Relativistic transformation of the wave packet length Let us suppose we have an excited atom at rest. It has a certain mean lifetime $\tau_0$. If we wait sufficiently long time $t>>\tau_0$, we will find a deactivated atom and a (spherical) electromagnetic wave function of photon with about $\tau_0\cdot c$ long layer with non zero probability to find a photon within. Something like a fast expanding probability "ring" with a $\tau_0\cdot c$ width of the ring.
Now, let us consider this system in a moving reference frame. It seems to me that this width $\tau_0\cdot c$ is relativistic invariant: it is a difference between two "fronts" of electromagnetic wave rather than a length of a material body subjected to the Lorentz contraction. Is it correct? In other words, whether this picture relativistic invariant?

 A: No, the probabilistic density isn't Lorentz-invariant. First of all, the probabilistic density  isn't a Lorentz scalar: it is a time component of a 4-vector whose spatial components encode the probability current.
Second, it is obviously not true that the ring will be equally thick in all directions. While it's true that the photons are moving by the speed of light in any inertial system, they're moving from a different center because the atom itself is moving (from the moving frame's viewpoint).
So the ring from a moving reference frame's viewpoint will look like one of the black or purple squeezed annuli on the picture below:

A: First, just as a note, I would want to say that the problem will be somewhat ill defined because in QM it may be simple to define transition probabilities for going from one state from another, but, the actual process over time which should give you the Dirac transition current from charge and spin, the current which is responsible for the electromagnetic field, is on a whole different level.  
Going back to your question: The circles you are drawing correspond to "Spheres of Simultaneity". In other reference frames they become "Ellipsoids of Simultaneity", so they are not Lorentz invariant.
I did put a significant amount of work in all the 3D images (4.9 through 4.24) in the introductory chapter: "Non-Simultaneity from the wave equation" to visualize why and how this happens.
http://physics-quest.org/Book_Chapter_Non_Simultaneity.pdf
for instance:

Regards, Hans
