Massive Scalar Field in 2+1 Dimensions

[xGAME-OVERx]

Homework Statement

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.

Homework 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.

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.

Thanks
Scott

 

[mathfeel]

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.

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.

 

[xGAME-OVERx]

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?

Thanks
Scott

EDIT: Sorry, found it in Abramowitz & Stegun...

 

[mathfeel]

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?

Thanks
Scott

EDIT: Sorry, found it in Abramowitz & Stegun...

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/

 

[paranormal]

I would suggest looking into the properties of the massive scalar field in 2+1 dimensions and how it differs from the 3+1 dimensional case. This could provide insight into how the equations and integrals may need to be modified in order to find the analogue of E = - \frac{1}{4\pi r} e^{-mr} in 2+1 dimensions. Additionally, exploring different mathematical techniques and approximations may also be helpful in solving the integral. Collaborating with other experts in the field may also lead to new ideas and approaches for finding a solution.