Complex scalar field: antiparticles and the charge operator

Units throughout: ħ = c = 1, so energy, momentum and mass are in MeV and length in MeV⁻¹, converted with ħc = 197.327 MeV fm; the metric is (+, −, −, −). Topic 9.02 is used, not repeated: a real field puts aₖ on e^(−ik⋅x) and aₖ† on e^(ik⋅x), the second forced by φ = φ†. A complex field obeys (∂_μ∂^μ + m²)φ = 0 with no reality condition, so the coefficients of the two frequencies are independent; call them aₖ and bₖ†. The conjugate momenta are π = ∂ℒ/∂(∂₀φ) = ∂₀φ† and π† = ∂₀φ, and [φ(r, t), π(r′, t)] = [φ†(r, t), π†(r′, t)] = iδ³(r − r′), with every other equal-time pair commuting, give two independent oscillator algebras per mode. Nothing new is assumed: with φ = (φ₁ + iφ₂)/√2 for two real fields of one mass, aₖ = (a₁ₖ + ia₂ₖ)/√2 and bₖ = (a₁ₖ − ia₂ₖ)/√2, and every commutator follows from topic 9.02's. Dimension check: φ has the dimension of 1/√(ωₖ V), MeV in these units.

The Legendre transform gives ℋ = ∂₀φ† ∂₀φ + ∇φ†⋅∇φ + m²φ†φ, a sum of squares, each written with the daggered field on the left. Substituting the modes, each oscillating cross term carries −ωₖ² + k² + m² = 0 and cancels, leaving H = Σₖ ωₖ (aₖ†aₖ + bₖ bₖ†) = Σₖ ωₖ (aₖ†aₖ + bₖ†bₖ) + Σₖ ωₖ. Normal ordering drops the last sum, ωₖ per mode, which is ½ωₖ for each of the two real fields: topic 9.02's zero-point energy, counted twice. What is left gives every quantum positive energy, and the operator on the negative frequency is a creator: [H, bₖ†] = ωₖ bₖ†. The momentum, derived the same way, is P = Σₖ k (aₖ†aₖ + bₖ†bₖ); its zero-point sum Σₖ k vanishes because k and −k pair off. So bₖ† adds energy ωₖ and momentum +k exactly as aₖ† does: a π⁻ of momentum 200 MeV has energy √(200² + 139.57²) = 243.9 MeV, the same as a π⁺.

ℒ is unchanged by φ → e^(−iα)φ with φ† → e^(iα)φ†, and Noether's theorem from topic 9.01 gives j^μ = i(φ† ∂^μ φ − (∂^μ φ†) φ), conserved because ∂_μ j^μ = i(φ† ∂_μ∂^μ φ − (∂_μ∂^μ φ†) φ) = i(−m²φ†φ + m²φ†φ) = 0. In modes, with each daggered field written on the left, Q = q∫d³r j⁰ becomes q Σₖ (aₖ†aₖ − bₖ bₖ†), its oscillating terms cancelling as a conserved charge's must. Two different things sit in that line. The minus sign is structural: it gives [Q, aₖ†] = q aₖ† and [Q, bₖ†] = −q bₖ†, so π⁺ and π⁻ carry +e and −e, and equally [Q, φ] = −qφ, so both terms of φ lower the charge by q, one by removing a particle, the other by adding an antiparticle. The constant is not structural: the two orderings give −qΣₖ 1 and +qΣₖ 1, and their average, the one that makes Q odd under charge conjugation, gives zero, so Q = q Σₖ (aₖ†aₖ − bₖ†bₖ) holds exactly.

In the real components the same operator reads Q = iq Σₖ (a₁ₖ†a₂ₖ − a₂ₖ†a₁ₖ), the generator of rotations in the (φ₁, φ₂) plane, and the phase symmetry is that rotation. Each mode is therefore the two-dimensional isotropic oscillator of Quantum Mechanics I: H = ωₖ(nₐ + n_b) is its energy, Q/q = nₐ − n_b its angular momentum, and a and b its circular quanta. A level holding N quanta in one mode is (N + 1)-fold degenerate, with Q = Ne, (N − 2)e, …, −Ne. This settles what the real-field quanta are: a₁ₖ† = (aₖ† + bₖ†)/√2 makes an equal superposition of π⁺ and π⁻, of definite energy and ⟨Q⟩ = 0, whose charge is found to be +e or −e, never 0. Such states are legitimate in the free field. Once the field couples to electromagnetism, the charge superselection rule, quoted here, admits only states of definite Q, which is why laboratories see π⁺ and π⁻ and never a φ₁ quantum.

Unit 3 refused j⁰ as a probability density because it turns negative on negative-frequency solutions. Here it is an operator, and its sign is the charge. In a one-particle packet |f⟩, ⟨f|:j⁰(x):|f⟩ is the bilinear i(F*∂ₜF − F∂ₜF*) of the positive-frequency wavefunction F(x) = ⟨0|φ(x)|f⟩, Unit 3's density up to its 1/2m; in an antiparticle packet |g⟩ it is minus the same bilinear of G(x) = ⟨0|φ†(x)|g⟩. Nothing had to be positive, because nothing was a probability: Pauli and Weisskopf's 1934 proposal, now derived. The Klein-Gordon equation survives as the equation of motion of an operator field, and its negative frequencies need no sea. A bosonic sea could not even be filled, since without exclusion every level takes more quanta, so spin-0 antiparticles sat on Unit 8's list of what hole theory could not do; this field does it. Non-relativistic check: a slow packet has F ≈ e^(−imt)ψ/√(2m), which turns the bilinear into |ψ|², Schrödinger's density, and one-particle quantum mechanics serves a slow pion because a pair costs 2m = 279.14 MeV.

Three checks close the derivation. The real limit: set bₖ = aₖ, so that φ = φ†, and Q = q Σₖ (aₖ†aₖ − aₖ†aₖ) vanishes identically. A real field has no current and carries no charge, and its quantum is its own antiparticle, as the π⁰ is. Charge conjugation: the unitary C with C aₖ C⁻¹ = bₖ and C bₖ C⁻¹ = aₖ leaves the vacuum and H unchanged and sends φ to φ† and Q to −Q, so particle and antiparticle have the same ωₖ at every k and therefore one mass. That is derived for this free field; for interacting fields the equality follows from the CPT theorem, quoted. It is tested: the charged pions have m = 139.57039 ± 0.00018 MeV, and (m(π⁺) − m(π⁻))/m = (2 ± 5) × 10⁻⁴, a difference of (0.03 ± 0.07) MeV, zero within its uncertainty. The coupling to electromagnetism, made by replacing ∂_μ with a covariant derivative containing qA_μ, turns this j^μ into the source of the photon field; it is named here and built in Unit 10.