Can virtual particles be thought of as off-shell Fourier components of a field?

I just found this blog post, which gives an interpretation of virtual particles I haven't seen before.

Consider a 1D system of springs and masses, where the springs are slightly nonlinear. A "real particle" is a regular $\cos(kx-\omega t)$ wavepacket moving through the line, where $\omega$ satisfies the dispersion relation $\omega = \omega(k)$. When two real particles collide, the region where they collide temporarily looks really weird, as they interact nonlinearly, pulling and pushing on each other.

Formally, we can write this weird region as the sum of a bunch of $\cos(kx-\omega t)$ waves, but there's no guarantee they'll have the right dispersion relation. Thus, each term in the resulting expansion is a off-shell "virtual particle". If you add up all the virtual particles, you get the actual intermediate field state, which is just a weird ripple in the field where the two particles are interacting.

As another example, consider the statement "a static EM field is made of virtual particles". Under this interpretation, what that really means is, "a static field (e.g. $1/r^2$) is not equal to $\cos(kx-\omega t)$, but may be expanded in terms of such sinusoids", which is much less mysterious sounding. In fact, this is exactly what we do when we consider scattering off a potential in normal QM, e.g. in the Born approximation.

The above gives some intuition for what a 'virtual particle' means in classical field theory. They are the Fourier components of the field with $(\omega, k)$ not satisfying the dispersion relation, and they are useful in classical perturbation theory. However, they are not propagating degrees of freedom, so they only appear during interactions.

The picture above is entirely classical. Does this picture generalize to quantum field theory, giving a physical intuition for virtual particles there?