I'm studying the canonical quantization of the Klein-Gordon real scalar quantum field theory, given by the classical Lagrangian density $$\mathscr L = {1\over 2}\partial_\mu\phi\partial^\mu\phi-{1\over 2}m^2\phi^2.$$
The plane wave solutions to the Euler-Lagrange equation (which becomes the KG equation) are, of course, of the form $$\phi(t,\vec x)\backsim e^{-i\vec k\cdot\vec x}$$
where $\vec k\cdot\vec x = k_\mu x^\mu$ and $k_0\equiv\omega = \sqrt{k^2+m^2}$. To find arbitrary solutions you take superpositions over the three spatial dimensions of $k$, and thus most of your integrals start with something looking like
$$\int \frac{d^3 k}{(2\pi)^3 2\omega}...$$
Indeed, the Hamiltonian and the creation and annihilation operators have such integral forms, and thus when you add bosons to the vacuum the parameter is this mysterious vector $\vec k$. Is there any sort of intuitive physical meaning to this (maybe having to do with the bosons for which it is an initial parameter) or is it simply a byproduct of abstract mathematics?