Canonical Quantization of Fields
Promoting fields to operators — building Fock space from oscillators.
Treat each Fourier mode of the field as a harmonic oscillator. Expand
$$\hat \phi(x) = \int \frac{d^3 k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf k}}}\left(\hat a_{\mathbf k} e^{-ik\cdot x} + \hat a^\dagger_{\mathbf k} e^{ik\cdot x}\right),$$with $[\hat a_{\mathbf k}, \hat a^\dagger_{\mathbf k'}] = (2\pi)^3 \delta^3(\mathbf k - \mathbf k')$. The vacuum $|0\rangle$ is annihilated by every $\hat a_{\mathbf k}$. Fock space is built by repeated application of creation operators: $|n_{\mathbf k_1}, n_{\mathbf k_2}, \ldots\rangle$. Each excitation is a particle of mass $m$.
Fields are operator-valued distributions — particles are emergent quanta of the underlying field, not fundamental objects.
Interactive: Fock-state occupation numbers
Each bar is the occupation of a momentum mode. Click bars to add/remove quanta.