Electric charge, U(1) phase and superselection

Take every field that carries charge q and multiply it by the same phase, ψ_q → exp(iqα/e) ψ_q, with one angle α for the whole system. A neutral field is untouched; an electron field turns by exp(−iα), a proton field by exp(+iα), a Δ⁺⁺ field by exp(+2iα). Every observable is built from products such as ψ_q* ψ_q, or the neutron-decay term (ψₚ* ψₙ)(ψₑ* ψ_ν), in which the charge created equals the charge destroyed, so the angle cancels and nothing measurable changes: the rotation is a symmetry. Because α is continuous the rotations form a one-parameter unitary group, U(α) = exp(iαQ/e), and Stone's theorem hands you a Hermitian generator — the charge operator Q, whose eigenvalue on a state holding n elementary charges is ne. Two features matter. The angle lives on a circle, α ≡ α + 2π, so the group is the compact U(1), which is why the eigenvalues of Q/e can all be integers. And this is not the overall phase of the state vector, which is unobservable for any system and generates nothing; it is a relative rotation between components of different charge.

Noether's theorem promises a conserved current for every continuous symmetry, and for the Schrödinger field you can check it by hand. With iℏ ∂ψ/∂t = −(ℏ²/2m)∇²ψ + Vψ and V real, differentiate |ψ|²: ∂(ψ*ψ)/∂t = (iℏ/2m)(ψ*∇²ψ − ψ∇²ψ*), the potential terms cancelling exactly because V is real — the condition that makes H Hermitian. A complex, absorptive V is just as blind to the phase rotation yet drains |ψ|², so the symmetry alone is not enough: Noether's theorem also needs a real action. The bracket is a divergence, ∇⋅(ψ*∇ψ − ψ∇ψ*), so with ρ = q|ψ|² and j = (qℏ/m) Im(ψ*∇ψ) = (qℏ/2mi)(ψ*∇ψ − ψ∇ψ*) you have ∂ρ/∂t + ∇⋅j = 0 identically, whatever the state. Test it on a plane wave ψ = A exp(i(kx − ωt)): ρ = q|A|² and j = ρ(ℏk/m) x̂, charge density times velocity as a current should be, and both are constant, so the equation reads 0 + 0. Multiply ψ by exp(iα) and neither ρ nor j notices; that blindness is the symmetry, and the current is its bookkeeping. Relativistically the same content is the one equation ∂_μ j^μ = 0 with j^μ = (cρ, j), and for a Dirac field j^μ = qc ψ̄ γ^μ ψ.

In operator language the argument is one line. The symmetry is U(α) = exp(iαQ/e); H is invariant when U H U† = H for every α; expand to first order in α and that is [H, Q] = 0. The Heisenberg equation then gives d⟨Q⟩/dt = (i/ℏ)⟨[H, Q]⟩ = 0 in every state, and because H and Q commute they share an eigenbasis, so a state of definite charge stays one under time evolution. The S-matrix is built from H, so it commutes with Q too: every reaction balances the charge column. Now look at what a violation would be. In the two-state space of |ν⟩ (Q = 0) and |e⁻⟩ (Q = −e), Q = (e/2)(σ_z − 1), and a mixing term H′ = εσₓ gives [H′, Q] = −iεe σ_y, so d⟨Q⟩/dt = (εe/ℏ)⟨σ_y⟩ — for ε = 1 μeV that is 1.5 × 10⁹ e per second in a state with ⟨σ_y⟩ = 1. A conservation law is not approximately true when the breaking term is small; it is false, at a rate the term sets, and a rate is what experiments bound. The commutator does not say why the eigenvalues of Q are multiples of e, or why the proton and positron charges agree to a part in 10²¹. Those facts need more than a global U(1).

Let the angle vary from point to point, ψ → exp(iqχ(x)/ℏ) ψ. The derivative now picks up an unwanted piece, ∂_μ ψ → exp(iqχ/ℏ)(∂_μ + (iq/ℏ) ∂_μ χ)ψ, and the kinetic term is no longer invariant. Repair it with a vector field that shifts to cancel the intruder: with D_μ = ∂_μ + (iq/ℏ)A_μ and A_μ → A_μ − ∂_μ χ, the combination D_μ ψ rotates exactly as ψ does. The field strength F_μν = ∂_μ A_ν − ∂_ν A_μ is invariant, so a Maxwell term is allowed; a mass term m²A_μ A^μ is not, so the photon is massless — the present limit is m_γ < 10⁻¹⁸ eV. The field equation ∂_μ F^μν = μ₀ j^ν is where the payoff sits: F is antisymmetric, so ∂_ν ∂_μ F^μν = 0 identically, and ∂_ν j^ν = 0 is forced by the equations themselves. Charge conservation is no longer one symmetry among many; it is a consistency condition of electrodynamics, and a non-conserved source could not be coupled to the photon at all. The same structure gives Gauss's law, Q = ε₀ ∮E⋅dA over any surface enclosing the system, which means the total charge can be read from arbitrarily far away.

Gauss's law makes Q measurable on a sphere at spatial infinity, spacelike to any apparatus, so no local operation can change it: every physical observable A commutes with Q. Sandwich [Q, A] = 0 between charge eigenstates: (Q₁ − Q₂)⟨Q₁|A|Q₂⟩ = 0, so ⟨Q₁|A|Q₂⟩ = 0 whenever Q₁ ≠ Q₂. Now take |ψ⟩ = a|Q₁⟩ + b exp(iφ)|Q₂⟩ with |a|² = 0.6 and |b|² = 0.4, and a probe with ⟨A⟩₁ = 2 and ⟨A⟩₂ = −1. The cross term vanishes, so ⟨A⟩ = 0.6(2) + 0.4(−1) = 0.8 for every φ: the relative phase is invisible, and the state cannot be told from the mixture 0.6|Q₁⟩⟨Q₁| + 0.4|Q₂⟩⟨Q₂|. Compare spin, where S_z is also conserved by a rotationally symmetric H but Sₓ connects the sectors: the same weights give ⟨σₓ⟩ = 2√(0.6 × 0.4) cos φ = 0.98 cos φ, and the phase is plainly there. Conservation does not imply superselection; the extra ingredient is that nothing connects the sectors. The rule concerns the total charge of a closed system. A subsystem may sit in a superposition of charges, as a BCS condensate carries an indefinite number of Cooper pairs, with the environment holding the complement.

Because the rule is exact, its test is a search for an absence. The cleanest charge-violating process is e⁻ → ν + γ: it balances every other ledger line and fails only charge. Two-body kinematics fixes the signature — an electron at rest shares mₑ c² = 511.0 keV equally between two massless products, so the photon carries 255.5 keV. Borexino watched the 2.5 × 10³¹ electrons of a 75.5 t fiducial volume for 408 days, saw no excess there, and set τ > 6.6 × 10²⁸ years at 90% confidence, some 5 × 10¹⁸ times the age of the universe. The number does not prove conservation; it bounds the coupling of any charge-changing term, and background at 255 keV, not the electron count, sets it — a background-free search of the same exposure would reach 10³¹ years. Theory expects zero, not small: Okun and Zeldovich argued in 1978 that a charge-changing vertex with a massless photon drags in an unbounded cloud of soft photons, so a small violation is not even consistent. It does not test quantisation: the equality of proton and electron charges to a part in 10²¹ is a separate question the global U(1) leaves open.