- #1
pines-demon
Gold Member
2024 Award
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- Homework Statement
- Use Wick's theorem to simplify [this fermionic operation]
- Relevant Equations
- ##\langle 0|c^\dagger_{p_1-q}c^\dagger_{p_2+q}c_{p_2}c_{p_1}|0\rangle##
In Lancaster&Blundell QFT for the Gifted Amateur, Chaper 18, the authors introduce Wick's theorem. I have already seen it Fetter&Walecka and in here, but my problem with the theorem is that it is usually announced for interaction picture operators with time-dependences. Nevertheless in the exercises they ask us to use Wick's theorem to rewrite different chains of operators that are not time-dependent. This problem (18.5) struck me specfically because it is completely ambiguous, not only one has to understand that ##|0\rangle## is the ground state (GS) and not the full vacuum, but also how does Wick theorem work here? I mean the operators are already normal ordered...
From previous problems I figured out how to use the Wick theorem for time independent operators. One can use that version of the theorem naively and say that
$$\langle 0|c^\dagger_{p_1-q}c^\dagger_{p_2+q}c_{p_2}c_{p_1}|0\rangle=\text{all contracted pairs}$$
Which provides the right result but it is fishy because this operation contains normal orderings like the original one ##c^\dagger_{p_1-q}c^\dagger_{p_2+q}c_{p_2}c_{p_1}## that are not zero by definition of GS. So why does this work? Does the normal ordering has to be redefined here?
A more natural way I have seen this before is to divide the fermion operators into particle-antiparticle operators so that the ground state works like a true vacuum but it seems that there is no need under this more-seemingly-naive use of Wick's theorem.
From previous problems I figured out how to use the Wick theorem for time independent operators. One can use that version of the theorem naively and say that
$$\langle 0|c^\dagger_{p_1-q}c^\dagger_{p_2+q}c_{p_2}c_{p_1}|0\rangle=\text{all contracted pairs}$$
Which provides the right result but it is fishy because this operation contains normal orderings like the original one ##c^\dagger_{p_1-q}c^\dagger_{p_2+q}c_{p_2}c_{p_1}## that are not zero by definition of GS. So why does this work? Does the normal ordering has to be redefined here?
A more natural way I have seen this before is to divide the fermion operators into particle-antiparticle operators so that the ground state works like a true vacuum but it seems that there is no need under this more-seemingly-naive use of Wick's theorem.
