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Two stupid questions about Peskin's QFT

  1. Apr 2, 2010 #1
    1. The problem statement, all variables and given/known data
    Hi,

    I have two stupid questions about Peskin's QFT book.

    (1) P23, How to derive from (2.35) to (2.36)
    (2) P30, How to derive (2.54)

    2. Relevant equations

    (1)
    peskin_23.gif
    (2)
    Perskin_30.gif

    3. The attempt at a solution

    (1) If I consider the dual-space vector, [tex] \langle \mathbf{q} | = \sqrt{2 E_{\mathbf{q} }} \langle 0 | a_{\mathbf{q}} [/tex]

    Combine with the ket (2.35), obtain
    [tex]

    \langle\mathbf{q} | \mathbf{p} \rangle = 2 \sqrt{ E_{\mathbf{q}} E_{\mathbf{p} } } \langle 0 | a_{\mathbf{q}} a_{\mathbf{p}}^{\dag} | 0 \rangle
    = 2 \sqrt{ E_{\mathbf{q}} E_{\mathbf{p} } } (2 \pi)^{3} \delta^{(3)} (\mathbf{p} - \mathbf{q})
    [/tex]

    Therefore
    [tex]
    \langle \mathbf{p} | \mathbf{q} \rangle = \langle \mathbf{q} | \mathbf{p} \rangle^* = 2 \sqrt{ E_{\mathbf{q}} E_{\mathbf{p} } } (2 \pi)^{3} \delta^{(3)} (\mathbf{p} - \mathbf{q})
    [/tex]

    But Peskin's (2.36) has a prefactor [tex] 2 E_{\mathbf{p}} [/tex] instead of [tex] 2 \sqrt{ E_{\mathbf{q}} E_{\mathbf{p} } } [/tex], is that made to be the convention?

    (2) Is that the principal value of integral [tex] \int_{- \infty}^{+\infty} d p^0 [/tex] or including the little semi-cycles around [tex] -E_{\mathbf{p}} [/tex] and[tex] +E_{\mathbf{p}} [/tex] ? If includes the semi-cycles, i can get the result

    Thank you ^_^
     
  2. jcsd
  3. Apr 2, 2010 #2

    vela

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    The delta function is non-zero only when p=q, so Ep=Eq.
     
  4. Apr 6, 2010 #3
    Thank you!
     
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