RELATIVISTIC INVARIANCE OF THE VACUUM

The question whether the vacuum is invariant under relativistic (Lorentz) transformation of reference frames is fundamental. Moreover, one must discuss continuous transformation and discrete like time or space reversal. The naive, classical vacuum is empty so it would be indeed the same in every (inertial) reference frame. However, the quantum vacuum is not empty, it has zero-point fluctuations (the vacuum corresponds to zero temperature). Still, in free theories the vacuum is the state with no particles and one can show that this "emptiness" is preserved in every frame. However, in interacting theories the particles can virtually exist even in the vacuum. One can prove that scattering amplitudes are invariant but it does not imply general invariance. The general proof needs complex time path formalism. The time runs on the path that goes downward in imaginary part and has real (flat) sections that can go forwards and backwards several times. The path is flexible but the jump is constant, equal ℏ/k_BT, where ℏ is Planck constant, k_B is Boltzmann constant and T is temperature.

The vacuum corresponds to zero temperature so the jumps become infinite

Lorentz transformations of time t and position x are linear and preserve the speed of light c, as required by Einstein's theory of special relativity so that (ct)²-x² is the same in every frame. Continuous Lorentz transformations preserve also the sign of t. Time reversal means replacement t by − t.

The key of the proof of invariance is the consistence of infinitesimal Lorentz transformations with complex time. It is perturbative, i.e. applied to a series in powers of interaction strength. It is by no means trivial, there is no way to find a general "simpler proof"! Moreover, there is no time reversal symmetry because the complex path is asymmetric in time (it has a preferred time arrow). It is closely connected to general quantum time asymmetry.

Before publishing in The European Physical Journal C, my result underwent long peer-review, with several rejections, often based on false claims, e.g. of Physical Review D Editorial Board Member V. P. Nair and Editor-in-Chief G.D. Sprouse, that the proof is not new (no referee provided a reference to a complete and correct proof). HERE I present excerpts form the reports/editorial decisions, with a comment. If you are interested in full correspondence, it is available on request by email: abednorz[at]fuw.edu.pl.