Studying Green's function in many body physics

[Jeff Chen]

Hi,everyone. Recently, I am studying green's function in many body physics and suffer from trouble.Following are my problems.
(1) What is the origin of the definition of green's function in many body physics?
(2) What is the physical meaning of self energy ? It seems like it is the correction due to interaction with environment and is it similar to the concept of quasi-particle and mass renormolization ?
(3)If we know the green's function for a certain system,what physical quantity can we obtain?

 

[king vitamin]

It seems that all three of your questions concern what the relation is between the Green's function in many-body physics and physical/experimental observables. One very clear connection is the fact that the spectral function is obtained from the retarded two-point correlation function as
$$
\rho(\omega) = \mathrm{Im} G_{\mathrm{R}}(\omega) = \pi \sum_{\alpha} | \langle \alpha | \psi | 0 \rangle|^2 \left[ \delta(\omega - E_{\alpha} + E_0) \mp \delta(\omega + E_{\alpha} - E_0) \right]
$$
where the upper (lower) sign is for bosons (fermions). But this form - a matrix element times a delta function constraining energy conservation - is precisely the form of Fermi's Golden Rule which computes the transition rate of time-dependent processes which couple to the operator ##\psi#. In addition, this object is only nonzero at precisely the frequencies where the many-body system has energy levels, so it tells you about the spectrum of your system. These two facts result in a lot of relations between experimental observables and spectral functions.

For more details, Piers Coleman's many-body textbook has about half a chapter devoted to relating spectral functions to various experimental observables in different systems. It is far more detailed and clear than anything I could write up here, so I highly recommend checking it out.

 

[ZapperZ]

You get the single-particle spectral function A(k,ω) from the imaginary part of the Green's function. This spectral function is useful because it is accessible via experiment (example: ARPES).

The self-energies in A(k,ω) contains the type of scattering or interactions that surrounds the quasiparticle. The imaginary part of the self-energy, for example, allows us to see the origin of the broadening of A(k,ω) peaks and gives us information about the underlying interactions that are going on.

Zz.