Annihilation operator acting on a Fock state

[L.W.C]

I'm trying to show:

a(p)|q₁,q₂,...,q_N> =
$\sum$^N_i=1(2pi)³2E_p\delta⁽³⁾(p-q_i)x|q_i,...,q_i-1,q_i+1,...,q_N>

I'm pretty sure you have to turn the ket into a series of creation operators acting on the vacuum |0>, but then not sure what relations need to be invoked for it to be clear.

Any help would be appreciated.

 

[strangerep]

I'm trying to show:

a(p)|q₁,q₂,...,q_N> =
$\sum$^N_i=1(2pi)³2E_p\delta⁽³⁾(p-q_i)x|q_i,...,q_i-1,q_i+1,...,q_N>

I'm pretty sure you have to turn the ket into a series of creation operators acting on the vacuum |0>, but then not sure what relations need to be invoked for it to be clear.

Any help would be appreciated.

You might get more help if you attempt a little more yourself, according to the PF rules for homework help. For starters, write down the canonical commutation relations between the a/c operators, and the rules for how they act individually on the vacuum. Then try and do the N=2 case.

It might also help if you mention which textbook you're working from.

 

[L.W.C]

O.K., so you just use the relation:

a(p)a⁺(q) = [a(p),a⁺(q)] + a⁺(q)a(p)

for each case.

Also can I just clarify that in the answer when it says:

|q_i-1> if i=1, is that just ignored?

I' learning form these notes:

http://www.hep.man.ac.uk/u/pilaftsi/QFT/qft.pdf

With the aide of peskin and schroeder.

Thank you

 

[strangerep]

O.K., so you just use the relation:

a(p)a⁺(q) = [a(p),a⁺(q)] + a⁺(q)a(p)

for each case.

OK, so now try to do the N=1 case explicitly, using equations (2.29), (2.35) and maybe (2.36) from P&S.

Also can I just clarify that in the answer when it says:

|q_i-1> if i=1, is that just ignored?

I think it's the vacuum $|0\rangle$.