# Studying Green's function in many body physics

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• Jeff Chen
In summary, the conversation discusses the Green's function in many-body physics and its relation to physical observables. The origin of the definition of Green's function is explained, along with its physical meaning as a correction due to interaction with the environment. The Green's function can be used to obtain the spectral function, which is related to experimental observables. The self-energy in the spectral function provides information about the interactions surrounding the quasiparticles. Piers Coleman's many-body textbook is recommended for further details on the subject.
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?

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.

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.

## 1. What is a Green's function in many body physics?

A Green's function is a mathematical tool used to describe the response of a system to an external perturbation. In many body physics, it is used to study the behavior of a system composed of many interacting particles.

## 2. Why is studying Green's function important in many body physics?

Studying Green's function allows us to understand the behavior and properties of complex systems, such as solids, liquids, and gases, which are made up of many interacting particles. It also provides a powerful framework for calculating physical quantities, such as energy and correlation functions, in these systems.

## 3. How is Green's function calculated in many body physics?

Green's function is typically calculated using advanced mathematical techniques, such as perturbation theory, Feynman diagrams, and numerical simulations. These methods allow us to solve the complex equations that describe the behavior of many body systems.

## 4. What are some applications of Green's function in many body physics?

Green's function has a wide range of applications in many body physics, including the study of electronic properties of materials, quantum field theory, and nuclear physics. It is also used in condensed matter physics to understand the behavior of materials at the atomic scale.

## 5. How does studying Green's function contribute to our understanding of many body systems?

By studying Green's function, we can gain insights into the fundamental principles that govern the behavior of many body systems. It also allows us to make predictions and test theories about the properties and interactions of particles, leading to a deeper understanding of the physical world.

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