# Two stupid questions about Peskin's QFT

1. Apr 2, 2010

### Beginner_2010

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)

(2)

3. The attempt at a solution

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

Combine with the ket (2.35), obtain
$$\langle\mathbf{q} | \mathbf{p} \rangle = 2 \sqrt{ E_{\mathbf{q}} E_{\mathbf{p} } } \langle 0 | a_{\mathbf{q}} a_{\mathbf{p}}^{\dag} | 0 \rangle = 2 \sqrt{ E_{\mathbf{q}} E_{\mathbf{p} } } (2 \pi)^{3} \delta^{(3)} (\mathbf{p} - \mathbf{q})$$

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

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

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

Thank you ^_^

2. Apr 2, 2010

### vela

Staff Emeritus
The delta function is non-zero only when p=q, so Ep=Eq.

3. Apr 6, 2010

Thank you!