According to our earlier equation for a k λ (t) the free field is the only field in existence at t = 0 as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system at t = 0 is therefore of the form | Ψ Ψ ⟩ ⟩ = | vac ⟩ ⟩ | ψ ψ D ⟩ ⟩ , where |vac⟩ is the vacuum state of the field and | ψ D ⟩ is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero: ⟨ ⟨ E 0 ( t ) ⟩ ⟩ = ⟨ ⟨ Ψ Ψ | E 0 ( t ) | Ψ Ψ ⟩ ⟩ = 0 since a k λ (0)|vac⟩ = 0. however, the energy density associated with the free field is infinite: 1 4 π π ⟨ E 0 2 ( t ) ⟩ = 1 4 π π ∑ ∑ k λ λ ∑ ∑ k ′ λ λ ′ 2 π π ℏ ℏ ω ω k V 2 π π ℏ ℏ ω ω k ′ V × × ⟨ a k λ λ ( 0 ) a k ′ λ λ ′ † † ( 0 ) ⟩ = 1 4 π π ∑ ∑ k λ λ ( 2 π π ℏ ℏ ω ω k V ) = ∫ ∫ 0 ∞ ∞ d w ρ ρ 0 ( ω ω )