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Factor in equation (3.49) of Peskin and Schroeder

  1. Apr 27, 2012 #1
    My question concerns the 1/2 factor in the exponential of Eq. (3.49) of Peskin and Schroeder.

    This equation concerns the Lorentz boost transformation of a spinor along the z-axis (or 3-direction).

    According to Eq. (3.26):

    [itex]S^{03} = -\frac{i}{2}\begin{bmatrix}\sigma^3 & 0 \\0 & -\sigma^3\end{bmatrix}[/itex]

    and Eq. (3.30):

    [itex]\Lambda_{1/2} = exp(-\frac{i}{2}\omega_{03} S^{03})[/itex]

    Combining these two expressions and using the infinitesimal boost (according to Eq. (3.48) and Eq. (3.21)): [itex]\omega_{03} = \eta[/itex] (here [itex]\eta[/itex] is the rapidity):

    [itex]\Lambda_{1/2} = exp(-\frac{1}{4}\eta \begin{bmatrix}\sigma^3 & 0 \\0 & -\sigma^3\end{bmatrix})[/itex]

    My problem is that Peskin and Schroeder have a 1/2 factor where my simple substitution gives a 1/4!?!

    Am I misinterpresting the meaning of [itex]\omega_{03}[/itex]? I am assuming that [itex]\eta = \beta[/itex] (v/c) from my understanding of Eq. (3.21)... this might be a problem?!?

    Thanks for your help!
     
  2. jcsd
  3. Apr 27, 2012 #2

    strangerep

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    Science Advisor

    In my version of P&S, eq(3.30) is
    $$
    \Lambda_{1/2} = exp(-\frac{i}{2}\omega_{\mu\nu} S^{\mu\nu})
    $$
    and you must sum over ##\mu, \nu##. So you need an exponent involving
    $$
    (\omega_{03} S^{03} + \omega_{30} S^{30})
    $$
    Is that enough for you to figure out the rest...?
     
  4. Apr 29, 2012 #3
    Thanks Strangerep! I just completely forgot the summation over the indices... That was also enough for me to figure out the rest!
     
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