- #1
furdun
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[SOLVED] Klein-Gordon Causality calculation
In Peskin and Schroeder on page 27 it is stated that when we compute the Klien-Gordon propagator in terms of creation and annihilation operators the only term that survived the expansion is
[tex]
<0|a_{\textbf{p}}a^{\dagger}_{\textbf{q}}|0> \ \ (1).
[/tex]
I am unsure of why the term
[tex]
<0|a^{\dagger}_{\textbf{p}}a^{\dagger}_{\textbf{q}}|0>
[/tex]
would vanish.
The expansion of the field is given by
[tex]
\phi (x) = \int \frac{d^{3}p}{(2 \pi)^{3}} \frac{1}{\sqrt{2E_{\textbf{p}}}}(a_{\textbf{p}}}e^{-ip\cdot x} + a^{\dagger}_{\textbf{p}}e^{ip\cdot x})
[/tex]
and the normalization condition for states is
[tex]
<\textbf{p}|\textbf{q}> = (2\pi)^{3}\delta^{3}(\textbf{p}-\textbf{q}).
[/tex]
Looking at the normalization condition given above I got,
[tex]
<0|a^{\dagger}_{\textbf{p}}a^{\dagger}_{\textbf{q}}|0> = <0|\textbf{p}+\textbf{q}> = (2\pi)^{3}\delta^{3}(\textbf{p}+\textbf{q}).
[/tex]
However this mean that (1) is not the only surviving term, and from my calculations this also gives a factor of 2 that should not be there. I am unsure of how this term vanishes.
Homework Statement
In Peskin and Schroeder on page 27 it is stated that when we compute the Klien-Gordon propagator in terms of creation and annihilation operators the only term that survived the expansion is
[tex]
<0|a_{\textbf{p}}a^{\dagger}_{\textbf{q}}|0> \ \ (1).
[/tex]
I am unsure of why the term
[tex]
<0|a^{\dagger}_{\textbf{p}}a^{\dagger}_{\textbf{q}}|0>
[/tex]
would vanish.
Homework Equations
The expansion of the field is given by
[tex]
\phi (x) = \int \frac{d^{3}p}{(2 \pi)^{3}} \frac{1}{\sqrt{2E_{\textbf{p}}}}(a_{\textbf{p}}}e^{-ip\cdot x} + a^{\dagger}_{\textbf{p}}e^{ip\cdot x})
[/tex]
and the normalization condition for states is
[tex]
<\textbf{p}|\textbf{q}> = (2\pi)^{3}\delta^{3}(\textbf{p}-\textbf{q}).
[/tex]
The Attempt at a Solution
Looking at the normalization condition given above I got,
[tex]
<0|a^{\dagger}_{\textbf{p}}a^{\dagger}_{\textbf{q}}|0> = <0|\textbf{p}+\textbf{q}> = (2\pi)^{3}\delta^{3}(\textbf{p}+\textbf{q}).
[/tex]
However this mean that (1) is not the only surviving term, and from my calculations this also gives a factor of 2 that should not be there. I am unsure of how this term vanishes.