> This is why the quantum vacuum is filled with particles

The quantum vacuum is defined as the no-particle state.

The particle number operator works by matching annihilation and creation operators. Those operators are fixed on particular coordinates.

When annihilation and creation operators match up there is no particle.

For inertial observers in flat spacetime, a Lorentz transform relates the annihilation and creation operators for any pair of observers no matter what their relative velocities or which way they face relative to one another.

A Bogoliubov transformation can relate annihilation and creation operators for a wider range of observers in flat spacetime, including those who are accelerated. However, a natural choice of coordinates for extremely accelerated observers relates poorly with a natural choice of coordinates for inertial obervers (for example, in each case using polar coordinates with the origin always at the respective observers), with a result that there is a disagreement about where in a set of shared coordinates one finds frequency modes which relate to annihilation and creation operators.

As a result, in flat spacetime, the no-particle vacuum of a(n inertial) Minkowski observer looks to an accelerated observer as if there are particle creations not matched by particle annihilations, and thus the particle number goes up. However, conversely, the no-particle vacuum of a (n accelerated) Rindler observer looks to all Minkowski observers as having particles.

Since Special Relativity does not privilege frames (even though its focus is on relating inertial observers' observations), neither type of observer is really "more correct".

However, since the accelerations required to have a detectable number of particles is extreme (you need to maintain 10^20 gees of acceleration to see a thermal bath of 1 kelvin) and since accelerations do not last indefinitely, it is fair to pseudo-privilege the inertial observers, and conclude that (a) the Rindler observer is counting annihiliation operators source by whatever is locally powering the uniform acceleration and (b) the Minkowski view that the accelerating observer is therefore emitting a thermal bath into the Minkowski vacuum is more correct.

Additionally, in SR one can tell an extreme acceleration from mere inertial (even ultrarelativistic) movement, using an accelerometer.

In general curved spacetime one expects observers not moving on a geodesic (one hovering at a fixed height above the surface of a planet, for instance) to disagree about the particle numbers in the local area compared to an observer freely falling on a geodesic as that observer passes infinitesimally close to the non-geodesic (accelerated) observer, when the particle numbers are defined using annihilation and creation operators set against "natural" coordinate choices for these observers and relating those choices through a Bogoliubov transformation.

Two mutually accelerating observers in flat spacetime or two observers in general curved spacetime may not be able to find a shared coordinate system, and when that is the case, no one can really say which of the two is "more correct" about her or his no-particle vacuum.

Consequently, the quantum vacuum may have particles in it, but only for a peculiar choice of quantum vacuum (i.e. you look at someone else's claimed no-particle state and see particles in it).

When looking at one's own no-particle state, one cannot determine the vacuum energy. One can conclude that at each coordinate point there is the same energy, but not the exact energy -- this ground state / minimum energy can be arbitrarily high.

However, the uncertainty inequality \Delta E\Delta t \geq {h \ over 4\pi} applies and one can treat this as momentarily separating annihiliation and creation operators. One can treat this formally (e.g. in Feynman diagrams of this process) as the production of virtual particle/anti-particle pairs, or alternatively as a realization of the non-commutation of the field's energy operator with the particle number operator, and thus the no-particle state is a superposition of particle number eigenstates.

Trying to turn these superposed states into real particles is taken less or more seriously depending on the writer. I'm fairly confident that many physical cosmologists are fine with the idea when considering the possibility that the early universe is a fluctuation of a high-energy no-particle state into a still-high-energy many-particle state. The problem is in suppressing these fluctuations in later regions of space that are free of particles (from our perspective in our solar system now). Physicists in other areas like to poke many holes in these ideas.

However, in both cases, "filled with particles" invites horror-show ideas of larger scale structures fluctuating into existence and causing problems. This doesn't happen as far as we can tell, so we should preserve the no-particles idea of the vacuum conceptually.

> the Casimir effect

does not really rely on the particle count operator. One can cast it as the (idealized) plates screening longer positive and negative frequency modes that when combined linearly and treated as creation and annihilation operators, simply produces more of these on the outsides of the plates. The "more particles" outside the plates push them together through the "no particles" vacuum between them.

Mathematically this is OK, but conceptually it has problems.

Decades ago a quantum electrodynamics approach was taken to the Casimir effect showing that it arises due to the relativistic retarded intermolecular forces within the plates themselves. This view fully reproduces the "vacuum fluctuation" observables for idealized plates and generalizes to non-idealized plates in a way that the fluctuation view does not (yet, as far as I know). There have been updates on this view of the Casimir force since, and it holds up well, and has the advantage of not needing any particular value for "zero point energy" (which is just the vacuum energy as I described a few paragraphs above, and which ought to be unprobe-able by observers similar to us, which is in fairly serious conflict with the vacuum fluctuation idea taken to extremes).

This is elucidated with citations under "Relativistic van der Waals force" and under "Generalities" in your [1].

Taking virtual particles too seriously can be misleading, even if it works out mathematically.