A QFT, what are the transformation rules for

[Jonsson]

I'm taking an introductory course in QFT. During quantization of the Dirac field, my textbook gives a lot of information on how annihilation and creation operators act on vacuum, but nothing about how they act on non-vacuum states. I need these to compute
$$
\int \frac{d^3 p}{(2\pi)^3} \sum_s ( {a^s_ {{\vec{p}}}}^\dagger a^s_ {{\vec{p}}} - {b^s_ {{\vec{p}}}}^\dagger b^s_ {{\vec{p}}} ) |\vec{k},s \rangle,
$$
I have searched google, but I couldn't find anything after about 1 hour of searching.

Are you able to tell me how the annihilation and creation operators from Dirac theory act on non-vacuum?

 

[PeterDonis]

Are you able to tell me how the annihilation and creation operators from Dirac theory act on non-vacuum?

Take the state you get when you apply a creation operator to the vacuum, and then apply another creation operator to it. What do you get? Notice that to answer this question, you need to figure out what operator corresponds to applying two creation operators in succession. Does your textbook talk about that?

 

[Jonsson]

Take the state you get when you apply a creation operator to the vacuum, and then apply another creation operator to it. What do you get? Notice that to answer this question, you need to figure out what operator corresponds to applying two creation operators in succession. Does your textbook talk about that?

I am following Peskin. As far as I can tell by reading though Dirac quantization, it doesn't.

 

[PeterDonis]

I am following Peskin. As far as I can tell by reading though Dirac quantization, it doesn't.

I assume it talks about what state you get when you apply a creation operator to the vacuum. What state is that, and have you tried applying a second creation operator to that state?

 

[Jonsson]

I assume it talks about what state you get when you apply a creation operator to the vacuum. What state is that, and have you tried applying a second creation operator to that state?

Yes

 

[PeterDonis]

Yes

So what did you get?

 

[Jonsson]

I didnt recognize the operator you're advertising :/

 

[PeterDonis]

I didnt recognize the operator you're advertising :/

Um, what? You said "yes" when I asked if you had tried applying a creation operator to the vacuum state, and then taking the state you obtained from that and applying a creation operator to it a second time. I am asking you to give the details of what happened when you did that. I can't help you unless I can see the actual mathematical steps that you're doing.

 

[Jonsson]

Nothing happens magically. I didn't know how to go on from there.

 

[PeterDonis]

I didn't know how to go on from there

From where?

You haven't posted any math that you personally have done. I can't help you if I can't see what you've already tried. I don't care if you don't think it got you anywhere; I need to see what you've tried.

 

[Jonsson]

I wrote up
$$
b^\dagger_s(\vec{p}) | \vec{k}, r \rangle = (2 E(\vec{k}))^{1/2}b^\dagger_s(\vec{p})b^\dagger_r(\vec{k}) | 0 \rangle
$$
I thought that perhaps i could get an equation by considering the commutation relation, but the operators commute, so that doesn't help. I didn't know how to go on from here.

 

[PeterDonis]

I wrote up
$$
b^\dagger_s(\vec{p}) | \vec{k}, r \rangle = (2 E(\vec{k}))^{1/2}b^\dagger_s(\vec{p})b^\dagger_r(\vec{k}) | 0 \rangle
$$

What if you take the state $| \vec{k}, r \rangle$ and apply an annihilation operator to it? The sum you said you were trying to compute in the OP applies annihilation operators to a non-vacuum state, then creation operators to the result.

Also, you are applying two different creation operators. What about if you apply the same creation operator twice? Or, even better, apply the corresponding annihilation operator, i.e., take $b^\dagger_r(\vec{k}) | 0 \rangle$ and apply $b_r(\vec{k})$ to it. That's what the sum in your OP does. (And similarly for $a$ and $a^\dagger$.)