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Dirac delta function

  1. Nov 29, 2006 #1
    I found this equation in a field theory book, which I can't figure how it was derived:

    [tex] \delta(x-a) \delta(x-a) = \delta(0) \delta(x-a)[/tex]
     
  2. jcsd
  3. Nov 29, 2006 #2

    OlderDan

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    From the definition of the delta function

    [tex]
    \int_{ - \infty }^\infty {f(x)\delta (x - a)dx = f(a)}
    [/tex]

    you get

    [tex]
    \int_{ - \infty }^\infty {\delta (x - a)\delta (x - a)dx = \delta (a - a)} = \delta (0)
    [/tex]

    I expect this is part of the answer.
     
  4. Nov 29, 2006 #3

    Hurkyl

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    Ugh. If that's supposed to be the dirac delta distribution, then both sides of that equation are gibberish. What is the context in which you saw it?
     
  5. Nov 30, 2006 #4

    arildno

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    From what I know (very slightly!), products of distributions like the delta "function" cannot be defined in a consistent manner.
     
  6. Nov 30, 2006 #5
    I was reading a section dealing with cross sections and scattering. He calculated some amplitude, A (first order) for a Feynman diagram which contains four 4-momentum delta functions. And with that amplitude, we need to get this invariant amplitude, iM which is the square of A.
    Squaring A yields us 8 delta functions.
    He states that 8 delta functions is bad news, and basically he gave a simple example, which was the one I posted in this forum.

    It's from Gauge Theories in Particle Physics Volume I by I J R Aitchison, page 152.
     
  7. Nov 30, 2006 #6
    Equations such as [tex] f(x) \delta(x-a) = f(a)[/tex] are supposed to be read as [tex] \int_{-\infty}^{\infty} f(x) \delta(x-a) dx= f(a)[/tex], I think dropping the integral sign is just some sort of convention, not one I'm a fan of though... I think people keep swapping limits and integral signs too, I think things like that should be made more consistent, as you can't do stuff like that in general.

    But yeah don't forget the integral sign!
     
  8. Nov 30, 2006 #7

    Meir Achuz

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    delta(0) can sometimes be taken to mean the volume of space (with appropriate 2pi factors) in relating delta function normalization with box normalization.
     
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