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Context

Please note: I am basically a beginner to QFT, so please bear with me.

I am currently reading Dr. David Tong's Quantum Field Theory notes and on the bottom of page 34 (or page 40 in the PDF viewer) eq. 2.75, I see this operator:

enter image description here

This is the difference in the number of particles to anti-particles on a complex scalar field.


My question is: why do we need to write down the annihilation operator for both the particle and the anti-particle when we are just creating a particle?

Shouldn't we just write the creation operator without the annihilation operator for both?

Something like this in order to create particles:

enter image description here


I see equations like this written often in QFT texts and not just here. In general, why do we need the extra annihilation operator when creating particles? To me, it seems redundant, but I am probably wrong.

What I think it means

To my limited understanding, I think it is to destroy an already existing particle so that it has the energy to create another particle, which is required because of the conservation of energy. Is this correct?

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    $\begingroup$ In my opinion, you should beware of trying to assign some sort of physical process to each of these operators. The annihilation operator does not actually "destroy a particle", and the creation operator does not actually "create a particle". They are mathematical operators that act on states. Instead, just note that that product of operators makes an operator that counts the total number of particles in the state. Make sure you are intimately familiar with the harmonic oscillator algebra and its interpretation before looking at QFT. $\endgroup$
    – march
    Sep 3 at 22:57
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    $\begingroup$ The operator in your question isn't creating particles—or annihilating them, for that matter—it's just counting them. We ought to have the same number of particles before and after counting them, so the operator that does the counting better have the same number of creation and annihilation operators. $\endgroup$
    – d_b
    Sep 3 at 22:58
  • $\begingroup$ That makes sense, I should have thought of that. So then, if you want to create a particle or destroy one, then you would use only one operator? Assuming you make them act on the vacuum. @march $\endgroup$
    – Tachyon
    Sep 3 at 23:02
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    $\begingroup$ Yes: a creation operator acting on the vacuum yields a single-particle state. The annihilation operator acting on a state of well-defined particle number (i.e., a Fock state $|n\rangle$) yields a state of one fewer particle, i.e. $|n-1\rangle$. This is the sense in which the operators "create" and "destroy" particles, but they don't necessarily correspond to actual physical processes. $\endgroup$
    – march
    Sep 3 at 23:07
  • $\begingroup$ @march Thank you, I think I understand now. If you could put that as an answer to my question, I will gladly accept. $\endgroup$
    – Tachyon
    Sep 3 at 23:10
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A creation operator acting on the vacuum $|0\rangle$ yields a single-particle state Fock state $|1\rangle$. The annihilation operator acting on a state of well-defined particle number (i.e., a Fock state $|n\rangle$) yields a state of one fewer particle, i.e. $|n-1\rangle$. This is the sense in which the operators "create" and "destroy" particles, but they don't necessarily correspond to actual physical processes. In my opinion, you should beware of trying to assign some sort of physical process to each of these operators.

(Although: there are times when we do use them to represent physical processes. For instance, in the theory of a measurement in which a photon is absorbed by a photodetector, we represent the process of absorption with an annihilation operator acting on the pre-measurement state of the field.)

Generally speaking, the creation and annihilation operators are mathematical objects that act on mathematical states. When constructing the operator in the OP, the operators merely count the number of particles in the state.

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  • $\begingroup$ What exactly is meant by a well-defined particle number in a Fock state? $\endgroup$
    – Tachyon
    Sep 3 at 23:37
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    $\begingroup$ By "state of well-defined particle number" I mean a state that is an eigenvector of the particle-number operator. A general state can be a superposition of states with different number of particles. A Fock state is another name given to the eigenstates of the number operators. $\endgroup$
    – march
    Sep 3 at 23:39

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