1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Massive Scalar Field in 2+1 Dimensions

  1. Sep 26, 2011 #1
    1. The problem statement, all variables and given/known data

    We wish to find, in 2+1 dimensions, the analogue of [itex] E = - \frac{1}{4\pi r} e^{-mr} [/itex] found in 3+1 dimensions. Here r is the spatial distance between two stationary disturbances in the field.

    2. Relevant equations

    In 3+1 we start from [itex] E = - \int \frac{ d^3 k }{(2\pi)^3} \frac{1}{ {\bf{k}}^2 + m^2 } e^{ i {\bf{k}} \cdot ( {\bf{x}}_1 - {\bf{x}}_2 ) } [/itex] where [itex] \bf{k} [/itex] is momentum, and [itex] \bf{x}_i [/itex] are the spatial locations of the two disturbances.

    3. The attempt at a solution

    I think in 2+1 we must use the equation [itex] E = - \int \frac{ d^2 k }{(2\pi)^2} \frac{1}{ {\bf{k}}^2 + m^2 } e^{ i {\bf{k}} \cdot ( {\bf{x}}_1 - {\bf{x}}_2 ) } [/itex]. I begin by transforming to polar coordinates, i.e. [itex] E = - \frac{1}{(2\pi)^2} \int_{0}^{\infty} dk \int_{0}^{2\pi} d\theta \frac{k}{ k^2 + m^2 } e^{ i k r \cos\theta } [/itex].

    However, I am not sure what to do with this. As far as I know the theta integral can't be done in this form, and the r integral extends only down to 0, preventing it from being amenable to countour integration methods.

    I tried a common trick of writing:

    [itex] E = - \frac{1}{(2\pi)^2} \int_{0}^{\infty} dk \int_{0}^{2\pi} d\theta \frac{\partial}{\partial r} \frac{1}{i\cos\theta} \frac{1}{ k^2 + m^2 } e^{ i k r \cos\theta } [/itex]

    Which just makes the integral worse (I think). Any pointers would be greatly appreciated.

  2. jcsd
  3. Sep 26, 2011 #2
    Integration over [itex]\theta[/itex] gives you the Bessel function of the first kind.

    This leave you with [itex] E = - \frac{1}{2\pi} \int_0^{\infty} dk \frac{k}{k^2 + m^2} J_0 (kr)[/itex]

    This I think is another Bessel function...Second Kind...I think. Look it up in a table.
    Last edited: Sep 26, 2011
  4. Sep 26, 2011 #3
    Thank you for your reply, but that doesn't match any of the definitions I have seen for Bessel functions. May I ask which definition you are using?


    EDIT: Sorry, found it in Abramowitz & Stegun...
    Last edited: Sep 26, 2011
  5. Sep 26, 2011 #4
    Rewrite the integral in term of a new variable [itex]t = \cos\theta[/itex]. Then it conforms to the first integral representation here: http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/07/01/01/
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook