Consider the imaginary time Greens function of a fermionion field $Ψ(x,τ)$ at zero temperature

$$ G^τ = -\langle θ(τ)Ψ(x,τ)Ψ^\dagger(0,0) - θ(-τ)Ψ^\dagger(0,0)Ψ(x,τ) \rangle $$

It is well known that we can obtain the retarded Greens function by performing Fourier transformation into frequency space and performing the analytic continuation $iω \to ω + i\eta$.

What I would like to do is to perform the analytic continuation directly in the form $iτ \to t$, but I don't know how to deal with the $θ(τ)$ terms.

How to perform the analytic continuation $iτ \to t$ of the step function $θ(τ)$?

In my case, I am dealing with a chiral Luttinger liquid, giving something like

$$ G^τ(x,τ) = -\left[θ(τ)\frac i{iλ + ivτ - x} - θ(-τ)\frac i{iλ - ivτ - x}\right] $$

where $λ \approx 0$ is an infinitesimal but important regularization. Of course, the analytic continuation into the time domain is going to look something like

$$ \frac1{iλ + vt - x} $$

but I'm interested in the precise form.

Also, I'm ultimately interested in the spectral function, so I don't mind if analytic continuation gives me yet another variant of a Greens function, but I would like to obtain it precisely from the imaginary time Greens function without going through a tedious Fourier transform. For instance, Giuliani and Vignale's book "Quantum Theory of the Electron Liquid" uses the Greens function $G_{>}(x,t)$ to great effect (equation (9.133)).

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