Zero-Point Energy per Mode
E_ZPE = ħω/2 — cannot be removed!
ΔE · Δt ≥ ħ/2 → virtual pairs live Δt ≈ ħ/2mc²
Renormalization: subtract divergence, keep differences
The vacuum is not empty. Each electromagnetic mode carries zero-point energy ½ħω. Virtual photon pairs constantly pop in and out. This isn't metaphor — it's measurable via the Casimir effect (two plates attract), the Lamb shift (2s₁/₂ level shifted by ~1 GHz), and spontaneous emission rates.
The total zero-point energy diverges — an ultraviolet catastrophe resolved only by renormalization or a physical UV cutoff. The cosmological constant problem asks why gravity doesn't respond to this ~10¹²⁰ overestimate.