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What is the importance of deriving the results of perturbation theory in condensed matter physics in terms of spectral functions ?

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In nuclear physics spectral functions can be measured and used when we don't have a detailed theory of the underling phenomena. Lots of phenomenological stuff is that way. – dmckee Jul 10 '13 at 15:38

The spectral function can be measured directly in certain experiments such as ARPES and other scattering experiments.

It is also related to linear responses, such as conductivity, via the Green-Kubo formalism

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In condensed matter physics (or many-body physics in general), it is extremely impractical to work with the many-body wavefunction, since it will generally depend on $\sim 10^{23}$ coordinates, and is therefore impossible to write down, let alone perform calculations with. This necessitates use of Green functions, which are directly related to experimentally measurable quantities, and usually depend only on the coordinates of a few particles at a time. In most cases, physical measurements are related to the retarded Green function, $G_R(r,t)$, which is usually difficult to compute directly. Depending on the situation, you may by able to compute, instead, the advanced Green function $G_A(r,t)$, or the finite temperature, Matsubara Green function $G_M(r,\tau)$, etc. Anyway, the point is that all these Green functions are related to the same spectral function. So, in practice, the spectral function can act as a "go-between", that allows you to compute one type of Green function from the others.

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Add to it the fact that using Green's function allows you to get the transport properties and Susceptibilities without having to know the Eigenvalues of the Hamiltonian. And, depending on the problem, due to Dyson's equation, you do not need to go too far to get the first order results which show up directly by taking the imaginary part of the Green's function (times -1/pi) is the Spectral density and Local DOS.

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