Understanding QFT's Operators and Fock Space: Demystifying Quantum Superposition

[Xezlec]

Hello and sorry for the following dumb question.

I was reading about quantum field theory out of general curiosity about the subject and I was confused by the way it seems like the web pages I've read imply that the operators we define in QFT (say, the annihilation operator, or the Hamiltonian) operate "on the Fock space". That sounds like it implies that the argument we are passing to those operators (i.e. what plays the role of the "wavefunction" from first quantization) is a point in Fock space. But if that is so, then doesn't that mean that the system cannot be in a superposition of states in which it is both at one point in Fock space and also another? I thought it could?

Another way to say this: I would have expected that the QFT version of $H\Psi=E\Psi$ would have $\Psi$ as something like $\Psi:F\rightarrow\mathbb{C}$ where $F$ is the Fock space, so that a probability amplitude and phase are assigned to each possible set of numbers of quanta. But what I'm reading seems to imply instead that the $\Psi$ is just a point in $F$, not a function on it, so that $\mid 1,2,0,...\rangle + \mid 2,1,0,...\rangle = \mid 3,3,0,...\rangle$. Surely that can't be right, right?

 

[Jolb]

Hello and sorry for the following dumb question.

I was reading about quantum field theory out of general curiosity about the subject and I was confused by the way it seems like the web pages I've read imply that the operators we define in QFT (say, the annihilation operator, or the Hamiltonian) operate "on the Fock space". That sounds like it implies that the argument we are passing to those operators (i.e. what plays the role of the "wavefunction" from first quantization) is a point in Fock space. But if that is so, then doesn't that mean that the system cannot be in a superposition of states in which it is both at one point in Fock space and also another? I thought it could?

As to your first paragraph, I think your question has more to do with quantum mechanics than quantum field theory. Let's talk about a Hilbert space for a single particle rather than a Fock space of arbitrary particle number. As a simple example, let's take the spin state of an electron: it lives in a two-dimensional Hilbert space where the basis vectors are |up> and |down> with respect to some axis. A superposition of states is a point in Hilbert space that lies off the axes--it has a component of |up> and a component of |down>. The states on the axes are eigenstates. So a superposition is just an off-axis point in a single space--we don't need more than one point in the space to make a superposition. The same reasoning holds for Fock spaces.

Maybe it is too late at night but I don't really follow your second paragraph.

[Sorry if I was being a little loose with my terminology but you seem to have a good enough background to follow. Please let me know if I said anything that's too jargony.]

Edit: It looks to me like you might be thinking of a "mixed state" rather than a superposition. This would be a quantum statistical mechanics topic. Here's what wiki has to say: http://en.wikipedia.org/wiki/Mixed_state_(physics)#Mixed_states

 

[Xezlec]

As to your first paragraph, I think your question has more to do with quantum mechanics than quantum field theory. Let's talk about a Hilbert space for a single particle rather than a Fock space of arbitrary particle number. As a simple example, let's take the spin state of an electron: it lives in a two-dimensional Hilbert space where the basis vectors are |up> and |down> with respect to some axis. A superposition of states is a point in Hilbert space that lies off the axes--it has a component of |up> and a component of |down>. The states on the axes are eigenstates. So a superposition is just an off-axis point in a single space--we don't need more than one point in the space to make a superposition. The same reasoning holds for Fock spaces.

So you're saying we define the term "Fock space" to refer not to the space of all states that can be reached through some combination of creation operators on $\mid 0\rangle$, but to the space spanned by the axes made up of multiples of those states? In that case it's the terminology on Wikipedia that is wrong, I guess, because it said:

The vacuum $|0\rangle$ is taken to be annihilated by all of the $a_k$, and $\mathcal{H}$ is the Hilbert space constructed by applying any combination of the infinite collection of creation operators $a_k^\dagger$ to $|0\rangle$. This Hilbert space is called Fock space.

Just to be clear: you're saying that the above text is flat-out wrong, right?

If so, that fully answers my question. It means my suspicions were right, but the problem was with the definition of "Fock space", not the way that elements of the Fock space were being used.

(EDIT: actually, maybe the problem is with the word "combination"? Does a sum of products count as a "combination" for the purposes of that definition?)

Maybe it is too late at night but I don't really follow your second paragraph.

Hmm. What if I remove all the extra explanatory sentences and just focus on the key part:

I think the following should be nonsense: |1,2,0,0...> + |2,1,0,0...> = |3,3,0,0...>.

If you agree that equation is complete, ridiculous nonsense, then you agree with what I was trying to say in the second paragraph.

Edit: It looks to me like you might be thinking of a "mixed state" rather than a superposition. Here's what wiki has to say: http://en.wikipedia.org/wiki/Mixed_state_(physics)#Mixed_states

Nope, not asking about mixed states.

 

[Jolb]

So you're saying we define the term "Fock space" to refer not to the space of all states that can be reached through some combination of creation operators on $\mid 0 \rangle$, but to the space spanned by the axes made up of multiples of those states? In that case it's the terminology on Wikipedia that is wrong, I guess, because it said:
Just to be clear: you're saying that the above text is flat-out wrong, right?

First I think it's important to remember that sometimes physicists use mathematical terminology so loosely that terms like "Hilbert space" get applied to things that aren't actually Hilbert spaces. Of course this all depends on your definitions, but normally a Fock space is thought of as a generalization of Hilbert spaces--in Hilbert spaces particle number is usually conserved. So the Wiki article is being a little loose with its terminology when it says "this Hilbert space is called Fock space"--in fact it would be inconsistent with wiki's definition of Fock space.

I think the following should be nonsense: |1,2,0,0...> + |2,1,0,0...> = |3,3,0,0...>.

If you agree that equation is complete, ridiculous nonsense, then you agree with what I was trying to say in the second paragraph.

Yes, I agree that it is absolute nonsense. The indices label particle number, not components that follow some sort of addition rule.

 

[Xezlec]

Thanks!

I also finally found this magical page on Wikipedia which clarified some of the terminology that was confusing me. Now I see that a Fock state is not just any element of Fock space, but rather, only a state that is "on an axis", so to speak, while the majority of Fock space consists of points that are not "Fock states". That clears it up nicely.

http://en.wikipedia.org/wiki/Fock_state