## Self-Consistent Time Travel

Posted by on Thursday, June 19, 2014

*As I wrote in yesterday's article, I was recently disappointed in a public lecture given by a supposed college physics instructor in which he spoke of ideas which are impossible according to the laws of physics. However speaking as a trained theorist who does keep up to date with the latest research, I must say that his list of impossibilities only served to show how little he himself knew of modern physics. As such I have written a series of articles explaining loopholes in the laws of physics which I consider to be interesting, but also accessible to the general public. Today's topic is on time travel paradoxes.*

One of the most common ideas raised in science fiction is time travel. It is interesting to consider the possibility of a mechanism that would allow a traveler to go back in time to an earlier era, and see a very different world from the one we know today.

Unfortunately such a mechanism also leads to bizarre paradoxical situations. The most famous of these is the

*grandfather paradox*, which asks the question of what happens if you go back in history and kill your grandfather before your parents are born. Since your parents were never born, you were never born and therefore cannot go back to kill your grandfather, meaning that he survives and you are born.

Some people argue that because of such paradoxes, time travel must be impossible. I consider this a weak argument, because it is essentially saying that anything we do not understand must not exist, which is clearly untrue. And of greater importance, there are resolutions of this paradox which do not require any violation of the known laws of physics. The simplest of these, which I have written about before and won't go into great detail here, is that we may live in one of multiple parallel universes, and time travelers simply enter an alternate history. Then you are not murdering your own grandfather, but the grandfather of a parallel universe copy of yourself. Although this may sound fanciful, many of the leading experts in the foundations of quantum mechanics believe that this is exactly what does happen, with the universe splitting into multiple copies of itself countless billions of times per second.

However instead I am going to present another resolution of time travel paradoxes using generally accepted methods of quantum mechanics (based loosely on work by Seth Lloyd at MIT). I must also give a disclaimer that this model cannot handle the complexity of the human mind or body, but is intended more as an indicator of how time travel paradoxes

*might*be resolved in quantum mechanics. But first I must provide a very basic review the path integral formalism.

Consider a simple system in which an electron is sent towards a screen that contains two slits in it. According to quantum mechanics, the electron effectively splits into two virtual particles, and each travels through one of the two slits. (This has been measured experimentally). To calculate where the electron hits the target, one calculates all possible paths from the source to the target, and assigns to each path a wave. The waves of all the paths interfere with each other, creating regions of peaks and valleys on the target. The amplitude of the wave when it lands at the target determines the probability of the electron hitting each point.

In general, this works for all quantum mechanical systems. If you want to know where a particle will end up, you must calculate every possible path that it could take, assign a wave to each path, and then add up all of the waves to find the final location of the particle. This is the path integral formulation of quantum mechanics, and it has been accepted as a valid formulation of quantum mechanics for the past sixty years.

So what happens if you include a time machine? To calculate how a particle will behave when a time machine is present, you would again calculate every possible path it could take, including paths that include multiple loops through the time machine. You would need to calculate a path that misses the time machine, one that loops once, that loops twice, that loops one thousand times, and up to infinite loops of the time machine.

The equivalent of the grandfather paradox in this case is a wave which is sent back in time, in such a way that it destructively interferes with itself to cancel itself out. However anyone who has completed any university mathematics or physics courses knows that waves on a closed loop are constrained to only certain frequencies. The same holds for a time loop and the quantum mechanical paths of a particle. If the path and its wave have the wrong shape, or the wrong frequency, then it cannot exist and therefore is not part of the path integral.

Consider this simple formula using quantum mechanics operators (this paragraph can be skipped by those who do not like mathematics). Let S be the operator that evolves the system from time T0 to time T1 including all effects from time travel, let P be the evolution operator over the same time without time travel present, and let X be the time-travel operator that moves the system backwards from T1 to T0. Then it follows that

And so in this very simple model, the grandfather paradox is resolved because quantum mechanics says that the probability of a paradox is zero. In a more realistic scenario with humans, this would be equivalent to the claim that something will always go wrong with the murder. The probability of a murder is zero, while the probability of missing, or of the grandfather surviving, or of the gun jamming, (or of the discovery that your grandfather wasn't actually your grandfather) are all non-zero. Something will always happen to prevent the paradox.

Of course the single particle model and arguments are not automatically valid for complex situations, but in my opinion it is clear that quantum mechanics has the potential at least to resolve all such time travel paradoxes. It certainly doesn't make time travel impossible!

So what happens if you include a time machine? To calculate how a particle will behave when a time machine is present, you would again calculate every possible path it could take, including paths that include multiple loops through the time machine. You would need to calculate a path that misses the time machine, one that loops once, that loops twice, that loops one thousand times, and up to infinite loops of the time machine.

The equivalent of the grandfather paradox in this case is a wave which is sent back in time, in such a way that it destructively interferes with itself to cancel itself out. However anyone who has completed any university mathematics or physics courses knows that waves on a closed loop are constrained to only certain frequencies. The same holds for a time loop and the quantum mechanical paths of a particle. If the path and its wave have the wrong shape, or the wrong frequency, then it cannot exist and therefore is not part of the path integral.

Consider this simple formula using quantum mechanics operators (this paragraph can be skipped by those who do not like mathematics). Let S be the operator that evolves the system from time T0 to time T1 including all effects from time travel, let P be the evolution operator over the same time without time travel present, and let X be the time-travel operator that moves the system backwards from T1 to T0. Then it follows that

S = P + PXP + PXPXP + ...

where each term on the right represents some number of loops through the time machine. However this sum is easy to calculate,

S = P / (1 - XP)

which is very large when XP ~ 1, and since the operator must be normalized this means that it is virtually zero for all other values. So only systems for which the combined operator XP leaves the system unchanged have a non-zero probability, which means only systems which are non-paradoxical can ever occur.

And so in this very simple model, the grandfather paradox is resolved because quantum mechanics says that the probability of a paradox is zero. In a more realistic scenario with humans, this would be equivalent to the claim that something will always go wrong with the murder. The probability of a murder is zero, while the probability of missing, or of the grandfather surviving, or of the gun jamming, (or of the discovery that your grandfather wasn't actually your grandfather) are all non-zero. Something will always happen to prevent the paradox.

Of course the single particle model and arguments are not automatically valid for complex situations, but in my opinion it is clear that quantum mechanics has the potential at least to resolve all such time travel paradoxes. It certainly doesn't make time travel impossible!

Tags: "time travel" "path integral formalism" "quantum mechanics" "manyworlds interpretation"