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Deriving an influence functional is deriving me mad! (Help)

  1. Mar 27, 2017 #1
    1. The problem statement, all variables and given/known data

    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).


    2. Relevant equations

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

    CLM1b.png

    [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...

    CLM2.png

    3. The attempt at a solution

    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...

    CLM3.png
    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,

    CLM4.png

    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.
     
  2. jcsd
  3. Apr 1, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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