Trouble understanding a Peskin-Schroeder text

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The forum discussion centers on understanding specific concepts from the Peskin-Schroeder text regarding the Dirac Field, particularly the invariance of the ground state |0> under translations and the manipulation of operators in quantum field theory. The user seeks clarification on the equation |0> = e^{i \textbf{P.x}} |0> and the expression <0| a^r_p {a^s_q}^\dagger e^{i P.x} |0>. The discussion concludes with the user expressing satisfaction after receiving clarification on these points.

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praharmitra
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I am having some trouble understanding the paragraph after eq (3.93) in Peskin Schroeder (Page 55). The chapter is on the Dirac Field. These are the parts where I am not following:

1. If the ground state |0&gt; is to be invariant under translations, we must have

|0&gt; = e^{i \textbf{P.x}} |0&gt;.

How?!

2. How does
&lt;0| a^r_p {a^s_q}^\dagger e^{i P.x} |0&gt;= e^{i(p-q).x} &lt;0| e^{i P.x} a^r_p{a^s_q}^ \dagger |0&gt;.
?!

Thanks :)
 
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got it, thanks! :)
 

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