Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Path integral in momentum space

  1. Dec 20, 2007 #1
    using the hamiltonian to derive the pathintegral is well known (see schulman), but i have only seen it for diagonal momenta and coupled coordinates:
    G(x,t;y) = <x|exp(-itH/hbar)|y> using the trotter formula etc one arrives at:
    G(x,t;y) = lim_N->infinity Int dx1...dx_N-1Prod_{j=0}^{N-1}<x_j+1|exp[-itT/(N hbar)|x_j>exp[-itV/(hbar N)]

    with H=T+V and V diagonal in coordinate space

    inserting a full set of momenta 1=int dp |p><p| one is able to solve this problem and express the argument of exp as iS/hbar.

    BUT i am not dealing with a coordinate coupling (so <x_j+1|...|x_j> makes no sense. i have a T of the form T_L, T_R, T_D and T_(D,L), T_(D,R) where T_L has a full set of momentum states (normalized) called p_L and the same for T_R and T_D. the coupling (T_(D,L) and T_(D,R)) looks like p_D(p_L+p_R).
    has anyone an idea how to reach S (argument of exp with factors) as done by the above method WITHOUT legendre transforming the hamiltonian from the start and deriving the path integral with the lagrangian????

    Considering the Problem (Interaction terms) for G in momentum space one gets <p_(M+1)|exp(-itp_M/(hbar N)[p_L+p_R]|p_M-1> so the p_M operator is no problem but the coupling of p_R (p_M+1) and p_L (p_M-1)
    Last edited: Dec 20, 2007
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Threads - Path integral momentum Date
I Random measurements in QED? Jan 25, 2018
A Path Integral of a Spontaneously Broken Theory Jan 7, 2018
I Feynman path integral Nov 14, 2017
I Does the path Integral contain virtual particles? Nov 5, 2017
Path integral in momentum representation Aug 19, 2007