In real time, we will explicitly indicate the time-ordered function with a superscript T: G T ( x t ∣ ∣ x ′ t ′ ) = ∫ ∫ k d k ∫ ∫ d ω ω 2 π π G T ( k , ω ω ) e i k ⋅ ⋅ ( x − − x ′ ) − − i ω ω ( t − − t ′ ) .