A Lehmann Kallen and spectral representation

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I'd like to better understand the lehmann kallen representation of the propagator.
Hello,

My question pertains to the formula below:
1644723952025.png


In particular, I would like to ask about the spectral density function shown below:
1644724002731.png
What does the spectral function physically represent? Is there any interpretation of its meaning, whether it has a relation to the physical spectrum of the theory.

My work:

The propagator real poles correspond to particles (a particle is a lump of energy that doesn't decay or split), it seems to suggest that the spectral function would have delta functions at 1 particle state at rest. Obviously, in a lorentz invariant QFT, one can boost this one particle state to get a continuous spectrum, so \rho(p^2) is not the spectrum of the theory, but it has some relation to it.

Thank you.
 
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