Green's function, path intergals, aaarrrgghh

[GSXR750]

Hello,

I have a question concerning path integrals. I have seen it for the first time in a course "many-particle physics", so if the path integral has a wider use: I don't know anything about it. Please think about that when you anser .

As far as I understood:
1.Green's function is the solution to the Schrödinger eqation. And the time evolution of a state can be expressed via the Green's function.

2.When you define: G= exp(i/h * S), the S is the quantum analog of the classical action.

3.The Green's function can be written in a path intergral by dividing the time in small enough steps and (?) the non commutativity of operators can be neglected.

Are these three points correct? And also: why is the connection between S and the classical action mentioned in my course? I suspect that a quantum system can be understood by considering the classical paths or something like that.

But I am not sure at all!

Please clarify all this a bit for me. Thanks.









jii

 

[Javier]

1.Green's function is the solution to the Schrödinger eqation. And the time evolution of a state can be expressed via the Green's function

A Green function is more generally a type of solution to a differential equation like Schroedinger's equation. In particular, let O be an operator acting on a function g. The type of equations we are looking at are of the form Og=0. For example, Sch. eqn is (H+d/dt)g=0, i.e., a wave equation. Or, we could consider the Laplace equation for electrostatics: D^{2} g =0. Such equations are said to be homogeneous.

In the case of electrostatics, you know that if you provide a function on the right hand side of the Laplace equation: D^{2} g = f, then f is a source of the field g (the inhomogeneous Laplace eqn is, btw, also called the Poisson eqn). If you consider a distribution on the RHS instead of a function (e.g. use the Dirac delta distribution), then the solution for the field g is called a Green function. This is interpreted as saying that the value of the field g at a point x is due to a point charge source at x'. In fact, g is 1/|x-x'| in this case (ignoring constants).

Similarly, we may consider any inhomogeneous form of an operator equation, and look for the solutions when the "source" is a distribution like the Dirac delta one. Instead of getting solutions, g, that tell you the value of a quantity at one location in space due to the presence of a source at another, we can find examples where we are determining the value of something at a point in time due to a source at another time (usually at time in the past for causal reasons..the solution is then called the retarded Green function).

Thus, Green functions provide a correlation between a source at one point in configuration space to the value of some quantity at another point in configuration space.

In quantum field theory, a process typically involves the propagation of a particle from a "creation" point in spacetime to an "annihilation" point in spacetime. The thing that correlates the value of a quantum field at the start (a source) to the value at the finish is the Green function for the particular field. For example, Klein-Gordon equation is (D^{2}+m^{2})g=0. But if we put a "source" Dirac delta distribution on the RHS, the solution g will be such a Green function for the scalar field.

2.When you define: G= exp(i/h * S), the S is the quantum analog of the classical action

This is the quantum mechanical analog of the partition functional in statistical mechanics. S is the *classical action*, and G here is a functional of S. You can use this functional to define the path integral and from that obtain information about correlations between events separated in space and time. See a text relevant to your course for the details. Even though S is the classical action (which, using the least action principle would give classical paths for particles), the path integral says that the value of the exp functional is evaluated for all possible "paths" of a particle, and when combined, do so in a constructive and destructive fashion such that some paths contribute to the amplitude for the process to occur, while others do not contribute.
So it is the functional G that is used in the quantum analog of the least action principle (which you might call an "almost least action principle" since it allows multiple particle trajectories that are "near" the classical ones).

 

[GSXR750]

I think I understand 1. But for 2.:

Classically, there are a number of paths in configuration space, and every one of them has an "action". The path with minimal action is what will happen.

So you say (correct me if I'm wrong) that if you minimise G, I will find a number of paths which will occur with a certain probability and the "sum" of those paths is ?. But I still don't see the connection between QM and classical mechanics.

 

[ZapperZ]

I would strongly recommend you get Mattuck's book "A Guide to Feynman Diagrams in the Many-Body Problem" if you're doing stuff in many-body physics. It is published by Dover, so it isn't terribly expensive. This book saved me many times through graduate school. He actually tried to explain IN WORDS many aspects of the subject, especially the use of Feynman diagrams and how they translate into the corresponding mathematical expansions.

Zz.

 

[GSXR750]

Ah, ok I'll consider it. Thanks for the tip.