# What is more fundamental: Fock space or Hilbert space? And why?

Consider the following state for some bosons represented in Fock space:

$$|2\rangle_{k_1}|1\rangle_{k_2}$$

where $$k_i$$ is some distinguishing index. You may think of these as the two different wavevectors for photons.

Now, if we use the Hilbert space representation of each individual boson, the same normalized state is written as

$$\frac{1}{\sqrt{3}} \left( |k_1,k_1,k_2\rangle+|k_1,k_2,k_1\rangle+|k_2,k_1,k_1\rangle \right)$$

Is this correct? Are the two expressions above equivalent? Now, consider that the particle in mode $$k_2$$ is transferred to the mode $$k_1$$ (photon addition from one mode into another, for example). This would give the state

$$\frac{3}{\sqrt{3}} |k_1,k_1,k_1\rangle$$

which is clearly not normalized. How is this possible? Does it mean Fock space is more fundamental? However, the Hilbert space picture appears more intuitive at first glance.

In any case, how do we make sense of the prefactor of $$\sqrt{3}$$ intuitively?

I am trying to somehow relate it to the prefactors that appear due to the action of creation operator in mode $$k_1$$:

$$\hat{a}^{\dagger}|2\rangle=\sqrt{3}|3\rangle$$

• "However, the Hilbert space picture appears more intuitive at first glance." That's odd. The (anti)symmetrized wave function picture should seem considerably less intuitive. See this post. May 25, 2020 at 23:12
• Let me pose a counter-question that may be enlightening: Suppose we have the spin state $\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$. Now, if we let $|\uparrow\rangle=|\downarrow\rangle$, our state becomes $\frac{2}{\sqrt{2}}|\uparrow\rangle$, which is not normalized. How is this possible? May 25, 2020 at 23:17
• Or, for another, even simpler counter-question: Suppose we have the vector $\frac{1}{\sqrt{2}}\hat{x}+\frac{1}{\sqrt{2}}\hat{y}$. If we let $\hat{x}=\hat{y}$, the vector then becomes $\frac{2}{\sqrt{2}}\hat{x}$, which is no longer of length 1. How is this possible? May 25, 2020 at 23:19
• @probably_someone I don't know. The reason I used $k_2$ goes to $k_1$ was that I was picturing photon addition where we add a photon from one mode into another. May 26, 2020 at 0:12
• The reason that $k_2 \to k_1$ breaks the normalization is that the inner product is not continuous as a function of $k$ (because it is a delta funciton). May 26, 2020 at 0:20

Fock space is the Hilbert space of multiparticle systems.

The difference you're alluding to is a difference of representation. The one you're naturally associating with Fock space is the occupation number representation, which is obviously more natural to describe multiparticle systems due to the indistinguishability of particles.

However, notice that the basis states of the occupation number representation are the same basis states that you get when you construct the full Hilbert space as a product of single-particle Hilbert spaces and write down the full basis as a product of bases of single-particle Hilbert spaces (see, for example, $$|1\rangle_k|1\rangle_K$$ is the same as $$\frac{1}{\sqrt{2}}(|k\rangle |K\rangle + |K\rangle |k\rangle)$$.

Hence, the occupation number representation just writes all of the same basis states in a new notation, a much helpful one, of course). So, not only is Fock space the same Hilbert space, the occupation number basis is not even some new basis used to span the same space.

The $$k_1$$ going to $$k_2$$ trick is misleading for the reason that we're talking about $$\underline{\text{basis}}$$ and you're mixing them with $$\textit{states}$$. One can perform some unitary transformations on the basis to get to a new basis but keeping everything else the same and just letting $$k_1$$ go to $$k_2$$ is not such a transformation.

If you want to talk about a physical state labeled by arbitrary $$k_1$$ and $$k_2$$, then you can do such a trick, and say that when this happens (in the expansion of the state into basis vectors), the coefficient in front of the basis vector with different $$k_1$$ and $$k_2$$ become zero and those coefficient which have all three particles having the same momentum i.e. ($$k_1$$ = $$k_2$$) goes to one.

• What about the case of fermions and the following reason given as an explanation for the Pauli exclusion principle: We have two identical fermions in a completely asymmetric state $\frac{|k\rangle |K\rangle -|K\rangle |k\rangle}{\sqrt 2}$. Now if we imagine that we put both fermions in the same state $|k\rangle$, then this expression tells that it disappears and hence cannot exist. May 26, 2020 at 22:37
• @SaurabhUShringarpure The state you wrote is explicitly normalized for different $k$ and $K$ so you cannot let $k$ go to $K$ in that expression. The idea is that to construct the basis for the full Hilbert space, you need to consider anti-symmetric superpositions of product of basis. And the anti-symmetric version of $|k\rangle |k\rangle$ vanishes so you simply don't have such basis in a Fermionic Hilbert space.
– ACat
May 26, 2020 at 22:46
• Ok, that makes sense. But, it is ok to say that the boson creation operator, that is applied to the Fock space representations, would act in the manner I described in the question in the Hilbert space representation. The creation operator is nonunitary and so I might be safe? May 26, 2020 at 22:55
• @SaurabhU.Shringarpure Yes, that's right. A creation operator is under no obligation to preserve the norm.
– ACat
May 26, 2020 at 23:00