Local U(1) Gauge Invariance and Minimal Coupling

Derivation

Local U(1) Gauge Invariance and Minimal Coupling

D-374 Home PU-402 Threads symmetry · force · fields Depends on Noether's First Theorem, Maxwell's Equations from −¼F², Dirac Equation from the Spinor Lagrangian

Statement

Take the free Dirac Lagrangian \( \mathcal{L}_0=\bar\psi\left(i\gamma^\mu\partial_\mu-m\right)\psi \), invariant under the global phase rotation \( \psi\to e^{ie\alpha}\psi \). Promoting the phase to a spacetime-dependent function, \( \alpha\to\alpha(x) \), and demanding the action stay invariant forces the introduction of a vector field \( A_\mu \) with transformation law \( A_\mu\to A_\mu+\partial_\mu\alpha \), the replacement of the ordinary derivative by the covariant derivative \( D_\mu=\partial_\mu-ieA_\mu \), and hence a unique interaction term \( \mathcal{L}_{\text{int}}=e\,\bar\psi\gamma^\mu\psi\,A_\mu=e\,j^\mu A_\mu \) (minimal coupling). The Noether current of the global symmetry, \( j^\mu=\bar\psi\gamma^\mu\psi \), is the very object that sources the gauge field, and gauge invariance of the full action is equivalent to its conservation \( \partial_\mu j^\mu=0 \).

Why it matters

This is the single most productive idea in modern physics: it turns a symmetry principle into a dynamical prescription. Rather than writing down electromagnetism by hand, one demands that the phase of a charged field be a local (unobservable) convention, and the entire structure of QED — the photon, its coupling to matter, and the conservation of electric charge — drops out uniquely. The same recipe, applied to non-Abelian groups, yields the weak and strong interactions.

Conceptually it inverts the usual logic. Charge conservation is normally read off Noether's theorem as a consequence of a global symmetry. Here the deeper statement is that local gauge invariance requires the coupling \( e\,j^\mu A_\mu \), and consistency of that coupling (via the field equations of \( A_\mu \)) demands \( \partial_\mu j^\mu=0 \). Symmetry and conservation become two faces of one requirement.

Assumptions

The matter field is a complex (charged) Dirac spinor.A real field carries no phase to gauge; there is nothing to make local and no minimal coupling to generate. The construction is specific to fields transforming under a non-trivial U(1) representation.

The gauge group is U(1) and the charge \( e \) is a single fixed number.Drop single-valuedness and you cannot assign a consistent charge; allow a larger group and the gauge field becomes matrix-valued, \( D_\mu=\partial_\mu-igA_\mu^aT^a \), and the field strength acquires a self-interaction \( gf^{abc}A_\mu^bA_\nu^c \) (non-Abelian, Yang–Mills).

The action is built from \( \psi,\bar\psi \) and \( D_\mu\psi \) only, with the lowest-dimension gauge-invariant terms.Admitting non-minimal, higher-dimension invariants (e.g. a Pauli term \( \bar\psi\sigma^{\mu\nu}\psi F_{\mu\nu} \)) still respects gauge symmetry but spoils the uniqueness of the coupling and renormalisability; "minimal" coupling is the statement that we keep only \( D_\mu \).

The transformation parameter \( \alpha(x) \) is a smooth, single-valued function on a topologically trivial region.On multiply-connected regions \( \alpha \) can wind, and gauge-inequivalent \( A_\mu \) with the same \( F_{\mu\nu} \) become physically distinguishable (Aharonov–Bohm). The local argument is unaffected but the global classification of \( A_\mu \) is richer.

Derivation

\[ \psi(x)\to\psi'(x)=e^{ie\alpha}\psi(x),\qquad \bar\psi\to e^{-ie\alpha}\bar\psi,\qquad \alpha=\text{const}. \]

Global U(1): \( \mathcal{L}_0 \) depends on \( \psi \) only through the invariant bilinears \( \bar\psi\psi \) and \( \bar\psi\gamma^\mu\partial_\mu\psi \); the constant phases cancel, so \( \mathcal{L}_0 \) is unchanged. A

\[ \partial_\mu\to?\qquad \psi\to e^{ie\alpha(x)}\psi,\qquad \partial_\mu\psi\to e^{ie\alpha}\left(\partial_\mu\psi+ie(\partial_\mu\alpha)\psi\right). \]

Now let \( \alpha=\alpha(x) \). The Leibniz rule produces an extra inhomogeneous piece \( ie(\partial_\mu\alpha)\psi \): the ordinary derivative does not transform covariantly, so \( \bar\psi\gamma^\mu\partial_\mu\psi \) is no longer invariant. A

\[ D_\mu\equiv\partial_\mu-ieA_\mu,\qquad A_\mu\to A_\mu'=A_\mu+\partial_\mu\alpha. \]

Postulate a compensating vector field \( A_\mu \) whose gauge transformation is engineered to absorb the offending gradient. This is a definition to be verified in Step 4, not yet a result. B

\[ D_\mu\psi\to\left(\partial_\mu-ieA_\mu'\right)e^{ie\alpha}\psi =e^{ie\alpha}\Big[\partial_\mu\psi+ie(\partial_\mu\alpha)\psi-ie(A_\mu+\partial_\mu\alpha)\psi\Big]. \]

Expand using Step 2 for \( \partial_\mu(e^{ie\alpha}\psi) \) and the postulated \( A_\mu' \) from Step 3. Every factor is exact; no term has been dropped. B

\[ D_\mu\psi\to e^{ie\alpha}\left(\partial_\mu-ieA_\mu\right)\psi=e^{ie\alpha}\,D_\mu\psi. \]

The \( +ie(\partial_\mu\alpha)\psi \) and \( -ie(\partial_\mu\alpha)\psi \) cancel identically. Thus \( D_\mu\psi \) transforms exactly like \( \psi \) itself — it is covariant. This fixes the transformation law of \( A_\mu \) up to the overall normalisation absorbed into \( e \). C

\[ \bar\psi\gamma^\mu D_\mu\psi\to e^{-ie\alpha}\bar\psi\,\gamma^\mu\,e^{ie\alpha}D_\mu\psi=\bar\psi\gamma^\mu D_\mu\psi. \]

Because \( \bar\psi \) carries the opposite phase, the pair \( \bar\psi\,\Gamma\,D_\mu\psi \) is gauge invariant for any \( \Gamma \). Replacing \( \partial_\mu\to D_\mu \) in \( \mathcal{L}_0 \) restores local invariance. B

\[ \mathcal{L}=\bar\psi\left(i\gamma^\mu D_\mu-m\right)\psi =\underbrace{\bar\psi\left(i\gamma^\mu\partial_\mu-m\right)\psi}_{\mathcal{L}_0} +\underbrace{e\,\bar\psi\gamma^\mu\psi\,A_\mu}_{\mathcal{L}_{\text{int}}}. \]

Substitute \( D_\mu=\partial_\mu-ieA_\mu \) and expand \( i\gamma^\mu(-ieA_\mu)=+e\gamma^\mu A_\mu \). The interaction is generated, not assumed: this is minimal coupling. A

\[ \mathcal{L}_{\text{int}}=e\,j^\mu A_\mu,\qquad j^\mu\equiv\bar\psi\gamma^\mu\psi. \]

Read off the current: \( j^\mu \) is precisely the Noether current of the global U(1) of Step 1 (Noether's first theorem). The gauge field couples to its own source current. B

\[ \mathcal{L}_{\text{Maxwell}}=-\tfrac14 F_{\mu\nu}F^{\mu\nu},\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu. \]

To give \( A_\mu \) dynamics we need a gauge-invariant kinetic term. \( F_{\mu\nu} \) is invariant because \( \partial_\mu\partial_\nu\alpha-\partial_\nu\partial_\mu\alpha=0 \); it is the unique lowest-dimension invariant built from \( A_\mu \) (prior result: Maxwell from a gauge Lagrangian). B

\[ \partial_\mu F^{\mu\nu}=e\,j^\nu\ \Longrightarrow\ \partial_\nu\partial_\mu F^{\mu\nu}=e\,\partial_\nu j^\nu=0 \ \Longrightarrow\ \boxed{\partial_\mu j^\mu=0}. \]

Vary the full action \( \mathcal{L}_{\text{Maxwell}}+\mathcal{L} \) with respect to \( A_\nu \) (inhomogeneous Maxwell). \( F^{\mu\nu} \) is antisymmetric, so \( \partial_\nu\partial_\mu F^{\mu\nu}\equiv0 \) identically; consistency forces current conservation. Equivalently, gauge invariance \( \delta\mathcal{L}=e\,j^\mu\partial_\mu\alpha \) integrates by parts to \( -e(\partial_\mu j^\mu)\alpha \), which must vanish for arbitrary \( \alpha \). C

Result

\[ D_\mu=\partial_\mu-ieA_\mu,\qquad \mathcal{L}=\bar\psi\left(i\gamma^\mu D_\mu-m\right)\psi-\tfrac14F_{\mu\nu}F^{\mu\nu},\qquad \partial_\mu j^\mu=0. \]

Reading. Local U(1) invariance is not free: it demands a compensating gauge field \( A_\mu \), packaged with \( \partial_\mu \) into a covariant derivative that transforms like the field it acts on. Expanding \( D_\mu \) manufactures the coupling \( e\,j^\mu A_\mu \) — the photon talks to the electron only through the conserved current \( j^\mu=\bar\psi\gamma^\mu\psi \). Because \( F^{\mu\nu} \) is antisymmetric, the field equation \( \partial_\mu F^{\mu\nu}=e\,j^\nu \) can only be consistent if \( \partial_\mu j^\mu=0 \): gauge symmetry and charge conservation are the same statement.

Units check. In natural units \( (\hbar=c=1) \) a Lagrangian density has mass dimension \( 4 \). With \( [\psi]=\tfrac32 \), \( [\partial_\mu]=[A_\mu]=1 \), and \( [e]=0 \): the free term \( \bar\psi\gamma^\mu\partial_\mu\psi \) has \( \tfrac32+1+\tfrac32=4 \), and the coupling \( e\,\bar\psi\gamma^\mu A_\mu\psi \) has \( 0+\tfrac32+1+\tfrac32=4 \). Both match, confirming \( eA_\mu \) is dimensionally identical to \( \partial_\mu \) — as it must be to sit inside \( D_\mu \). (In SI the same object is \( D_\mu=\partial_\mu+\tfrac{iq}{\hbar}A_\mu \), and \( j^\mu \) carries units of number-current density.)

Limiting cases

\( \alpha=\text{const} \) (global limit): \( \partial_\mu\alpha=0 \), so \( D_\mu\to\partial_\mu \) and \( A_\mu \) decouples; one recovers ordinary Noether charge conservation without any gauge field.
\( e\to 0 \): the coupling \( e\,j^\mu A_\mu \) vanishes and matter and radiation become independent free theories; \( A_\mu \) is a free (Maxwell) photon.
Non-relativistic reduction: minimal coupling descends to \( \hat{H}=\dfrac{(\hat{\mathbf p}-e\mathbf A)^2}{2m}+e\phi \); the same \( D_\mu \) that generated QED becomes the Schrödinger/Pauli minimal substitution \( \mathbf p\to\mathbf p-e\mathbf A \).
Pure-gauge field \( A_\mu=\partial_\mu\alpha \): then \( F_{\mu\nu}=0 \) and locally \( D_\mu\psi \) is just a phase-rotated \( \partial_\mu\psi \); no physical field is present (yet global holonomies can survive — Aharonov–Bohm).

Breaks when

The gauge symmetry is anomalous. A U(1) that is a symmetry of the classical action can fail to survive quantisation: the triangle diagram gives \( \partial_\mu j^\mu_5=\dfrac{e^2}{16\pi^2}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\neq0 \) (chiral/ABJ anomaly). If the gauged current is anomalous the theory is inconsistent; the derivation's classical conservation no longer holds.
The gauge symmetry is spontaneously broken. With a charged scalar acquiring \( \langle\phi\rangle\neq0 \), the photon eats a Goldstone mode and becomes massive (Higgs mechanism); \( A_\mu \) is no longer a free gauge field and the naive "unique massless photon" conclusion fails, though \( \partial_\mu j^\mu=0 \) still holds.
Topologically non-trivial configurations. On multiply-connected space, or with Dirac monopoles, a single global \( A_\mu \) and single-valued \( \alpha(x) \) do not exist; one needs patched gauge potentials with transition functions, and charge quantisation \( eg=2\pi n \) appears — outside the smooth local argument used here.
Non-Abelian promotion. If the phase group is enlarged (SU(N)), \( [D_\mu,D_\nu]\neq -ieF_{\mu\nu} \) as a c-number; \( F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+ig[A_\mu,A_\nu] \) and the current is only covariantly conserved, \( D_\mu j^\mu=0 \), not \( \partial_\mu j^\mu=0 \).

Failure modes

Sign/charge convention muddle. Writing \( D_\mu=\partial_\mu+ieA_\mu \) together with \( \psi\to e^{ie\alpha}\psi \) and \( A_\mu\to A_\mu+\partial_\mu\alpha \) gives an uncancelled gradient. The three conventions (phase sign, \( D_\mu \) sign, \( A_\mu \) shift sign) are locked together; you may fix any one, not all independently.
Thinking the interaction was added by hand. Students often "add" \( e\bar\psi\slashed{A}\psi \) as a separate postulate. The whole point is that it is forced by expanding \( D_\mu \); it is not a free choice.
Forgetting the \( \bar\psi \) phase. Treating \( \bar\psi \) as inert makes \( \bar\psi\psi \) appear non-invariant; \( \bar\psi \) rotates by \( e^{-ie\alpha} \), which is exactly what makes bilinears invariant.
Believing global symmetry already needs a gauge field. A global U(1) needs nothing new; only the local version requires \( A_\mu \). Conflating the two erases the content of the argument.
Deriving \( \partial_\mu j^\mu=0 \) only "on-shell for \( \psi \)". That is the Noether route and is fine, but the gauge argument gives a stronger, kinematic reason: antisymmetry of \( F^{\mu\nu} \) forces it regardless of the matter equations of motion.
Assuming \( A_\mu \) is gauge invariant. Only \( F_{\mu\nu} \) and closed-loop holonomies \( \oint A_\mu dx^\mu \) are physical; \( A_\mu \) itself is convention-dependent.

Discussion

The engine of the whole construction is the mismatch in Step 2: \( \partial_\mu \) compares field values at neighbouring points, but a local phase rotation makes "same phase here and there" a meaningless statement. The covariant derivative repairs this by supplying a connection \( A_\mu \) that tells you how to parallel-transport the phase convention from one point to the next. In this geometric language \( ieA_\mu \) is a U(1) connection, \( D_\mu\psi \) is the covariant derivative of a section of a line bundle, and \( -ieF_{\mu\nu}=[D_\mu,D_\nu] \) is its curvature. Electromagnetism is the curvature of the phase of the electron field.

The logical direction is worth dwelling on. Noether's first theorem (a prior result) reads charge conservation off a global symmetry. The gauge argument gives a complementary and in some ways deeper statement: once the symmetry is local, the field equation \( \partial_\mu F^{\mu\nu}=e j^\nu \) makes conservation a near-identity, riding on the antisymmetry of \( F^{\mu\nu} \) rather than on the dynamics of \( \psi \). Charge is conserved because the photon couples to a current, and the photon can only couple consistently to a conserved one. This is why a massless spin-1 field is obliged to couple to a conserved current — the same reason a massless spin-2 field must couple to the conserved stress tensor (general relativity).

Minimal coupling is also the sharpest available statement of the correspondence between symmetry and interaction. The strength \( e \) is the single free parameter; every vertex, every Feynman rule of QED, is dictated by \( D_\mu \). Enlarging the group from U(1) to SU(2)×U(1) or SU(3) reruns exactly this argument with matrix-valued \( A_\mu \), and the extra term \( ig[A_\mu,A_\nu] \) in the field strength — absent here only because U(1) is Abelian — is what makes gluons self-interacting and gives QCD its asymptotic freedom.

At the quantum level the story acquires a crucial caveat. Gauge invariance is a redundancy of description, not a symmetry of states, and it must survive quantisation exactly or the theory is sick. The Ward–Takahashi identities that enforce this are the quantum shadow of \( \partial_\mu j^\mu=0 \), guaranteeing photon transversality \( k_\mu M^\mu=0 \) and the equality of charge and wavefunction renormalisation \( Z_1=Z_2 \). When a gauged current suffers a triangle anomaly, this fails catastrophically — which is precisely why the Standard Model's hypercharges are tuned so that gauge anomalies cancel generation by generation. The innocuous-looking demand "let the phase be local" thus reaches all the way to the quark and lepton charge assignments.

Common misconceptions. (i) "Gauge symmetry is a physical symmetry that could be broken by experiment" — no; it is a redundancy in our variables, and \( A_\mu \) is not measurable, only \( F_{\mu\nu} \) and holonomies are. (ii) "The photon is massless because of gauge symmetry" — more precisely, a mass term \( m^2A_\mu A^\mu \) is not gauge invariant, so masslessness is protected by the symmetry unless it is spontaneously broken. (iii) "Minimal coupling is an approximation" — it is the exact, unique lowest-dimension gauge-invariant coupling; corrections like the anomalous magnetic moment arise from quantum loops, not from a better classical ansatz.

Worked examples

\[ \textbf{Aharonov–Bohm phase from the connection.}\quad \Delta\varphi=e\oint_C A_\mu\,dx^\mu=e\,\Phi_B. \]

A charged particle encircling a confined magnetic flux \( \Phi_B \) picks up a phase equal to the charge times the holonomy of \( A_\mu \), even where \( F_{\mu\nu}=0 \) along its path. Symbols first. B

\[ \Delta\varphi=\frac{q\,\Phi_B}{\hbar}\quad(\text{SI, restoring }\hbar). \]

Restore units: the dimensionless phase is \( q\Phi_B/\hbar \), with \( q \) the particle charge and \( \Phi_B \) in webers. A

\[ \Delta\varphi=\frac{(1.602\times10^{-19}\,\mathrm{C})(1.00\times10^{-15}\,\mathrm{Wb})}{1.055\times10^{-34}\,\mathrm{J\,s}}. \]

Insert an electron \( q=e=1.602\times10^{-19}\,\mathrm C \) encircling a solenoid carrying \( \Phi_B=1.00\times10^{-15}\,\mathrm{Wb} \) (about half a flux quantum, \( \Phi_0=h/e=4.14\times10^{-15}\,\mathrm{Wb} \)). A

\[ \Delta\varphi\approx1.52\ \text{rad}\ \ (\approx0.24\,\text{of a full turn}). \]

Reading. The interference fringe shifts by \( \Delta\varphi/2\pi\approx0.24 \) of a period even though the electron never enters the field region — direct experimental evidence that the connection \( A_\mu \), through its holonomy, is physical while \( A_\mu \) pointwise is not. Units. \( \mathrm{C\cdot Wb/(J\,s)}=\mathrm{C\cdot(V\,s)/(J\,s)}=\mathrm{C\,V/J}=1 \), correctly dimensionless.

\[ \textbf{Minimal coupling in the non-relativistic limit: cyclotron gap.}\quad \hat H=\frac{(\hat{\mathbf p}-e\mathbf A)^2}{2m}. \]

The relativistic \( D_\mu \) reduces to the substitution \( \hat{\mathbf p}\to\hat{\mathbf p}-e\mathbf A \). For uniform \( \mathbf B=B\hat{\mathbf z} \) the spectrum is Landau levels \( E_n=\hbar\omega_c\left(n+\tfrac12\right) \). Symbols first. B

\[ \omega_c=\frac{eB}{m},\qquad \Delta E=\hbar\omega_c=\frac{\hbar eB}{m}. \]

The level spacing is \( \hbar\omega_c \) with cyclotron frequency \( \omega_c=eB/m \); this frequency is entirely a consequence of minimal coupling. A

\[ \Delta E=\frac{(1.055\times10^{-34})(1.602\times10^{-19})(1.00)}{9.11\times10^{-31}}\ \mathrm{J}. \]

Take an electron \( m=9.11\times10^{-31}\,\mathrm{kg} \) in \( B=1.00\,\mathrm T \). Numbers only now that the symbolic form is fixed. A

\[ \omega_c\approx1.76\times10^{11}\,\mathrm{rad/s},\qquad \Delta E\approx1.86\times10^{-23}\,\mathrm J\approx0.116\ \mathrm{meV}. \]

Reading. The observable Landau-level ladder — seen in cyclotron resonance, the quantum Hall effect, and de Haas–van Alphen oscillations — is set entirely by the \( \mathbf p\to\mathbf p-e\mathbf A \) rule inherited from \( D_\mu \). A 0.116 meV gap corresponds to \( \sim28\,\mathrm{GHz} \), squarely in the microwave band used for cyclotron-resonance spectroscopy. Units. \( \mathrm{(J\,s)(C)(T)/kg}=\mathrm{(J\,s\,C\cdot kg\,s^{-2}A^{-1})/kg}=\mathrm{J} \), since \( \mathrm{T=kg\,s^{-2}A^{-1}} \) and \( \mathrm{C=A\,s} \).

Problems

Starting from \( \psi\to e^{ie\alpha(x)}\psi \) and demanding that \( D_\mu\psi=(\partial_\mu-ieA_\mu)\psi \) transform as \( D_\mu\psi\to e^{ie\alpha}D_\mu\psi \), derive the required transformation law of \( A_\mu \).
Solution
Require \( D_\mu'\psi'=e^{ie\alpha}D_\mu\psi \). Compute \( D_\mu'\psi'=(\partial_\mu-ieA_\mu')e^{ie\alpha}\psi=e^{ie\alpha}[\partial_\mu\psi+ie(\partial_\mu\alpha)\psi-ieA_\mu'\psi] \). Setting this equal to \( e^{ie\alpha}(\partial_\mu-ieA_\mu)\psi \) and cancelling \( e^{ie\alpha}\psi \)-factors gives \( ie(\partial_\mu\alpha)-ieA_\mu'=-ieA_\mu \), i.e. \( \boxed{A_\mu'=A_\mu+\partial_\mu\alpha} \). The gradient shift of \( A_\mu \) exactly compensates the gradient of the local phase.
The Heaviside–Lorentz fine-structure constant is \( \alpha_{\rm em}=\dfrac{e^2}{4\pi\hbar c}\approx\dfrac{1}{137.036} \). Compute the dimensionless coupling \( e \) that appears in \( D_\mu \) (natural units).
Solution
In natural units \( \hbar=c=1 \), so \( \alpha_{\rm em}=e^2/4\pi \). Thus \( e=\sqrt{4\pi\alpha_{\rm em}}=\sqrt{4\pi/137.036}=\sqrt{0.09170}\approx\boxed{0.3028} \). This is the electromagnetic gauge coupling at low energy; it runs slowly upward with energy scale (e.g. \( \alpha_{\rm em}^{-1}\approx128 \) at \( M_Z \)).
Show directly from the Dirac equation \( (i\gamma^\mu\partial_\mu-m)\psi=0 \) and its conjugate that \( j^\mu=\bar\psi\gamma^\mu\psi \) satisfies \( \partial_\mu j^\mu=0 \).
Solution
From the Dirac equation, \( \gamma^\mu\partial_\mu\psi=-im\psi \). The Dirac conjugate obeys \( \partial_\mu\bar\psi\,\gamma^\mu=+im\bar\psi \) (take the Hermitian conjugate of the Dirac equation, multiply by \( \gamma^0 \), using \( \gamma^{0}\gamma^{\mu\dagger}\gamma^{0}=\gamma^\mu \)). Then \( \partial_\mu j^\mu=(\partial_\mu\bar\psi)\gamma^\mu\psi+\bar\psi\gamma^\mu(\partial_\mu\psi)=(im\bar\psi)\psi+\bar\psi(-im\psi)=im\bar\psi\psi-im\bar\psi\psi=0 \). Hence \( \boxed{\partial_\mu j^\mu=0} \); the mass terms cancel, so the current is conserved for any \( m \).
Verify that \( F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu \) is gauge invariant under \( A_\mu\to A_\mu+\partial_\mu\alpha \), and explain why a photon mass term \( \tfrac12 m_\gamma^2 A_\mu A^\mu \) is forbidden.
Solution
Under the shift, \( F_{\mu\nu}\to\partial_\mu(A_\nu+\partial_\nu\alpha)-\partial_\nu(A_\mu+\partial_\mu\alpha)=F_{\mu\nu}+(\partial_\mu\partial_\nu-\partial_\nu\partial_\mu)\alpha=F_{\mu\nu} \), since mixed partials commute. So \( F_{\mu\nu} \) — and hence \( -\tfrac14F_{\mu\nu}F^{\mu\nu} \) — is invariant. A mass term transforms as \( A_\mu A^\mu\to(A_\mu+\partial_\mu\alpha)(A^\mu+\partial^\mu\alpha)=A_\mu A^\mu+2A_\mu\partial^\mu\alpha+\partial_\mu\alpha\,\partial^\mu\alpha\neq A_\mu A^\mu \). It is not gauge invariant, so \( m_\gamma=0 \) is enforced by the symmetry (unless it is spontaneously broken via the Higgs mechanism, which supplies the missing longitudinal mode).
A superconducting ring of area \( A=1.0\,\mathrm{mm}^2 \) sits in a perpendicular field. Using flux quantisation \( \Phi=n\Phi_0 \) with \( \Phi_0=h/(2e) \) (Cooper-pair charge \( 2e \)), find the field spacing \( \Delta B \) between adjacent quantised flux states, and comment on why the charge is \( 2e \).
Solution
Flux quantum for charge \( 2e \): \( \Phi_0=h/(2e)=6.626\times10^{-34}/(2\times1.602\times10^{-19})=2.07\times10^{-15}\,\mathrm{Wb} \). Adjacent states differ by \( \Delta\Phi=\Phi_0=B\!\cdot\!A \)-spacing, so \( \Delta B=\Phi_0/A=2.07\times10^{-15}/(1.0\times10^{-6})=\boxed{2.07\times10^{-9}\,\mathrm T\approx2.1\ \mathrm{nT}} \). The charge is \( 2e \) because the gauge-coupled field in a superconductor is the charged Cooper-pair condensate, \( \psi\to e^{i(2e)\alpha}\psi \); the doubled charge in the minimal-coupling phase halves the flux quantum relative to single electrons, a direct experimental fingerprint of the \( D_\mu \) charge assignment.