Greens functions from path integral

aaaa202 · May 11, 2015

Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH))
Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's function is given by:
G(xx',τ-τ') = 1/ZTr[exp(-H(β-τ)c_xexp(-H(τ-τ'))c_x'exp(-Hτ')]
But how exactly does this trivally allow us to apply the formula for the partion function path integral?

stevendaryl · May 11, 2015

aaaa202 said:

Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH))
Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's function is given by:
G(xx',τ-τ') = 1/ZTr[exp(-H(β-τ)c_xexp(-H(τ-τ'))c_x'exp(-Hτ')]
But how exactly does this trivally allow us to apply the formula for the partion function path integral?

I'm not sure what you're asking. Are you asking how to get [itex]G[/itex] from [itex]Z[/itex], or how to derive the path-integral expression for [itex]G[/itex]?

aaaa202 · May 11, 2015

I'm asking how you end up with the equation 2.7 given that we know the path integral representation of Z.

Greens functions from path integral

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Related to Greens functions from path integral

1. What are Green's functions?

Green's functions are mathematical tools used to solve differential equations in physics and engineering. They represent the response of a system to a delta function input, and can be used to solve a variety of problems in quantum mechanics, electromagnetism, and other fields.

2. How are Green's functions related to path integrals?

Green's functions can be derived from path integrals, which are mathematical expressions that represent the sum of all possible paths of a particle between two points in space. In this context, the Green's function represents the contribution of each path to the overall probability amplitude.

3. What is the significance of Green's functions in quantum mechanics?

4. Can Green's functions be used for systems with multiple dimensions?

Yes, Green's functions can be extended to higher dimensions, making them applicable to a wide range of physical systems. In fact, the original derivation of Green's functions by mathematician George Green was in three dimensions, and they have since been extended to other dimensions as well.

5. What are some practical applications of Green's functions?

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