Interpretation and applications of Sokhotski–Plemelj theorem in physics

Question

Sokhotski–Plemelj theorem states: $$ \tag{1} \frac{1}{x + i0} = \text{P}\frac 1 x - i \pi\delta(x) $$ I have seen this theorem being used in QFT and in non relativistic QM (collision theory, Green functions). However all the applications I've seen so far were using (1) as simple mathematical trick.

I want to know if (1) has some kind of (simple) physical interpretation (meaning)? Such interpretation would hopefully make using (1) much more natural in particular example. Can you also give me some applications of (1) which illustrate given physical interpretation of (1) ?

Answer 1

One of the most intuitive and basic ways this theorem enters physics is through Fourier transforms. Either side of this function is the Fourier transform of the Heaviside step function.

This is particularly relevant when we think of time-dependent physics, where the Heaviside function often appears in initial value problems. We then have

$$ \int_{-\infty}^\infty \Theta (t) e^{i\omega t} \mathrm{d} t = \pi \delta(\omega) + \mathrm{P} \frac{i}{\omega}. $$

In practice, it is often convenient to work in frequency space and then transform to the time domain. In this context, the Sokhotski–Plemelj theorem allows you recover the so-called $i\varepsilon$ prescription. E.g.

$$\Theta (t) \propto \lim_{\varepsilon\rightarrow 0} \int_{-\infty}^\infty \frac{1}{\omega + i\varepsilon} e^{-i\omega t} \mathrm{d} \omega . $$

If followed through rigorously, this Fourier relation and the idea of solving initial (or final) value problems underlies many concepts in scattering theory (in QM and QFT alike).

Answer 2

Feynmann-Dyson expansion (i.e., using Feynman diagrams) routinely involves calculating the integrals of factors like $$\prod_\alpha\frac{1}{x - x_\alpha +i0^+}\prod_\beta\frac{1}{x - x_\beta +i0^-}.$$ The sign of the imaginary factor typically corresponds to the retarded or advanced nature of the (part of the) Green's function. I would be however cautious to attribute specific physical meaning to such a basic mathematical formula - it is akin attributing physical meaning to addition or multiplication - it obviously depends on the context.