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

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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 given by:

$\frac{|k_1,k_1,k_2\rangle+|k_1,k_2,k_1\rangle+|k_2,k_1,k_1\rangle}{\sqrt {3}}$

Is this correct? Are the two expressions above equivalent?

Now, consider that $k_2$ becomes equal to $k_1$.

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?

The statement that your state $\frac{|k_1,k_1,k_2\rangle+|k_1,k_2,k_1\rangle+|k_2,k_1,k_1\rangle}{\sqrt {3}}$ is normalized is based on $\langle k_1|k_2\rangle = 0$, which means you can't set $k_1 = k_2$.

I was wondering what happens in the context of photon addition. There we have a photon in one mode that gets transferred to/added to another mode.

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