Field Quantisation: Creation Operators and Particle Number

Take a free real scalar field on 0 < x < L with Dirichlet ends, φ(0, t) = φ(L, t) = 0. Expand in the orthonormal basis e_j(x) = √(2/L) sin(jπx/L) for j = 1, 2, …, so φ(x, t) = Σ_j q_j(t) e_j(x). Orthogonality, ∫₀ᴸ e_i e_j dx = δ_ij, makes each coefficient q_j an independent coordinate. The canonically normalized field density is ℒ = ½(φ_t² − c²φ_x² − (mc²/ħ)²φ²), with L > 0 the box length, c the wave speed, m the mass, and ħ the reduced Planck constant; the mass term vanishes for m = 0.

Integrating ℒ over 0 < x < L using orthogonality, the Lagrangian splits into Σ_j (½q̇_j² − ½ω_j² q_j²) with ω_j = c·jπ/L for the massless case, one unit harmonic oscillator per mode; a massive scalar has ω_j = √(c²(jπ/L)² + m²c⁴/ħ²). Levels are E_n = (n + ½)ħω_j for n = 0, 1, 2, …; n counts quanta above ground. The ground state keeps ½ħω_j of zero-point energy, which is energy of the mode, not half a particle. A number eigenstate has a sharp occupation number; a generic superposition or coherent state need not. Fermions differ: each mode holds at most one quantum, occupation 0 or 1.

Photons are the quanta of the electromagnetic field specifically. With a† the creation operator, a the annihilation operator, and N = a†a the number operator, the bosonic ladder is a†|n⟩ = √(n+1)|n+1⟩, a|n⟩ = √n|n−1⟩, a|0⟩ = 0. Equal-time commutation [a_i, a_j†] = δ_ij underlies bosonic symmetry: [a_i†, a_j†] = 0, so a_i†a_j†|0⟩ is unchanged under exchange. Fermions anticommute, {A, B} = AB + BA: {c_i, c_j†} = δ_ij and {c_i†, c_j†} = 0, so (c_i†)² = 0, a mode holds at most one quantum, and distinct-mode creators give an antisymmetric sign under exchange. Exchange symmetry comes from the operator algebra, not from never storing labels.

In the free theory, where modes are independent, the number operator N is time independent and commutes with the Hamiltonian, [H₀, N] = 0, so particle number is conserved. Interactions can couple modes. A time-dependent driving term or an interaction can make N fail to commute with H, and then occupation numbers can change; for the full closed system including any drivers or sources, energy, momentum, and conserved charges such as electric charge still balance. So the existence of creators does not by itself make number nonconserved; the criterion is whether the Hamiltonian commutes with N.