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On the book QFT in a Nutshell by A.Zee page 61 writes

Always create before we annihilate, not the other way around. —Anonymous

But in this Phys.SE question we are doing it the other way around.

So how should we interpret that phrase?

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    $\begingroup$ Who knows. Maybe the author means that the creation operators are to the left of the creation operators. $\endgroup$ – DanielSank Oct 17 '14 at 3:08
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    $\begingroup$ an annihilation on the vacuum is pointless? $\endgroup$ – anna v Oct 17 '14 at 4:03
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    $\begingroup$ In my 2003 edition the quote is at the top of page 64 $\endgroup$ – John Rennie Oct 17 '14 at 10:01
  • $\begingroup$ @annav Annihilation of vacuum makes sense in the particle number operator. If we would not annihilate vacuum, the particle number operator would not give zero for vacuum. So in this case we must first annihilate, then create - not the other way around. $\endgroup$ – mpv Oct 17 '14 at 11:51
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As others pointed out, the statement probably means that if you want to have a nonzero contribution from creation and annihilation operators acting on vacuum, you need to apply the creation operator first, since $\hat{a}|0\rangle = 0$, in other words, you cannot annihilate if you have nothing.

In the question you link to, the situation is different, as that deals with the number operator. Since we want the eigenvalues of the number operator to give the number of particles in given state, we need the vacuum state not to contribute. Hence, the order of the operators is changed and we start by annihilating.

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That's a proverb in the beginning of Chapter I.8 (in both editions).

Zee undoubtedly included it in the book because it reminded him of the fact that an annihilation operator annihilates the vacuum ket: $$\hat{a}|\Omega \rangle =0,$$ while this is not so for the creation operator $$\hat{a}^{\dagger}|\Omega \rangle \neq 0.$$

The proverb is not meant as a rigorous mathematical statement that applies to all situations, such as, e.g. normal ordering prescription, which seems to follow the opposite saying.

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