Travelers is one of those stories that assumes physical time travel to be impossible, but some mode of transmission (like tachyons) can be used to send a data stream into the past. In this case the complete contents of a person's mind. Whether or not you could actually overwrite the information in a human brain would depend on the properties of the tachyons, and how they interacted with ordinary matter. Right now such particles are theoretical and their existence has neither been proven nor ruled out. So I think we can give them our suspension of disbelief in that regard.
Quantum theory in its most basic form predicts that all possible outcomes of a sequence of events are real, each part of the solution to the Schrödinger wave equation for that system. The physics of lasers don't make sense unless you assume these multiple states are real and not just some mathematical gimmick. In all likelihood multiple timelines do indeed exist. They're not so much parallel realities as they are facets of a single more complicated reality, but the branching timeline picture is still a useful visualization.
So if you followed the famous "grandfather paradox" and killed your own grandfather before he met your grandmother, you wouldn't fade away or anything like that. Going forward you'd simply find yourself in a world where you (and probably everyone else you know) are never born. You would be this off-the-grid individual whom the authorities would assume was using an alias. The timeline you experienced the "first time around" is still there, you've just switched tracks so to speak, and your old life is now forever inaccessible. This picture of time travel is especially elegant because no matter how many time travelers there are, or what they do, it's impossible to generate a paradox.
I've never been able to swallow the idea that you could travel into the past, but no matter how you attempt to intervene the universe always conspires to frustrate you and make everything happen exactly the way it did before. Too contrived. You can easily come up with scenarios where a ridiculous chain of events would be required for that. Time travel being impossible is far more likely. Suppose you wanted to kill Julius Caesar as a teenager so he never becomes Emperor. You send one hundred special forces soldiers in full body armor, with automatic weapons, to a place and time where you know he'll be. What happens - every last one of them has their gun jam? Or each one suffers a completely different random misfortune? That strikes me as ludicrous. Either they're able to take out their target, or you can't send them there in the first place.
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