Deriving an influence functional is deriving me mad (Help)

[Tbonewillsone]

Homework Statement

I am attempting to derive Caldeira-Leggett's influence functional found in their paper "Path Integral Approach To Quantum Brownian Motion". If you find my following statements confusing, then pages 16-18 of http://web.science.uu.nl/itf/Teaching/2006/MxWakker.pdf show the problem clearly. I thought I'd found a neat little workaround to simplify the derivation... but I now believe that may have manifested into a fundamental misconception of the actual physics.

Despite the technical info I'm about to give, I think the issue is quite a fundamental one that shouldn't require too much inspection to solve.

So here's the deal:
We have a system of interest, System A, that is interacting with a second system, the reservoir, System B. The entire system is treated, as always, as a collection of harmonic oscillators. The main trick here, is that we are evaluating this system using Feynman's path integral approach, so unlike traditional quantum mechanics the action finds itself involved in the propogation of its respective system.

The Variables:
S_cl: Classical Action of the entire system.
R, Q: Coordinates of particles in the reservoir.
x(t), y(t): The paths followed by a system as a function of time, t.
ρ: The reduced density operator of the entire system.
K: Memory Kernel/Propagator, describes the evolution of a system through time (in Feynman's path integral formulism).
F: The influence function, it describes the interaction of System A with System B (in Feynman's path integral formulism).

Homework Equations

Because I can't work out how to put equations on here, I've organized all the equations in LaTeX

[Note: The prefactor of the density operator here is different to the one in the source provided. That is because in the three sources I have been going through, the prefactor changes every time, so I am going with the original paper, Caldeira-Leggett's prefactor.]

And after all of those definitions...

The Attempt at a Solution

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So, it seemed rather straight-forward to me, I substitute these elements into equation (3), whilst making the relevant changes to the conjugate Kernel, K[x(t), R, R'] becomes K*[y(t),R,Q']. This leads to...

I then noticed that because of the common factor of R_k in both actions, the second order R_k terms should cancel, leaving R_k terms of the first order multiplied by a bunch of terms and an imaginary unit, i. This, integrated over R_k should be equivalent to the dirac delta function,

This in turn should lead to a simple substitution with an integration over Q' or R', resulting in one single gaussian integral. I shan't post the answer I got here because it's incredibly messy and contributes nothing much. One thing I shall say is that the answer has a prefactor multiplied by the exponential, which is lacking in equation (6) AND a cosectant term in the exponent which is clearly not meant to be there. Without the cosectant term, however, the answer does look similar to that of the desired result. I am currently thinking that my error is in the single product term, Pi, and grouping all R_k terms together rather than having two products, producing R_k and R_j. That would suggest an error in the Wakker source I'm using, and I don't know how the integration would work if that was the case with different indices.

I have done this both by hand and with Mathematica to no avail. Any help is MASSIVELY appreciated.

Thanks.

 

[mark726]

Thank you for sharing your thoughts and progress on deriving Caldeira-Leggett's influence functional. It is clear that you have put a lot of effort into understanding the problem and finding a solution. However, I would like to point out a few things that may help you in your derivation.

Firstly, your approach seems to be correct in terms of substituting the relevant variables and making the necessary changes to the conjugate kernel. However, I believe the issue may lie in your integration over R_k. It is possible that the second-order R_k terms may not completely cancel out, leading to the discrepancies in your final result. I suggest re-checking your integration steps and possibly trying different integration techniques.

Secondly, it is important to note that the derivation of the influence functional is a complex and technical process, and it is not uncommon for different sources to have slightly different prefactors or formulations. It is important to carefully check your equations and make sure they are consistent with the source you are using. If you are still unsure, I suggest consulting with your peers or a mentor who may be able to provide some insights.

Lastly, it may also be helpful to break down the problem into smaller steps and try to simplify the equations as much as possible before attempting the full derivation. This can help identify any potential errors or misunderstandings in your approach.

I hope these suggestions will help you in your derivation. Keep up the good work and don't hesitate to ask for help if needed. Science is a collaborative effort and we all learn from each other's insights and mistakes.

Best of luck in your research.