We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as G ( x , τ τ ∣ ∣ x ′ , τ τ ′ ) = ⟨ ⟨ T ψ ψ ( x , τ τ ) ψ ψ ¯ ¯ ( x ′ , τ τ ′ ) ⟩ ⟩ .