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Re: Integrating Feynman integrals in mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg50854] Re: Integrating Feynman integrals in mathematica
• From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
• Date: Fri, 24 Sep 2004 04:41:27 -0400 (EDT)
• References: <ciu5iv$81d$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Unless I have missed the point of your question, this is standard material 
covered in textbooks on quantum field theory. I have Itzykson + Zuber's 
tome, and I find your first integral discussed in section 1-3-1 on Green 
functions. Your second integral is related to your first integral by the 
trick of differentiating w.r.t. an auxiliary parameter (i.e. make the 
replacement m^2-->a m^2 in your first integral, differentiate w.r.t. a, then 
set a-->1).

Steve Luttrell

"Xman" <pabird at supanet.com> wrote in message 
news:ciu5iv$81d$1 at smc.vnet.net...
>I am trying to find an explicit form of the following 4-dimensional
> fourier transforms. Can anyone help? ( x and k are 4 dimensional
> vectors) They are from physics.
>
> 1)
>
> f(x) =Intregral[    e^(i x.k) / (k.k -m^2)  ]dk^4
>
> 2)
>
> g(x)=Intregral[    e^(i x.k) / (k.k -m^2)^2  ]dk^4
>
> I know that the first is of the form:
>
> f(x) = 1/|x.x| + log|x.x| * P((m^2/4) |x.x|) + Q((m^2/4) |x.x|)
>
> (when m=0 this becomes 1/|x.x|)
>
> Where P and Q stand for infinite polynomial series and that I think
> P(y) = Sum( y^n /(n!(n+1)!) ,y=0..infinity )
>
> and that in the second one
> g(x)  =  log|x.x| * R((m^2/4) |x.x|) + S((m^2/4) |x.x|)
>
> (when m=0 this becomes log|x.x|)
>
> where R(y) = Sum( y^n /(n!n!) ,y=0..infinity )
>
> But the functions Q and S are more difficult to find.
> Plus does anyone know if the series P and R (=P') or Q and S can be
> written in terms of simple functions?
>
> It may help to know that f and g satisfy the following 4 dimensional
> wave equations:
>
> ( d/dx . d/dx - m^2) f(x) = delta(x)          (=0 for x=/=0)
> ( d/dx . d/dx - m^2)^2 g(x) = delta(x)          (=0 for x=/=0)
>
> I am particularly interested in g(x).
>