Recently, I've started studying the book "condensed matter field theory" by Alexander Altland and Ben Simons. At second chapter when he's trying to rationalize the one-body operators in Fock space, I can not get it. I googled it and found out this one. During this article we can start from equation 2.28 which it's been told that one-body operator can be written in the following form, F=∑α∑β|α⟩⟨α|F|β⟩⟨β|
and then it claims that the corresponding extension of this operator in N-particle space is simply, FN=F(1)+F(2)+...+F(N)=N∑i=1F(i)
which each operator F(i) acts only on particle i. Then he applys F(i) on a product state. F(i)|α1α2...αN)=|α1⟩|α2⟩...|αi−1⟩{∑βi|βi⟩⟨βi|F|αi⟩}|αi+1⟩...|αN⟩
=∑βi⟨βi|F|αi⟩|α1...αi−1βiαi+1...αN)
I can not understand this part that why he does believe that the two following terms are same? ⟨βi|F(i)|αi⟩=⟨βi|F|αi⟩
and it says, "Note that the matrix elements of F do not depend on which particle is considered". And another question for me would be that what's the form for F(i)?
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