where in the second equality we re-positioned x ^ ^ ( t ) using the cyclic property of trace. Next, in the third equality, we inserted e − − β β H ^ ^ e β β H ^ ^ next to the trace and interpreted e − − β β H ^ ^ as a time evolution operator e − − i ℏ ℏ H ^ ^ Δ Δ t with imaginary time interval Δ Δ t = − − i ℏ ℏ β β . The imaginary time shift turns into a e − − β β ℏ ℏ ω ω factor after Fourier transform