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?