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Please help; calculation phase space -> how do I use the Delta functions?

  1. Dec 2, 2006 #1
    Can somebody help me out? I'm reading about formulas for cross sections for spin1 particles but I don't understand the delta functions, in calculating the 2particle pahse space psi

    For example the interaction; A+B -> C+D has the formula;

    psi= (2pi)^2 delta(Pa+Pb-Pc-Pd) d3Pc d3Pd / 4EcEd

    then all the books say the same;

    delta(Pa+Pb-Pc-Pd) d3Pd = delta (Ea+Eb-Ec-Ed)
    :confused:

    can somebody explain this to me?

    the left side delta is zero everywhere pa+pb =not pc+pd.

    the integral over the delta function d3Pd is;
    1 for pd<pa+pb-pc and
    0 for pd>pa+pb-pc

    ??? how do I use this?
     
    Last edited: Dec 2, 2006
  2. jcsd
  3. Dec 3, 2006 #2
    Vergeet niet dat de je p3's en p4's hebt! (p4=(E,p)

    [tex]\delta^{4}(p1+p2-p3-p4) d^{3}[/tex]p4
    =[tex]\delta(E1+E2-E3-E4)
    \delta^{3}[/tex](p1+p2-p3-p4)[tex]d^{3}[/tex]p4
    =[tex] \delta(E1+E2-E3-E4) \delta^{3}[/tex](-p3-p4) [tex]d^{3} [/tex]p4 (CM frame: p1=-p2)
    =[tex]\delta(E1+E2-E3-E4) \delta[/tex](x) [tex]d[/tex]x (x= -p3-p4 = 0 CM frame)
    = [tex] \delta(E1+E2-E3-E4) [/tex] * [tex]1[/tex] = [tex]\delta(E1+E2-E3-E4) [/tex]
    QED
     
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