\to\boldsymbol{A}+\nabla\alpha\) and \(\phi\to\phi-\partial_t\alpha\).

The Gauge-Invariant Schrödinger Equation#

For reference, the nonrelativistic Schrödinger equation for a free particle (no electromagnetic potentials) is the familiar form

(147)#\[ \mathrm{i}\hbar\,\partial_{t}\psi(\boldsymbol{r},t) \;=\; -\frac{\hbar^{2}}{2m}\,\nabla^{2}\psi(\boldsymbol{r},t), \]

equivalently \(\mathrm{i}\hbar\,\partial_{t}\psi = \frac{\hat{\boldsymbol{p}}^{2}}{2m}\psi\) with \(\hat{\boldsymbol{p}}=-\mathrm{i}\hbar\nabla\). This is the dynamics we generalize below: the left side is a time derivative, the right side is kinetic energy from spatial derivatives of \(\psi\).

Minimal coupling replaces those bare derivatives by covariant ones so the equation keeps the same form under local phase redefinitions together with shifts of \((\phi,\boldsymbol{A})\):

\[\begin{split} \begin{split} \partial_{t} &\to D_{t} = \partial_{t} + \mathrm{i}(q/\hbar)\phi,\\ \nabla &\to \boldsymbol{D} = \nabla - \mathrm{i}(q/\hbar)\boldsymbol{A}. \end{split} \end{split}\]

The result, written first in compact covariant form and then in standard Hamiltonian form:

Schrödinger equation in a gauge field

(148)#\[ \mathrm{i}\hbar\,D_{t}\psi \;=\; \frac{1}{2m}(-\mathrm{i}\hbar\,\boldsymbol{D})^{2}\psi, \]

equivalent to \(\mathrm{i}\hbar\,\partial_{t}\psi \;=\; \hat{H}\psi\) with

(149)#\[ \boxed{ \hat{H} \;=\; \,\frac{(\hat{\boldsymbol{p}} - q\boldsymbol{A})^{2}}{2m} + q\phi.} \]

Both fields enter — neither is freely chosen. The vector potential modifies kinetic energy through \(\hat{\boldsymbol{p}} - q\boldsymbol{A}\), and the scalar potential adds an electrostatic energy \(q\phi\). The combination \(\hat{\boldsymbol{\pi}}=\hat{\boldsymbol{p}}-q\boldsymbol{A}\) is the kinetic momentum; we will identify it dynamically as the velocity operator in §4.1.2.

The lesson is striking: electromagnetism is not added to quantum mechanics by hand. Once the phase of \(\psi\) is granted to be local internal data, the connection \((\phi,\boldsymbol{A})\) — and its coupling to matter — is the geometric price of a self-consistent local description.

Emergent Gauge Fields in Matter#

The argument above is not specific to electromagnetism. Wherever a quantum system has local internal degrees of freedom that cannot be globally compared, the same logic forces a connection field — and that connection becomes a gauge field with its own dynamics.

Emergent gauge fields in condensed matter

Frustrated magnets, quantum dimer models, and spin liquids often host emergent gauge fields: low-energy excitations that look exactly like a \(U(1)\) gauge field \((\phi^{\mathrm{eff}},\boldsymbol{A}^{\mathrm{eff}})\) even though the underlying lattice has no fundamental electromagnetism. A few signatures:

Quantum spin liquids on the kagome or pyrochlore lattice. Local moments cannot order globally; the low-energy theory has emergent gauge bosons (“emergent photons”) that mediate interactions between fractionalized excitations (“spinons”).
Quantum dimer models on bipartite lattices. A \(U(1)\) gauge structure emerges from the local dimer-covering constraint; the “Coulomb phase” supports gapless photon-like modes.
Lattice gauge theory (the §4.1 project). One discretizes spacetime, places phases on links, and lets the system equilibrate. The same minimal-coupling structure derived above appears as a result of the lattice dynamics, not as an input.

In each case, “matter” (local DOF that need to be compared) and “gauge field” (the connection that makes the comparison possible) are not separate inputs — they are two facets of one structure. The gauge principle of this lecture is one instance of a broader pattern that pervades modern many-body physics.

The gauge principle (modern statement)

Phase is local internal data attached to each spacetime point of \(\psi\). Locality forbids globally enforced phase agreement, so a local phase rotation is a redundancy, not a symmetry. A redundancy at every point requires a connection to compare data at distinct points; that connection is the gauge field \((\phi,\boldsymbol{A})\).

Concrete consequences:

The gauge field transforms together with \(\psi\) as in (145).
Covariant derivatives \(\boldsymbol{D}, D_{t}\) ((146)) replace ordinary derivatives.
Minimal coupling (149) is forced — kinetic momentum becomes \(\hat{\boldsymbol{p}}-q\boldsymbol{A}\), and an electrostatic energy \(q\phi\) appears automatically.

The same logic, applied to any system with local internal DOF, produces an emergent gauge field — including in condensed matter systems with no fundamental \(U(1)\) charge.

Summary#

\(\vert\psi\vert^{2}\) is the only directly observable quantity; the phase of \(\psi\) at each point is local internal data, not a physical observable.
Local phase freedom forbids globally synchronized phase conventions, motivating covariant comparison across spacetime.
Comparing \(\psi\) at neighboring points requires a connection — exactly the gauge field \((\phi,\boldsymbol{A})\). Covariant derivatives \(\boldsymbol{D}, D_{t}\) implement this comparison and transform homogeneously under the gauge transformation.
Minimal coupling is forced by the redundancy + locality argument: electromagnetism is the geometric price of a local phase redundancy.
The same construction yields emergent gauge fields in any system whose local internal DOF cannot be globally synchronized — frustrated magnets, dimer models, and spin liquids.
The combination \(\hat{\boldsymbol{\pi}}=\hat{\boldsymbol{p}}-q\boldsymbol{A}\) — kinetic momentum — is fixed by gauge covariance.

Homework#

1. Diagnosing redundancy. For each pair \((\psi, \psi')\) below, compute \(\vert\psi\vert^2\), \(\arg\psi\), and \(\langle\hat{p}_x\rangle\), and decide whether the two states represent the same physical system or different ones.

(a) \(\psi(x) = \mathrm{e}^{\mathrm{i}kx}\) and \(\psi'(x) = \mathrm{e}^{\mathrm{i}kx+\mathrm{i}\theta_0}\) (constant overall phase).

(b) \(\psi(x) = \mathrm{e}^{\mathrm{i}kx}\) and \(\psi'(x) = \mathrm{e}^{\mathrm{i}(k+k_0)x}\) (different wavevector).

(c) \(\psi(\boldsymbol{r}) = f(\boldsymbol{r})\) and \(\psi'(\boldsymbol{r}) = \mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol{r})/\hbar}f(\boldsymbol{r})\), together with the simultaneous gauge transformation \(\boldsymbol{A}\to\boldsymbol{A}+\nabla\alpha\).

State which transformation is a symmetry and which is a redundancy, and identify the operational diagnostic.

2. Plane-wave gauge transform. Let \(\psi(\boldsymbol r,t)\) be an arbitrary state (not necessarily a plane wave). Use the single gauge function

\[ \alpha(\boldsymbol{r},t)=\boldsymbol{k}\cdot\boldsymbol{r}-\omega t, \]

with constant \(\boldsymbol{k}\) and \(\omega\), and apply

\[\begin{split} \begin{split} \psi' &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi,\\ \boldsymbol A' &= \boldsymbol A + \nabla\alpha,\\ \phi' &= \phi - \partial_t\alpha. \end{split} \end{split}\]

(a) Compute the shifts \(\boldsymbol{A}'-\boldsymbol{A}\) and \(\phi'-\phi\).

(b) Show directly that the fields are unchanged:

\[\begin{split} \begin{split} \boldsymbol E' &= -\nabla\phi' - \partial_t\boldsymbol A' = \boldsymbol E,\\ \boldsymbol B' &= \nabla\times\boldsymbol A' = \boldsymbol B. \end{split} \end{split}\]

(d) Give a short physical interpretation: what changes under this transformation (description/kinematic labels), and what does not (gauge-invariant fields and force law)?

3. Covariance of derivatives. Verify by direct computation that under (145),

\[\begin{split} \begin{split} D'_{i}\psi' &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}D_{i}\psi,\\ D'_{t}\psi' &= \mathrm{e}^{\mathrm{i}q\alpha/\hbar}D_{t}\psi. \end{split} \end{split}\]

(a) Carry out the spatial calculation explicitly in 1D, showing which \(\partial_{x}\alpha\) terms cancel.

(b) Repeat for the time derivative, showing how the shift \(\phi\to\phi-\partial_{t}\alpha\) cancels the \(\partial_{t}\alpha\) term from differentiating \(\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\).

(c) Note that the spatial gauge shift uses \(+\nabla\alpha\) while the temporal one uses \(-\partial_{t}\alpha\). Explain in one sentence why these signs are opposite — what does that have to do with the opposite signs in \(D_{i}\) vs \(D_{t}\)?

4. Gauge on punctured plane. Consider a particle restricted to the punctured plane \(\mathbb{R}^{2}\setminus\{0\}\) (the origin removed). Let the gauge function be

\[ \alpha(\rho,\varphi)=\kappa\,\varphi, \]

where \(\varphi\) is the azimuthal angle and \(\kappa\) is a constant.

(a) Compute \(\nabla\alpha\) in polar coordinates and write the transformed vector-potential shift \(\Delta\boldsymbol{A}=\nabla\alpha\).

(b) Show that \(\nabla\times\nabla\alpha=0\) only away from the origin, so \(\boldsymbol{B}\) is unchanged on the punctured plane. Then evaluate \(\oint \nabla\alpha\cdot \mathrm{d}\boldsymbol{\ell}\) around a loop enclosing the hole and use Stokes/distribution language to explain why a singular magnetic flux \(\Phi_{\mathrm{hole}}\) at the hole is required.

(c) Explain why this gauge function is not globally single-valued when \(\varphi\to\varphi+2\pi\). What condition on \(\kappa\) (in terms of \(q\) and \(\hbar\)) makes the wavefunction phase factor \(\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\) single-valued after one full loop?

(d) Summarize the physical picture: outside the hole, local fields are unchanged; at the hole, a singular flux \(\Phi_{\mathrm{hole}}\) carries the nontrivial holonomy. Explain why both statements are needed for a consistent global description.

5. What gauge transformations cannot do. A particle moves in a static gravitational potential \(V(\boldsymbol{r}) = mgz\), where \(\boldsymbol{r}=(x,y,z)\) and \(z\) is the vertical (height) coordinate. One might claim: “A spatial gauge transformation \(\psi \to \mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol{r})/\hbar}\psi\) should remove \(V\) from the Schrödinger equation, just as it removes a static \(\boldsymbol{A}\).”

(a) Try to find \(\alpha(\boldsymbol{r})\) that eliminates \(V\). Where does the attempt fail?

(b) Allow \(\alpha\) to depend on \(t\). The only way to cancel \(V\) in the \(q\phi\) slot of the Hamiltonian is \(-\partial_{t}\alpha = -V/q\), which forces \(\alpha(\boldsymbol{r}, t) = (V(\boldsymbol{r})/q)\,t + g(\boldsymbol{r})\) for some \(g\). Compute the resulting \(\boldsymbol{A}' = \boldsymbol{A} + \nabla\alpha\) and show it is now time-dependent. Conclude that \(V\) has not been removed — it has been shuffled into a time-dependent vector potential.

(c) Conclude: gauge transformations can shuffle physical content between \(\phi\) and \(\boldsymbol{A}\) but cannot erase it. The energy landscape of an external \(V(\boldsymbol{r})\) is gauge-invariant — gravity does not arise from a \(U(1)\) gauge redundancy.

6. Second-order bilinear covariance. Consider the same local phase redundancy

\[ \psi' = \mathrm{e}^{\mathrm{i}q\alpha(\boldsymbol r)/\hbar}\psi. \]

For second spatial derivatives, study the bilinear combination

\[ \partial_{ij}[\psi]\equiv \psi\,\partial_i\partial_j\psi-(\partial_i\psi)(\partial_j\psi). \]

(a) Expand \(\partial_{ij}[\psi']\) explicitly and show that the non-covariant remainder equals \((\mathrm{i}q/\hbar)(\partial_i\partial_j\alpha)\psi^2 \cdot \mathrm{e}^{2\mathrm{i}q\alpha/\hbar}\).

(b) Introduce a symmetric rank-2 gauge field \(A_{ij}=A_{ji}\) and define

\[ \mathcal{D}_{ij}[\psi]\equiv \psi\,\partial_i\partial_j\psi-(\partial_i\psi)(\partial_j\psi)-\mathrm{i}\frac{q}{\hbar}A_{ij}\psi^2. \]

Find the transformation law for \(A_{ij}\) such that

\[ \mathcal{D}'_{ij}[\psi'] = \mathrm{e}^{2\mathrm{i}q\alpha/\hbar}\,\mathcal{D}_{ij}[\psi]. \]

(d) Build one gauge-invariant scalar density from \(\mathcal{D}_{ij}[\psi]\) and \(\psi\) (for example using index contraction and complex conjugation), and state in one sentence what physical type of constrained motion this kind of second-derivative gauge structure is designed to capture.

7. Gauge connection on links. Consider a one-dimensional tight-binding model with hopping between neighboring sites \(n\) and \(n+1\):

\[ \hat{H}_{\text{hop}}=-t\sum_n \left(U_{n,n+1}\,\hat{c}_{n+1}^{\dagger}\hat{c}_n+\text{h.c.}\right), \]

where \(U_{n,n+1}=\mathrm{e}^{\mathrm{i}q a A_n/\hbar}\) is a link variable and \(a\) is the lattice spacing. Under a local phase redefinition,

\[ \hat{c}_n\to \mathrm{e}^{\mathrm{i}q\alpha_n/\hbar}\hat{c}_n. \]

(a) Find how \(U_{n,n+1}\) must transform so that each hopping term remains invariant.

(b) Translate your answer into a transformation law for \(A_n\). Show that, in the continuum limit, it becomes \(A\to A+\partial_x\alpha\).

(c) Explain why the link variable is the lattice version of a connection: what does it compare between neighboring sites, and why is it forced by local phase redundancy?