How time loops work, from the point of view of physics
Time loops are ubiquitous in modern fiction. People keep waking up and live the same day over and over (e.g., "Groundhog Day"); people get sent to the past, or to the future, then come back to their own "timeline" and so on. However, if time loops were real (closed timelike curves in a semiclassical spacetime evolving via Einstein's general relativity), their physical effects would be quite different than what is shown in the movies.


Time loops are not ruled out in physics. Hawking made a conjecture of "chronology protection": as a consequence of known or yet to be discovered physical laws, time loops should not occur. But this conjecture remains without proof. One can write down the gravitational field (the pseudo-Riemannian metric( describing a spacetime that contain time loops. One can then try solving for the motion of matter (classical or quantum) in those spacetimes. So, that is what we are assuming when we are talking about "time loops according to physics". To be more precise, we are talking about a "gravitational time loop".

The first thing that's different from (some) movies is that there will be no paradoxes due to time travel. The equations of motion will have solutions and will describe what happens at any given time. There is only one correct timeline, and it is always consistent. The actors may imagine that they are going to "change" the timeline - but they will in fact never be able to do so.

That said, the "correct" timeline may have to look quite weird. There are simple mechanical scenarios where perfect billiard balls in the presence of a time loop will have infinitely many solutions of the equations of motion ("Polchinsky's scenario", https://en.wikipedia.org/wiki/Novikov_self-consistency_principle). This is highly counter-intuitive and means that even the motion of perfectly elastic, point-like billiard balls - a very simple mechanical system - cannot be deterministic in the presence of a time loop. (Usually, mechanical equations of motion for perfectly elastic bodies have a unique, deterministic solution.)

Now, quantum mechanics will allow several outcomes with known probabilities in that scenario, and those probabilities are well-defined. This also means that quantum-mechanical properties may actually play a significant role in the behavior of perfectly classical systems; we may not get reasonable results without describing the situation with quantum mechanics.

In simpler terms, it means that the "correct timeline" may look quite weird: it might include events that are only explained as quantum fluctuations that are normally very unlikely but nevertheless keep happening.

To make this more visually clear, imagine that you go into the time machine in order to travel to the past and prevent that same time machine from being built. This is clearly a "paradox" that you intend to create. You go into the time loop and install, say, a remote-controlled explosive device in the room where the time machine is being built. However, when you press the button to trigger the explosion, a large quantum fluctuation happens and prevents the button from working properly. So, the time machine does not explode.

How did this quantum fluctuation happen? It happened, despite its expected low probability, because of the requirement that the timeline must be consistent. In the sum-over-histories formulation of quantum mechanics, we sum over all possible histories -- that is, only over all paradox-free scenarios (paradoxes are a mathematical contradiction and cannot enter the equation!). It could happen that all paradox-free scenarios involve some low-probability quantum fluctuations. Then we will certainly observe one of those quantum fluctuations, even though it would normally have a very low probability.

So, the presence of a time loop may "amplify" quantum fluctuations and make their probability large. We may see weird (not impossible but extremely rare) events happening, such as large objects quantum-tunneling through walls, small particles self-assembling "by chance" into macroscopic bodies, and so on, if those events are necessary to keep timelines consistent.

(The fact that those events have never been seen could be evidence towards the non-existence of gravitational time loops.)

What we certainly cannot see in a time loop is a "Groundhog Day" scenario where one person remembers entering a time loop many times but all other people remember nothing. The memory of one person follows the same physical laws as the memories of all other people. A gravitational time loop is no more than a special configuration of the gravitational field in a specially chosen spacetime. Physical bodies move in that spacetime according to the equations of motion. Time travel happens to all physical bodies that enter the loop (including bacteria, air molecules, etc.), so time travel cannot be limited to memories or the "consciousness" of one person. If some persons travel back in time via a time loop, they will meet previous copies of themselves in the past; they will not "become" their previous selves. It is impossible that only the memories of one person would travel to the past, replacing the memories of the previous copy of that person; this is not how a gravitational time loop works.

Another impossibility is that of "changing the timeline". Plot lines usually go around someone who wants to change the timeline. Let us weaken this condition and say that we don't need to be the agent of change; we just want to time-travel along and observe that the timeline has changed.

But even in this weakened formulation, the change of the timeline cannot be observed. In the memory of any given person, there exists only one timeline. The memory may travel with the person in a time loop but the events that the person observed at a given time must remain the same.

This is because the events remembered by a person depend on the (semi-classical) wave function of the world. The particular version of the person's memories is part of a particular branch of the wave function. There is only one set of (semi-classical) events that can be observed at a particular time.

So, it is not possible that a person remembering one timeline (where certain events happened) would travel to the past and observe another, slightly different timeline (say, where some of the events did not happen). If we do observe this, it would mean, rather, that we have traveled to a different part of the universe where, by chance, there is a copy of the Earth and the events are similar except the ones that were "changed". This kind of travel is certainly not impossible (especially in the "multiverse" scenario), but it does not correspond to time travel due to the presence of a time loop. It is not time travel at all; it is "space travel".

Quantum mechanics describes a world where each quantum event creates many possible futures or "branches" of the wave function. David Deutsch claimed at some point that time travel may go from one of the branches to another. However, I believe it was shown that Deutsch was mistaken. I would explain it like this: The branches are disconnected from each other but not separated in space; so, one cannot travel from one branch to another by going through a special place (the time machine) that connects different branches. The branches are nowhere connected. This means we cannot have a "branching" of the timeline where different copies of the same person remember different events because they are in a different branches of the quantum wave function.

In summary, time loops would have been much less weird than usually shown in movies, but nevertheless weird enough to be clearly observable.