Ground eigenstate of the quantum harmonic oscillator with the interacting vacuum $| \Omega \rangle$

Question

According to this video (at the timestamp), the professor writes down the derivation of the ground state of the 1D quantum harmonic oscillator.

Here is the screenshot from the slides of where she does it:

Now that is well and good. However, I am wondering what if we tried to derivate the ground state wavefunction for the interacting vacuum $| \Omega \rangle$ in the same way. Could we?

I know that in general

$$| \Omega \rangle \neq |0\rangle$$

And maybe we could do

$$\hat{a}| \Omega \rangle = 0$$

Just like in the screenshot and video to attain the ground state wavefunction, but this should not work since

$$\hat{a}| \Omega \rangle \neq 0$$

Because the annihilation operator $\hat{a}$ is from the free theory and we need the interacting annihilation operator instead because we are working with the interacting vacuum. I have seen this discussed here. Also, from the QFT wiki (the previous section from the one I linked, just before the Lagrangian), it says

However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory)

So I am not sure if these exist.

Now, I have never seen this done anywhere (from textbooks or videos) so could someone please provide an example of this or a link to some resources that do this?

Is there such a thing as the QHO in the interacting theory with the ground state wavefunction being

$$\Psi_\Omega (x) = \langle x|\Omega\rangle$$

Answer 1

There's a little bit of confusion here between notions from quantum mechanics and notions from quantum field theory, but this is a very interesting question that leads in interesting directions!

The basic problem is that the state of the QHO system that we represent by $|0\rangle$, which is what you mean when you say $a|0\rangle = 0$, is not the same as the vacuum of a free QFT, which is the one that you're referring to in $|\Omega\rangle \neq |0\rangle$. One reason that we use the same symbol for both is that it's sometimes useful to think of the vacuum of a free QFT, e.g. that of the free scalar theory, as a state that's annihilated by $\hat{a}_\mathbf{k}$ for any four-vector $\mathbf{k}$–the physical picture here is that the Hilbert space is spanned by states $$|\mathbf{k}_1, \mathbf{k}_2, \ldots\rangle = a^\dagger_{\mathbf{k}_1} a^\dagger_{\mathbf{k}_1} \ldots |0\rangle,$$ and we can conceive of a Hamiltonian that looks something like $$ H \sim \int d^4 \mathbf{k}\, \left(a_\mathbf{k}^\dagger a_\mathbf{k} + \frac{1}{2}\right),$$ which is diagonalized by this basis of states.

There are a few conceptual problems with the picture that I just presented. One of them is that the value of this Hamiltonian will diverge for all states, which is no good, but we can sort of fix this by just "changing our energy zero" and getting rid of the $1/2$–this is the first rumbling of troublesome infinities baked into QFT. The other problem is that when you turn on interactions, those states are no longer eigenstates of our interaction Hamiltonian, and not really clear what should take their place. One way to think about this is that in QED, for instance, a lone photon with a plane-wave wavefunction can't be a stationary state, because QED allows it to spontaneously turn into an electron-positron pair. You can still define the interacting vacuum $|\Omega\rangle$, though, but now we define it as "the unique state which is invariant under all translations and rotations and boosts," which is not very constructive but is still a helpful notion, and we can consider electrons and photons and such to be perturbations to this vacuum.

At this point you have to be very careful when applying intuition from quantum mechanics and its nice, well-defined Hilbert spaces. In QFT there is a bit of a shift away from the central QM quest to find the spectrum and eigenstates of given system, which is often either not really well-defined, not possible, or not helpful in terms of calculating actually observable quantities. In a QFT course you will build up machinery like the Dyson series and Feynman diagrams, and along the way you pivot towards calculating things like correlation functions and beta functions instead.

Part of the weirdness about trying to define interacting QFTs consistently and by analogy with free ones comes from Haag's theorem and related no-go theorems. Algebraic QFT is the field that tries to define these notions rigorously, and it's still very much an active research area.