- #1
Beginner_2010
- 2
- 0
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)
(2)
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 ^_^