In deriving these equations for x, p, and a k λ we have used the fact that equal-time particle and field operators commute. This follows from the assumption that particle and field operators commute at some time (say, t = 0) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture operator A (t) evolves in time as A (t) = U (t) A (0) U (t), where U (t) is the time evolution operator satisfying i ℏ ℏ U ˙ ˙ = H U , U † † ( t ) = U − − 1 ( t ) , U ( 0 ) = 1 .