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Even back when I wasn't happy about quantum mechanics, I recognized a significant point: it provided perhaps the only way for traditional notions of free will to sneak into our description of the universe. Before quantum mechanics, the universe seemed to run like clockwork: once you'd set up the initial conditions, every moment of the future was uniquely determined.
But quantum mechanics brings fundamental uncertainty into the mix: only probabilities could be predicted, so one could hope that "free will" could somehow slip into the picture. In particular, as I've discussed before, quantum mechanics implies that every possible history "happens" and gets an equal vote to determine the probabilities of what we'll actually observe. One could imagine that what we see as free will is the branching of the wave function at each moment of decision: when we decide "Should I flip this switch or not?", both histories "happen" and both have some chance of being observed. It's not entirely comforting, but it better than clockwork.
Shifting gears, I've also at times contemplated how time travel would fit into physics if it were actually possible. The framework for thinking about questions of time is general relativity, since the ability to loop back to an earlier time would imply specific things about the structure of space-time. Relativity is a classical (i.e. pre-quantum) theory, and in it one should view all of space-time as a single four dimensional object. If "loops" in time are possible, those must be built in to the structure of space-time from the start. In practice, that means that time travel paradoxes from sci-fi simply aren't possible: "If something happened, it happened", as Sean Carroll puts it. In other words, history will automatically and by definition be self-consistent. You can't specify a single space-time object that includes a person murdering their own grandfather as a child, so that can't happen. But again, that's a classical theory, so we wouldn't necessarily expect to have free will in that context. What happens when we bring quantum mechanics back into the mix?
Sadly, that still doesn't save the day. The "quantum histories" that we're summing over in this case must be self-consistent space-times! Yes, we still have a chance of seeing any possible history actually occur, but if the switch you're considering flipping would kill Grandpa, you simply don't have that option no matter how easy the action itself might be. So the comforting notion that free will is hidden within quantum mechanics doesn't hold up in a world with time travel: quantum histories have to be defined globally, not locally.
In fact, even if time travel isn't possible, I suspect that conclusion holds: the idea that every moment and every decision spawns a branching tree of quantum histories doesn't quite capture all of physics. There are global effects that give important contributions to quantum calculations, quite apart from time travel (I study some of those in my research). So if there is free will out there, we probably shouldn't look to quantum histories to provide it. I'm not sure that we're left with much wiggle room, though: perhaps traditional notions of what free will really means just aren't right.
But quantum mechanics brings fundamental uncertainty into the mix: only probabilities could be predicted, so one could hope that "free will" could somehow slip into the picture. In particular, as I've discussed before, quantum mechanics implies that every possible history "happens" and gets an equal vote to determine the probabilities of what we'll actually observe. One could imagine that what we see as free will is the branching of the wave function at each moment of decision: when we decide "Should I flip this switch or not?", both histories "happen" and both have some chance of being observed. It's not entirely comforting, but it better than clockwork.
Shifting gears, I've also at times contemplated how time travel would fit into physics if it were actually possible. The framework for thinking about questions of time is general relativity, since the ability to loop back to an earlier time would imply specific things about the structure of space-time. Relativity is a classical (i.e. pre-quantum) theory, and in it one should view all of space-time as a single four dimensional object. If "loops" in time are possible, those must be built in to the structure of space-time from the start. In practice, that means that time travel paradoxes from sci-fi simply aren't possible: "If something happened, it happened", as Sean Carroll puts it. In other words, history will automatically and by definition be self-consistent. You can't specify a single space-time object that includes a person murdering their own grandfather as a child, so that can't happen. But again, that's a classical theory, so we wouldn't necessarily expect to have free will in that context. What happens when we bring quantum mechanics back into the mix?
Sadly, that still doesn't save the day. The "quantum histories" that we're summing over in this case must be self-consistent space-times! Yes, we still have a chance of seeing any possible history actually occur, but if the switch you're considering flipping would kill Grandpa, you simply don't have that option no matter how easy the action itself might be. So the comforting notion that free will is hidden within quantum mechanics doesn't hold up in a world with time travel: quantum histories have to be defined globally, not locally.
In fact, even if time travel isn't possible, I suspect that conclusion holds: the idea that every moment and every decision spawns a branching tree of quantum histories doesn't quite capture all of physics. There are global effects that give important contributions to quantum calculations, quite apart from time travel (I study some of those in my research). So if there is free will out there, we probably shouldn't look to quantum histories to provide it. I'm not sure that we're left with much wiggle room, though: perhaps traditional notions of what free will really means just aren't right.
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I wasn't seeing that before because I was thinking of time travel as simply the ability to violate the "only move forward" rule, rather than adding a surface where "forward" goes a different direction. But it's not clear what that would entail. One possibility is just "vanish/reappear elsewhere", but you already discussed the problems inherent in that. I suppose in my original tree version, where I postulated you go back and are then forced onto a different branch, the "back" part was traveling up your own branch presumably in some sort of non-corporeal form that didn't interact with the normal world while moving (which is more or less what you often see in various time travel movies, nevermind the causality nonsense they usually get into later). But... what exactly would that *mean* in terms of physics? I have no idea, I suppose it's probably also nonsense.
The only other way out of this I'm seeing at this point is some sort of Hyperspace that lets you just plain leave the surface (or skim "over" it or whatever. You're still traveling a path to a different branch but in some way it doesn't "count" because you're not on a surface that is space-time as we currently know it. I'm not sure if there's any way for *that* to make sense either. It kind of reeks of "because I postulated so".
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Part of the problem (and one which I didn't explain very clearly earlier) is that (as I understand it) the "fractal tree" picture is more of an organizational scheme for thinking about the many branches rather than an actual history of how they develop. My tentative sense of this is that all the branches throughout all the history of the universe (past and future) have always existed simultaneously (to the extent that "always" makes sense for a structure outside of time), and moreover that a "fractal tree" organizational scheme will inevitably leave out any branches that differ topologically from the initial state that it's based on.
Moving on, yeah, simply stopping and reversing one's motion through space-time would seem to be the natural way to arrange time travel. Unfortunately, it's not clear what exactly that would mean. Relativity simply doesn't allow it (or rather, I think that moving backward within that "45 degree cone" is technically allowed, but there's no way to slow down or turn around so it's moot). There is a curious idea from quantum field theory that might almost be relevant (and that could make for an interesting movie): particles can be considered to move backward in time, but when they do, they're their own antiparticles.
So if you could arrange for a perfect anti-you to annihilate perfectly with you (and no cheating: left-shin positrons can only annihilate the corresponding left-shin electron, not one in your spleen), you could consider the anti-you to have been you but moving back in time. I don't pretend to know how consciousness would work through all this, but it's conceivable to me that your internal entropy would still progress the way you wanted it to. Of course, that anti-you would have had to have been created in a tremendously well-orchestrated incidence of pair-creation at whatever point in the past you wanted to travel to, and if you had the ability to create a future you from scratch back then (along with bonus future anti-you!), time travel is probably the least of your worries. :)
As for hyperspace, the trouble there is that our physical bodies are intrinsically linked to the structure of space-time that they're embedded in. Every time I've tried to picture how some sort of hyperspace would work, I'm inevitably led to the conclusion that it would have to at least temporarily attach onto our universe like that ribbon did (and that it would have to at least locally have the structure of space-time that our bodies require). But at that point, I'm back to the same ribbon-like topology change arguments: even if hyperspace in general isn't "space-time as we currently know it", your existence there more or less requires some portion of it to act enough like what we know to count. I think.
Stupid physics, always getting in the way of my beloved sci-fi tropes. :)