According to the Many Worlds Theory (MWT) interpretation of quantum mechanics, which is the least convoluted and most widely accepted interpretation among quantum physicists, every time a quantum-random event happens—every time the Schrödinger wavefunction collapses—multiple new branches of reality, or timelines, are created in which each possible outcome of the event is realized.
While the chances are unimaginably small for most outcomes, just about anything is possible. For example, there is a vanishingly small but nonzero probability that your head will suddenly turn into a basketball in the next split second. This possibility must therefore be realized in some particular timelines.
This, interestingly, has some logical implications regarding personal immortality.
In the event that you die in certain timelines, in other timelines branching off from just before your death, atoms just happen rearrange themselves in such a manner that you avoid death, or quantum events otherwise lead to that result. You won’t experience those timelines in which you died, because you’ll be nonexistent in those timelines, but in the remaining timelines you’ll experience your continuing to live. This will happen no matter how many times you die in certain realities or how long you happen to live in a minority of them.
Since all of the above timelines are necessarily the causal results of their trunk timelines in the same way we observe causation, or the passage of time, in everyday life, this means that the yous that survive experience a continuity of self progressing from the would-be time of death into the future, at least in the timelines where the quantum events causing the avoidance of your death are not so drastic that the resultant form of “you” has lost its sense of identity.
Since there’s on meaningful way to say that those yous that lived are any less you than those that died, therefore, it’s only logical to say that you will experience life continuing onward into the future indefinitely.
That’s quantum immortality in a nutshell. The paradox comes in when we consider your life from the perspective of outside observers.
From the perspective of the rest of the worlds in the realities in which you continue to live, the frequency at which individuals miraculously avert death N number of times is in perfect accordance with mundane statistical prediction. There will be no flooding in these realities of people who are apparently immortal, because the chances of those particular quantum events in those realities that led to your continuing to live are just as small as the ratio of branching timelines in which you survived is to the ones in which you died. For every centillion realities, there will be like one person who happened to quantum-randomly avert death for, say, 500 years.
So, from the individual’s perspective, his or her chances of quantum immortality are 100%, while from the public’s perspective, they’re virtually zero, and the likelihood matches that predicted according to normal causality.
So there’s the paradox: quantum reality must totally be a real thing for the individual, even though it’s not real, and will never be statistically observed as anything beyond the mundane, from the perspective of the rest of the world and science.