Quantum histories, time travel, and free will

[steuard]

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.

Comments

[kirin]

Hmm. My layman's notion of time travel in a quantum, multiple-histories universe was always that you'd end up jumping to different history paths. The very act of traveling would leave you with no way to "go home again". But you seem to be saying even that's not really possible.

I'm not entirely clear on why you can't have a *globally* consistent set of histories, wherein one particular line involves a guy getting killed by his grandson who somehow jumped over from a different potential history. In the history the killer now inhabits, he didn't "come from" grandpa... he came from a time machine. Which basic law am I violating (other than the timey-wimey physics needed to have the time machine exist in the first place)?

[steuard.livejournal.com]

The "parallel universe" possibility is another possibility to consider, certainly. Sean Carroll touches briefly on the idea toward the end of the blog entry I linked to. Michael Crichton's Timeline is based on something along these lines, too (though his attempts at scientific detail had so many flaws and internal contradictions that I couldn't really enjoy it).

Sean seems to think that the concept of somehow jumping to the past of a different branch of the quantum wave function is possible, or at least that its impossibility is of the same order as ordinary time travel would be. I think he's nuts. :) I'd be okay with the idea that you're actually jumping to a "parallel world" in some distant but near-identical region of our own universe (whose history is offset in time by just the right amount), but that would still be part of the same space-time manifold in the same branch of the wave function. And it would rely on somehow finding another region of our universe that looked nigh-identical to our own. (An earlier discussion on my blog has convinced me isn't mathematically impossible, but it still seems hopeless to find one).

But as for quantum multiple universes providing the opportunity, It it my strong impression that quantum decoherence prevents any two branches of the wave function that differ at a macroscopic level from interfering in the future. I have trouble even imagining what sort of structure could make it possible to cross to another branch (much less at a different time), and it's difficult to see how such an idea could be consistent with laws of physics that are unitary (that is, information preserving).

Not that I haven't been tempted by the idea of exchanging to a different branch of the wave function from time to time. :) But hey.

[kirin]

Hmmm... I have a feeling I'm still not really internalizing something - probably to do with the "summation" of histories - but it seems like if you focus only on the branching aspect, thinking of the multitude of histories as an uncountably huge fractal tree, and you're given the opportunity to travel backwards, you could end up on other branches without ever jumping "sideways" between parallel worlds. You skim back down your own branch (combining with countless others on the way), and as soon as you pop back into reality you end up - *inevitably*, I'd think, due to your own presence - going back up a different branch in real time. And one of the events in this branch is "a time traveler appeared". Anything that happened on the branch you came from is now irrelevant, except in that many of the same things are very likely to happen again.

I'm pretty sure that interpretation is missing something, but I'm not exactly sure what.

[steuard.livejournal.com]

Setting aside whatever Sean thinks he's talking about with cross-branch travel, let me refer back to my original description. My point is that based on my current (and somewhat recent) understanding, the "branching fractal tree" perspective on quantum histories is subtly incorrect, and the flaw becomes particularly apparent in the context of time travel.

Essentially, I think that the branching tree concept is based on the idea of locality: that whatever is going to happen next is based on what happens here and now. Locality is absolutely true in some sense, but there are ways in which it's not the whole story. Specifically, you'll never recognize different topological sectors if you only look at local branches, any more than you could deform a clay ball into a doughnut without ever linking two points that weren't touching to begin with.

My claim is that in relativity, a space-time that allows a time traveler is topologically different than one that doesn't (in some sense because time travel would create a closed loop of causality, and closed loops are like doughnuts). So one isn't just a small deformation of the other resulting from a different outcome of a single event: your sum over histories has to include not just local branching on a single tree but also a sum over entirely distinct trees. And each instance of time travel requires its own distinct tree.

That's still viewing time travel as something that happens within a single space-time, though. My issue with the notion of traveling from one branch to another (whether you view that as moving back down a branch of a tree or swapping two branches or whatever else) is that I can't even come up with a way for the idea to make sense on a technical level. (Contrast that with relativity, where I can easily describe a space-time that includes time travel even if understanding it is a challenge.) I think my sense of "history" is tied closely to the space-time manifold picture, while my sense of time in quantum mechanics is simply "unitary evolution" of information. Time travel seems to make more sense in the former picture than the latter.

[steuard.livejournal.com]

Actually, let me put that another way. The only scenarios I've been able to imagine for "moving to another branch of the wave function" seem indistinguishable from a combination of 1) a precise copy of all the particles and fields that compose your body coalescing randomly in that "other universe", and 2) all the particles and fields composing your body in "this universe" spontaneously vanishing.

That's certainly possible (and thus there are presumably branches of the wave function where it happens). But I can't see any mechanism by which you could create a causal connection between those two events (as I said before, decoherence would seem to make doing so impossible, at least on an effective level). And without a causal connection, they might as well be independent events. At that point, either one on its own would be vastly more likely than both together: no time travel necessary.

[patrissimo.livejournal.com]

Inspirational!

http://patrissimo.livejournal.com/1355535.html

:)

[steuard.livejournal.com]

My goodness I've said a lot over in that thread. :)

I'm happy to provide inspiration, even if it does mean that some of your less careful readers have decided that I'm an exemplar of failed reasoning. (I still think I was coming out right back when we argued about the rationality of voting, by the way, but I doubt most of your readers followed the thread that long. To be fair, though, real life responsibilities seemed to have forced you to leave the discussion before we'd entirely established that.)

[coldtortuga.livejournal.com]

Why would time-travel cause a "donut" topology, eh? I don't see why that is implied under either a relativistic understanding, or a sum-over-relativistically-consistent-possible-outcomes understanding.

[steuard.livejournal.com]

Oh, this is a purely (general) relativistic statement. Let's see if I can boil it down a little by starting with a simplified case.

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?

[kirin]

Ah, it helps me anyway. And it applies even in the fractal tree multiverse, I think... such a ribbons going back down the branches would still create a handle, even if it were attached such that you ended up going up a different branch rather than your original one once you got back onto the "regular" surface.

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".

[steuard.livejournal.com]

Actually, in the fractal tree multiverse (the one from a quantum "many worlds" scenario) I'm considerably less clear on how this sort of story would work. My initial take on it would be that you aren't really "traveling" down any given branch: the branches simple are, and (in some of them) a version of you exists and propagates through time within each one.

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"

[coldtortuga.livejournal.com]

I'm pretty sure this intuition of "handles" and "donuts" is incorrect, topologically speaking. For example, Mobius strips and Klein bottles have no "handles", and it is perfectly do-able to define a Lorentzian manifold (a spacetime) on these geometries.

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"

[steuard.livejournal.com]

I'm pretty sure this intuition of "handles" and "donuts" is incorrect, topologically speaking.

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"

[coldtortuga.livejournal.com]

... any Lorentzian manifold that allows closed causal curves must have a nontrivial fundamental group.
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"

[steuard.livejournal.com]

You're right: as far as I know, Case #1 is consistent with all the evidence we've got. That means that's it's absolutely possible, and we need to keep our eyes open for traces of it happening. But we shouldn't take that too far: time travel being possible would be extraordinary, and extraordinary claims require extraordinary evidence. Given that we have no evidence actively favoring a universe that allows time travel, substantial skepticism is justified.

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

[coldtortuga.livejournal.com]

Hmmm. I appear to be turning into one of those physics "crank callers". "Globally hyperbolic" does seem like the most reasonable hypothesis, yes :-)

Please I'd like to crank-call in ask 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

[steuard.livejournal.com]

I'm reasonably sure that there are manifolds with nontrivial topology that are still globally hyperbolic. It's just when those topologies involve closed timelike curves that things get messy. The sum over histories should include representatives of each allowed topological sector, regardless. The degree to which any two branches of the wave function will have an effect on each other depends on how similar they are: two branches that differ only in the configuration of the electric field (a "finite" difference, but possibly including topological changes in the field) will tend to have substantial quantum interference. Two branches that have a large number of differences (topological or not) will tend not to have any substantial interference at all. (Ok, technically those statements are mostly true only if you read them as "the neighborhoods of two branches", in the sense that I used the term in my earlier sum over histories post.)

Re: I call "BS" on any argument against free will based on "determinism from initial

[steuard.livejournal.com]

As for changing the topology of spacetime, that's an interesting question. I think that in general relativity (whether in a quantum framework or not), dynamical topology change is impossible just as you say. But string theory is weird, and my understanding is that some such processes appear to be possible. I'm pretty sure they'd only be changes that were compatible with unitarity (and probably global hyperbolicity), though.

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.