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In my condensed matter course there is written :

"In a non interacting system, particle/hole excitations have finite lifetime".

I would like to understand why.

I guess we are talking about of non interacting in the sense that the quasiparticle don't interact between them (but the electron interact with the Fermi sea).

Also, it is written that the inverse of the lifetime of an excitation is proportional to $\epsilon^2=(E-\mu)^2$ where $E$ is the energy of the particle and $\mu$ the chemical potential. Where does this formula come from ?

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Short google search brought up the following: The relaxation is based on the Landau theory of Fermi liquids. The proportionality was calculated by Quinn, see Phys. Rev. 126, p1453 (1962). It is a consequence of the Pauli-principle restrictions on the number of available unoccupied states for both electrons involved in a scattering process.

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