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Expectation of the Wilson Loop

  1. Jan 24, 2006 #1
    Hey, I've got this problem from Peskin & Schroeder (chapter 15). I'm not particularly confident with functional integration, as I'm pretty new to it, and working through such a book by myself is pretty tricky in places. Well here goes

    The Wilson Loop for QED is defined as

    [tex]U_p(z, z)=\exp \left[-ie\oint_pdx^{\mu}A_{\mu}\right][/tex]

    With the Wilson line defined similarly (just change it so that there's not a closed contour integral and with the end points (z,z) changed to (z, y), or whatever you like).

    Where A is the photon field, the gauge connection asociated with transformations in U(1).

    Now it says: using functional integration, show that the expectation of the Wilson loop for the electromagnetic field free of fermions is

    [tex]\langle U_p(z, z)\rangle =\exp \left[-e^2\oint_pdx^{\mu}\oint_pdy^{\nu}\frac{g_{\mu\nu}}{8\pi^2(x-y)^2}\right][/tex]

    Where x and y are integrated over the closed loop P.

    I think the Feynman propogator might be useful here, so to save anyone having to look it up,

    [tex]D_F^{\mu\nu}(x-y)=\int\frac{d^4q}{(2\pi )^4}\frac{-ig^{\mu\nu}e^{-ip\cdot (x-y)}}{q^2+i\epsilon }[/tex]

    (The imaginary term in the denominator of the integrand is the application of the Feynman boundary conditions, ensuring the convergence of the Gaussian integral involved in the derivation of the propogator.)

    I have a vague idea of how to go about it, but I'm not particularly confident about it, it's finding the relevant starting point that's causing me problems, i.e. putting together and computing the functional integral for the expectation. I'm just going through this to gain some confidence in functional integration etc. so if anyone can give a few pointers as to going about this it'd be much appreciated. This isn't a homework question, if that perturbs anyone from helping me, I doubt my A level teacher would set something like this :).

    Last edited: Jan 24, 2006
  2. jcsd
  3. Jan 26, 2006 #2
    *Shamelss Bump*

  4. Jul 20, 2009 #3
    This is a simple problem.

    1)Discretize time-space on a lattice with index i=(t,x); this is the only way to understand what's going on(!!)
    2)Functional integral is nothing but a gaussian integral with extra term (the Wilson loop) which is linear in the field:
    [tex]\frac{1}{2}\sum_{i,j}A_i G^{-1}_{ij}A_j+\sum_{i\in\p}A_i[tex].
    3)Properly normalized this is:
    [tex]e^{-\frac{1}{2}\sum_{i,j\in\p}G^{-1}_{ij}} [tex]
    4)Now the Green function of the D'Alambert operator is the analytic continuation of that of the Laplacian in 4d; namely:
    [tex]G^{-1}_{ij}=\frac{1}{4\pi^2}\frac{1}{|i-j|^2} [tex],
    where remember that i is a short notation for the point on the lattice (x_i, t_i) and |...| indicates the distance between the points in the usual Minkorwsky norm.

    Putting all together we get the result.
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