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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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Imagine a flat two dimensional universe where "north" is forward in time and "east/west" corresponds to travel through space. The rules of the game state that you can't stop walking forward and you can never turn more than 45 degrees away from north. It's clear that you're never going to be able to go back in time (i.e., farther south) as things stand: you simply can't turn around.
To be able to go back in time, what you need is for someone to attach a ribbon to your surface sort of like a roller coaster loop. If you aim right, you can walk forward onto the loop. At that point, your "north" is "along the ribbon": you'll happily walk up the slope, upside down, and back down to the surface. Depending on how far south from your starting point the tail end of the ribbon is attached, that's how far back in time you've gone.
But once you've got that ribbon attached, it's clear that what was originally a flat, simply connected surface now has a "handle", just like a doughnut does. And as long as you have just one timelike dimension, this will remain true in our 3+1 dimensional world or in the 10+1 dimensions of M-theory. The limitation that everyone must keep moving forward in time means that there's no way to turn around unless the shape of space somehow arranges to turn around for you. And I'm pretty sure that one could prove that any such shape of space will have non-trivial topology.
Does that help?
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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. :)
I call "BS" on any argument against free will based on "determinism from initial conditions"
Here is an alternative intuition. Suppose spacetime has a certain geometry, for example the simplest case of flat (Minkowski) 4-space. Suppose we specify a vector field on this spacetime with "timelike" orientation, for example all paths in this vector field "go forward in time" (every path's tangent in the time-like dimension is always positive). This vector field corresponds to the vector field of everything in the history of the universe. The intuitive argument that no such vector field can contain a closed path. But I don't think that's correct at all? Maybe if some constraints on the vector field like smoothness are imposed? ... But certainly not in general. See http://en.wikipedia.org/wiki/Causality_conditions for more details.
There is plenty of argument that time-travel is challenging semantically. But there doesn't seem to be anything about general relativity that physically prevents it. In other words, I don't see any reason why the geometry required by general relativity implies both unitary physical laws and hoped-for intuitions about "causality". I don't see any reason why unitary physical laws imply causality: the laws of physics are time-reversible.
Note that the strongest sort of causality proposed on that wikipedia page is "globally hyperbolic". In that case, one can set up initial conditions on the universe, and those initial conditions are sufficient to determine everything else. But weaker forms of causality do not imply this initial-conditions property, and these weaker forms are not inconsistent with general relativity or unitary physics.
Re: I call "BS" on any argument against free will based on "determinism from initial conditions"
I don't think I follow you here, quite possibly because we're both being casual with language. Let me make a more mathematical statement: I claim that any Lorentzian manifold that allows closed causal curves (i.e. any hope of time travel) must have a nontrivial fundamental group. (And therefore, the quantum path integral over histories must take include a discrete sum over representatives of each topological sector in addition to the usual integral over continuous deformations about each representative.) Both a Mobius strip and a Klein bottle contain non-contractible curves, so they are not topologically trivial.
I'm pretty sure that's all I need for my initial argument. Ah, but maybe you're talking about my very first comment, about the pre-quantum story. Your point is well taken in that case, and in fact come to think of it the individual branches of the wave function that I've been imagining later on are indeed self-consistent non-causal geometries of this sort. I think I was summarizing the classical worldview as of the period when it was believed, before GR or potentially non-globally-hyperbolic geometries were understood. But regardless of such possibilities, classical physics is still locally deterministic (that is, if you focus your attention on a small region that has no closed causal curves, you can solve the Cauchy problem there), with the consequent lack of free will (in the traditional sense).
Re: I call "BS" on any argument against free will based on "determinism from initial conditions"
Um; yep, I think that's correct (at the very least, correct modulo niggly details of little import) based on what I've been reading. Then if I'm following your argument properly, this means we should consider two cases when extending from GR to quantum mechanics. Case #1 the universe has a nontrivial fundamental group (requiring one sort of sum-of-histories, including enough terms to capture different "parts" of that topology). Case #2 the universe has a trivial fundamental group (requiring just a single such term (?), and definitely ruling out time-travel).
But I don't see that Case #1 is ruled out?
... classical physics is still locally deterministic...
Hah! Classical physicists do like to claim this sort of thing :-) But outside of tightly-controlled experiments --- which must be tightly controlled in order to "localize" a patch of spacetime --- this notion of "determined by initial conditions" does not bear upon free will/determinism arguments. (Unless the argument takes the whole universe as the local region; but that would presuppose Case #2, and I don't see we've gotten to ruling out Case #1.)
I once argued with a Mudd prof that "determinism from initial conditions" is easy to challenge. Suppose there is a closet-sized piece of spacetime somewhere (somewhen? somewherewhen?) that contains a basketball-sized lump of non-deterministic (or even just locally-not-obeying-the-usual-rules-of-physics) spacetime. The rest of the universe may well be deterministic from initial conditions. But if the closet door opens, then (even if the basketball now settles down and starts behaving itself properly) everything in its light-cone becomes "tainted". Unless one is willing to rule out such closets and basketballs from everywhere --- event horizons, Big Bangs, Planck-scale vacuum energy fluctuations, New Jersey --- then I don't see that the argument holds water. It is a vacuous over-generalization of the Platonic ideal of experimentally-confirmed results.
... I had to drop that class. Some people don't like their Platonic ideals coming out of the closet.
Re: I call "BS" on any argument against free will based on "determinism from initial conditions"
Your reasoning about the ease with which determinism could fail is exactly right: the entire future light cone of a non-causal region is non-causal. (There are some rather nice theorems about all this. Fun fact: Bob Geroch, who's mentioned in that Wikipedia article about causality conditions, is the guy who taught me quantum mechanics in grad school and convinced me to like the subject.) But I think most physicists expect that the universe actually is globally hyperbolic, and that there really aren't any holes in the spacetime manifold like what you've considered.
Re: I call "BS" on any argument against free will based on "determinism from initial
Please I'd like to
crank-call inask two more questions, just for my edification.Question the first: When doing sums-over-histories for quantum mechanics, is it necessary that all the possible spacetimes are exactly the same? (Meaning the topology of each manifold is the same, but of course not meaning the particular shape i.e. energy/matter distribution of each manifold is also identical.) I'm wondering what would happen if the vast majority of possible histories are evolving on a globally hyperbolic manifold, but a few are not --- what fraction would need to have an exotic spacetime manifold in order to have a "noticeable effect" on the superposition?
Question the second (or rather, confirmation of what I would intuitively guess): There is no way to change the topology of spacetime due to "intrinsic" dynamics. For example, light-cones are intuitively cone-shaped, but can become light-vuvuzelas given a particular energy/matter density --- but if spacetime has a well-behaved Cauchy surface, then light-donuts are out of the question.
The reason I ask those two questions is to consider the following. Suppose that the topology becomes fixed at some initial event (presumably the Big Bang). It does not seem unreasonable that the Big Bang could produce a variety of different topologies. Some might vanish i.e. collapse to some trivial point and hence fail to contribute to today's sums-over-histories. Most would appear to be globally hyperbolic :-) A small fraction might have a more exotic topology. What is the impact of sums-over-histories?
Re: I call "BS" on any argument against free will based on "determinism from initial
Re: I call "BS" on any argument against free will based on "determinism from initial
My take on all of this together is that it's complicated. :) But I would tend to suspect that branches of the wavefunction that include closed timelike curves would in general differ macroscopically from branches without them, and that quantum decoherence would come into play. That is, the sum total of those differences would mean that there would be negligible quantum interference between those branches and globally hyperbolic ones. But this stuff gets messy pretty fast; I wouldn't want to swear to that.