Are zero-point fluctuations (ZPF) of vacuum Lorentz invariant?

Question

According to the second postulate of Special Relativity the speed of light c must be invariant that consequently leads to the first postulate of Einstein the Lorentz invariance in all inertial frames of reference.

However, with ZPF energy fluctuations of vacuum space with virtual particles popping out of the vacuum from 'nothing', I find it difficult to define ZPF as Lorentz invariant.

Unless it is conclusively proven that the non-zero tiny ZPF noise energy is due to almost perfect QFT fields cancellation it is difficult to claim that ZPF of vacuum is Lorentz invariant.

Answer 1

In technical terms, the zero-point-energy is the counterterm of the form $\mathcal L_\text{ct}\supset\text{const}$. Given that the Lagrangian is a Lorentz scalar by definition, this makes it manifest that this counterterm is a scalar as well.

Being a counterterm, it is not scheme-independent and thus not measurable by itself, although for some suitable backgrounds it does have an observable effect. Flat spacetime is too boring of a background and does not allow you to detect any physical effects of this couterterm. That being said, in flat spacetime the contribution of this counterterm to the momentum operator is $$ P_\mu\sim \int \delta(p^2-m^2)p_\mu\mathrm dp $$ For $\mu=i$ this integral vanishes by spherical symmetry, and therefore only the $\mu=0$ part is non-trivial. This is the reason we call this a zero-point-energy. Lorentz invariance is perhaps non-manifest since only the $\mu=0$ part is non-zero. But of course in more interesting backgrounds, there is also a non-trivial contribution to the momentum (which is essential, for example, in $2d$ CFTs), and Lorentz invariance is more manifest. In any case, Lorentz invariance is always guaranteed to hold given the definition in terms of $\mathcal L_\text{ct}$.

Answer 2

Both postulates are independent claims, it is not true that the second leads to the first. This is easy to see when we apply the same logic to wave propagation in ether-like systems, e.g. sound propagation in atmosphere. Sound speed is independent of the relative motion of the source, but this does not imply that all frames in the atmosphere are equivalent.

Zero point EM radiation is usually described by a spectral function of frequency (Poynting energy density per unit frequency interval) that is proportional to 3rd power of frequency. People who studied the concept of zero point radiation in depth (e.g. Timothy Boyer) claim this spectral function is Lorentz-invariant [1]. Virtual particles are artifacts of perturbation formalism, they do not "pop out of the vacuum" in reality.

[1] T. H. Boyer, Random electrodynamics: The theory of classical electrodynamics with classical electromagnetic zero-point radiation, Phys. Rev. D 11, 790-808 (1975). (Lorentz invariance is discussed in Section D.)