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Two stupid questions about Peskin's QFT

  • #1

Homework Statement


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)

Homework Equations



(1)
peskin_23.gif

(2)
Perskin_30.gif


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 ^_^
 

Answers and Replies

  • #2
vela
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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?
The delta function is non-zero only when p=q, so Ep=Eq.
(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 it includes the semi-cycles, I can get the result.
 
  • #3
Thank you!
 

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