Confusion about the contour green's function

In summary, the conversation discusses the use of Green's functions in physics, specifically in non-equilibrium situations. The speakers mention that many books on Green's functions are overly formal and can be confusing, leading to a struggle to understand the various functions and their applications. The concept of a closed contour, used in solving non-equilibrium physics, is also brought up, with one speaker suggesting studying complex variable theory to understand it. Green's functions are described as solutions to differential equations with point sources, with the ability to use the superposition principle to compute solutions. The closed-contour technique is explained as relying on Cauchy's theorem from complex analysis. The conversation ends with a recommendation to study from textbooks such as Jackson's E&M or Ar
  • #1
wdlang
307
0
i find that most books on green's function are burdened with too much formalism

i am now reading the book by Rammer, which deals with non-equilibrium physics.

The formalism is so lengthy and so confusing. You have to strive hard to remenber the various green's functions, and only to find that there is little concrete applications.

I am haunted with the puzzle: the idea of a closed contour seems very naive, and how this trick can solve non-equilibrium physics?

I still cannot biuld a picture of the green's function
 
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  • #2
Study some complex variable theory to learn why Cauchy's theorem requires a closed contour.
 
  • #3
I am by no means an expert on non-equilibrium physics, and therefore probably can't answer your q's very well. But I can tell you about greens fcns in general:

A Green's fcn is basically the solution of a differential equation with a point-source. For example: the Green's function of electrostatics is just the Coulomb potential. The great thing about Green's functions is that as long as your equation is linear, there is a superposition principle (if f(x) and g(x) are solutions, so is f(x)+g(x)). Therefore if you can break up your source as a sum of "point particles", you can effectively compute your solution to the diffEQ as a sum of Green's functions. Of course, for continuum mechanics, the sums become integrals, but that's the basic idea.

As to the closed-contour technique: that's just Cauchy's theorem from complex analysis (see Pam's response). I see nothing "naive" about relying on a mathematical theorem! The idea is simple enough: if the integrand vanishes fast enough at infinity, if you add a term that is the contour-integral over the semicircle at infinity, you are just adding zero and that is perfectly allowed. But now that you have added that term, you have closed the contour and can use Cauchy's theorem to evaluate the integral. That's all there is to it.

As to "solving non-equilibrium physics": you got me! I don't know what you mean by this!

Green's functions are dealt with pretty hard-core in Jackson's famous E&M textbook. Also, most math method textbooks (Arken & Weber, for example) go into it a bit.

Hope this helps!
 

1. What is a contour green's function?

A contour green's function is a mathematical tool used in the field of quantum mechanics to describe and solve problems related to the behavior of particles. It represents the probability amplitude for a particle to move from one point to another in space and time, taking into account all possible paths that the particle could take.

2. Why is there confusion surrounding contour green's functions?

Contour green's functions can be quite complex and difficult to understand, especially for those who are not familiar with quantum mechanics and mathematical concepts such as complex analysis. Additionally, there are different approaches and formulations for contour green's functions, which can lead to confusion and conflicting information.

3. How is a contour green's function different from a regular green's function?

A regular green's function is used in classical mechanics to solve problems related to the motion of particles, while a contour green's function is used in quantum mechanics. The main difference between the two is that contour green's functions take into account the probabilistic nature of particles and the concept of quantum superposition.

4. What are some applications of contour green's functions?

Contour green's functions are used in a variety of applications, including solving problems in solid state physics, quantum field theory, and statistical mechanics. They are also used in the development of quantum computing algorithms and in the study of complex systems in biology and chemistry.

5. How can I better understand contour green's functions?

To better understand contour green's functions, it is important to have a strong background in mathematics, specifically complex analysis and differential equations. It can also be helpful to study the fundamentals of quantum mechanics and to practice solving problems using contour green's functions. Additionally, seeking guidance from experts in the field or attending lectures and workshops can also aid in understanding this complex concept.

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