The real-time two-point Green function can be written in terms of 'retarded' and 'advanced' Green functions, which will turn out to have simpler analyticity properties. The retarded and advanced Green functions are defined by G R ( x t ∣ ∣ x ′ t ′ ) = − − i ⟨ ⟨ [ ψ ψ ( x , t ) , ψ ψ ¯ ¯ ( x ′ , t ′ ) ] ζ ζ ⟩ ⟩ Θ Θ ( t − − t ′ ) and G A ( x t ∣ ∣ x ′ t ′ ) = i ⟨ ⟨ [ ψ ψ ( x , t ) , ψ ψ ¯ ¯ ( x ′ , t ′ ) ] ζ ζ ⟩ ⟩ Θ Θ ( t ′ − − t ) , respectively.