Using the fact that [ a k λ λ ( 0 ) , a k ′ λ λ ′ † † ( 0 ) ] = δ δ k k ′ 3 , δ δ λ λ λ λ ′ and that equal-time particle and field operators commute, we obtain: = [ z ( t ) , m z ˙ ˙ ( t ) ] + [ z ( t ) , e c A z ( t ) ] = [ z ( t ) , m z ˙ ˙ ( t ) ] = ( i ℏ ℏ e 2 2 π π 2 m c 3 ) ( 8 π π 3 ) ∫ ∫ 0 ∞ ∞ d ω ω ω ω 4 ( ω ω 2 − − ω ω 0 2 ) 2 + τ τ 2 ω ω 6