The Foundational Questions Institute is sponsoring an essay competition on “The Nature of Time.” Needless to say, I’m in. It’s as if they said: “Here, you keep talking about this stuff you are always talking about anyway, except that we will hold out the possibility of substantial cash prizes for doing so.” Hard to resist.
The deadline for submitting an entry is December 1, so there’s still plenty of time (if you will), for anyone out there who is interested and looking for something to do over Thanksgiving. They are asking for essays under 5000 words, on any of various aspects of the nature of time, pitched “between the level of Scientific American and a review article in Science or Nature.” That last part turns out to be the difficult one — you’re allowed to invoke some technical concepts, and in fact the essay might seem a little thin if you kept it strictly popular, but hopefully it should be accessible to a large range of non-experts. Most entries seem to include a few judicious equations while doing their best to tell a story in words.
All of the entries are put online here, and each comes with its own discussion forum where readers can leave comments. A departure from the usual protocols of scientific communication, but that’s a good thing. (Inevitably there is a great deal of chaff along with the wheat among the submitted essays, but that’s the price you pay.) What is more, in addition to a judging by a jury of experts, there is also a community vote, which comes with its own prizes. So feel free to drop by and vote for mine if you like — or vote for someone else’s if you think it’s better. There’s some good stuff there.
My essay is called “What if Time Really Exists?” A lot of people who think about time tend to emerge from their contemplations and declare that time is just an illusion, or (in modern guise) some sort of semi-classical approximation. And that might very well be true. But it also might not be true; from our experiences with duality in string theory, we have explicit examples of models of quantum gravity which are equivalent to conventional quantum-mechanical systems obeying the time-dependent Schrödinger equation with the time parameter right there where Schrödinger put it.
And from that humble beginning — maybe ordinary quantum mechanics is right, and there exists a formulation of the theory of everything that takes the form of a time-independent Hamiltonian acting on a time-dependent quantum state defined in some Hilbert space — you can actually reach some sweeping conclusions. The fulcrum, of course, is the observed arrow of time in our local universe. When thinking about the low-entropy conditions near the Big Bang, we tend to get caught up in the fact that the Bang is a singularity, forming a boundary to spacetime in classical general relativity. But classical general relativity is not right, and it’s perfectly plausible (although far from inevitable) that there was something before the Bang. If the universe really did come into existence out of nothing 14 billion years ago, we can at least imagine that there was something special about that event, and there is some deep reason for the entropy to have been so low. But if the ordinary rules of quantum mechanics are obeyed, there is no such thing as the “beginning of time”; the Big Bang would just be a transitional stage, for which our current theories don’t provide an adequate spacetime interpretation. In that case, the observed arrow of time in our local universe has to arise dynamically according to the laws of physics governing the evolution of a wave function for all eternity.
Interestingly, that has important implications. If the quantum state evolves in a finite-dimensional Hilbert space, it evolves ergodically through a torus of phases, and will exhibit all of the usual problems of Boltzmann brains and the like (as Dyson, Kleban, and Susskind have emphasized). So, at the very least, the Hilbert space (under these assumptions) must be infinite-dimensional. In fact you can go a bit farther than that, and argue that the spectrum of energy eigenvalues must be arbitrarily closely spaced — there must be at least one accumulation point.
Sexy, I know. The remarkable thing is that you can say anything at all about the Hilbert space of the universe just by making a few simple assumptions and observing that eggs always turn into omelets, never the other way around. Turning it into a respectable cosmological model with an explicit spacetime interpretation is, admittedly, more work, and all we have at the moment are some very speculative ideas. But in the course of the essay I got to name-check Parmenides, Heraclitus, Lucretius, Augustine, and Nietzsche, so overall it was well worth the effort.
Albert Einstein, in a letter to the widow of his best friend, Michael Besso:
“Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion.”
My coauthors Professor Philp V. Fellman, Prof. Christine M. Carmichael, Andrew C. Post, and others have a series of refereed papers that compare the theories of time of Dr. Sean Carroll (Senior Research Associate in Physics at Caltech), Hawking, Penrose, and the un-institutionally supported New Zealand iconoclast Peter Lynds (born May 17, 1975) is a New Zealander who first drew sudden attention in 2003 with the publication of a physics paper about time, mechanics and Zeno’s paradoxes.
Lynds attended university for only 6 months. His career as a physicist began in 2001 with his submission of an article entitled “Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity” to the journal Foundations of Physics Letters.
The papers that I coauthored can easily be found via GoogleScholar. The next will be at Complexity’09, February 2009 in Shanghai, China.
I am a big fan of Sean Carroll. But I have to keep a very open mind to possibilities such as that the cosmos as a whole in trapped in a closed time-like curve, with the “origin” and “end” of time being the same point. It is hard to analytically continue such curves, which is why I suspect ambiguities in quantum cosmology.
Lawrence, thanks for responce. Still, what is wrong with my biological analogy?
I mean: “Obviously, a sudden appearance of a fully-fledged math physisict on an otherwise abiotic Earth is numerically more possbile than the whole big biosphere with gazillions of different animals. But if you take evolution into account, biosphere looks almost inevitable while single m.ph. keep being an impossible example for discussion”
Rephrasing B.Brain problem – “It is vastly more probable to be a single human on Earth than see other people around”.
This all tends to point to the matter of the cosmological constant and the value of coupling gauge terms. The cosmological constant is probably not constant. It depends upon a Higgs-like field which inflated the spacetime rapidly into a flat (or near flat) geometry. The cosmological constant settled into the value we infer today which is involved with the latent accelerated expansion of the universe. We might imagine a universe with a large comsological constant which inflates early on much more rapidly and enters into a latent inflationary (accelerated expansion we observe) that is far more rapid. In such a spacetime mass-energy would rapidly expand away and little local structure (stars, galaxies, planets etc) would occur. Conversely suppose that that cosmological constant is very small. Such a universe might expand more slowly and have matter far more clumped together with far more black holes. In that case too much mass-energy would be tied up in black holes and local complex structure not as possible.
Sean’s thesis here, which BTW assumes time exist, has quantum systems entering states which accumulate or have some asymptotic bound. These ever closely packed states have a lot of detailed selection rules for transitions between them. In order for all of these states to be occupied it probably requires that the universe becomes colder and for the cosmological arrow of time to march forwards as entropy increases. Robert M. Wald in The Arrow of Time and the Initial Conditions of the Universe (Enrico Fermi Institute and Department of Physics University of Chicago, arXiv:gr-qc/050709 vl 21 Jul 2005) writes:
“There is no question that our present universe displays a thermodynamic arrow of time: We all have observed phenomena in our everyday lives where entropy is seen to increase significantly, but no one has ever reliably reported an observation of entropy decrease in a macroscopic system.”
So a universe which expands too fast or one which buries quantum bits into black holes by expanding too slow might not satisfy Sean’s thesis as well. So the running parameters in a renormalization group appear adjust so that the time scales for the universe are appropriate for the “Goldilocks” condition we observe around us. We might be seeing Liebniz thesis about this being the best of all possible world at work! Voltaire’s Pangloss aside 🙂 In fact, the renormalization group equations (Wilson, Polchinski etc) are remarkably similar to the Navier-Stokes equation! So there is some underlying sense of a flow. The forward march of time, whether that be an emergent macroscopic thing or something absolutely fundamental, appears wrapped up in the structure of the universe, the “fine tuning” of gauge coupling terms and the cosmological constant.
So this Liebnizian or Goldilock universe we observe appears to permit local inhomogeneities that give rise to structure on a wide range of scales: galaxies, stars, planets, complex chemistry, life and so forth. Some of these local clumps behave approximately as open thermodynamic systems which can give rise to very complex systems. It appears this is how the complex structure arises.
The Boltzmann brain thesis is still outstanding in some sence. In a sufficiently large sample space of states Einstein could emerge spontaneously. Yet the universe we observe appears to not let this possible. An ever expanding universe is one which prevents incredibly long Poincare recurrences and for any equilibrium system to enter into all possible states.
Lawrence B. Crowell
I think i understand now, thanks.
So, St.Augustine was right when he called the idea that history goes in circles totally preposterous? He used different arguments though 🙂
Sean seems to think that infinite time is possible. Let’s make ourselves immortal and check.
From now on we are immortal (regardless what happens to the universe) and we keep track of time. We will realize that after being around for 10^100000 years we can still be around a little longer. We see that there will never come time when we can say that we have lived for an infinite amount of time. Even though we live for ever we will never reach infinite age.
So time cannot be infinite. By reflection of the argument a physical reality cannot be eternal. There is not enough time that can pass form minus infinity to reach the present time.
One cannot escape creation from nothing. 🙂
I think it likely that time is infinite into the future, at least as measured by some standard clock which could measure that long. Time going back to the past appears to be bounded by the big bang, though there is room for arguments there whether time pushes further back to some pretunnelling state or other universe and so forth. The “time at infinity” is the attractor point in the (mini)superspace, which might be a Minkowski spacetime that is completely void or empty.
We humans of course will not be there to see much of it. In fact there is room to question whether we will survive this century. The demise of the sun strikes me as a drop dead end time limit. Paleontology indicates that most large mammalian species are on the Darwinian game table for only a few million years. That is far short of any cosmological time scale, and our historical time frame is far shorter still.
Lawrence B. Crowell
LC, you must realize that there will never come a time when the clock says: “An infinite amount of time has passed!” That time will simply never come even for a clock that runs forever. Jeez. Time is finite no matter how long.
In part I indicated this with the problem of time intervals. If there is an accumulation point of energy eigenvalues then the energy spacings between these levels becomes very small Delta E_{i,i+1} < Delta E_{i-1,i} for all i, so in the limit i goes to infinity these spacings converge in a Cauchy type of sequence. The corresponding time intervals associated with these transitions which this quantum system computes become larger and larger. This system in order to be a physical clock must exhibit decoherence similar to a measurement. So as the "pure time" goes to infinity what quantum clocks there are which compute this time do so with ever larger time intervals. So there appears to be an issue of detailed balance or ratios here. So depending on how these energy eigenvalues values converge it could be that what quantum clocks the universe provides for itself may in the end compute a finite number (albeit enormously large) of time intervals. I'd need to think more about this to be certain of this conjecture.
As for this so called pure time, this might be something considered by the Compton wavelength of an electron L = hbar/mc. As the universe expands in this accelerated manner there might in the distant future be some sort of "quantum solipcism," where every quanta which exists does so in isolation within a cosmological event horizon, which in the very distant future will retreat off to infinity. The electron appears absolutely stable, and for argument I will assume it is, even through vast times of 10^{10^{10^{…}}} years. So the quantum oscillations of an electron in its rest frame will beat with a T = hbar/mc^2 = 6.4×10^{-22}sec. So this might be seen as some parameter for the forward direction of time. If the universe expands endlessly and the electron is stable then there will be an infinite number of Compton oscillations.
Of course this might not count as a quantum clock, for there is no decoherence involved with state transitions. Maybe these lonely quanta become entangled across vast distances across cosmic event horizons, or there are some sort of physics involving transitions, but I will defer opinion on that for now.
Lawrence B. Crowell
LC, you don’t see. There will never be an infinite number of Compton oscillations. The number will keep increasing, from a finite n to n+1 to n+2 etc forever, but the number will NEVER become infinite.
A transition from a finite value to an infinite “value” cannot take place by finite increments. Such a transition can only happen for an infinite increment.
Sean,
I agree with you that this “departure from the usual protocols of scientific communication” is a “good thing.” When I asked Peter Woit if he would be inclined to participate, he didn’t think it would be a good idea, given that he objects to the Templeton funding.
Did that give you any pause and do you think it might be a more widely spread deterrant?
CarlN: your argument would then mean the real number line can’t be infinite because nobody could ever count them all. You might in a sense pop out of the picture and think as Einstein told us to and think of the whole spacetime. The temporal part then “goes to infinity” in much the same way that the real number line goes to infinity.
Lawrence B. Crowell
Hi Sean
Very interesting essay. I have a question, namely that if we are talking about the time evolution of a wave function, then the equations which describe the time evolution of such a wave function in quantum mechanics are invariant under Galilean transformation,
not Lorentz. Galilean invariance assumes a universal time, does this matter in your argument ?
LC, the correct wording is that the real number line is unbounded. For any number you write down I can write a larger one. Then you write one larger still etc. We could go on for all “eternity” without ever reaching the “infinite number”.
I guess you are starting to see what is wrong with Einsteins spacetime geometry.
FWPT– You can certainly have a Lorentz-invariant version of quantum time evolution (the Schrodinger equation). The only trick is that the fundamental variables are not the positions of particles, but the amplitudes of quantum fields. (You can look up “functional Schrodinger equation” in some textbooks.) The notion of a Hamiltonian demands that you choose a frame in which to define it, but the result is independent of the frame you choose, if the underlying theory is Lorentz invariant.
I discussed this more in the original version of the essay, but space constraints did not permit me to keep it. In a Lorentz-invariant theory, the time parameter is certainly not unique, but any choice is fine.
CarlN, indeed the real number line is unbounded, and in a deSitter type cosmology the same is the case. I am not sure I see anything particularly wrong with this. We might ponder whether there are really indeed physical systems which demark time intervals endlessly. Sean says they do with the accumulation of eigenvalues. I think the conjecture is fairly reasonable. I suppose I see nothing wrong with the prospect of a quantum system that oscillates endlessly and demarks and infinite number of time intervals. That the system will never count some “final” time does not bother me particularly.
Lawrence B. Crowell
LC, I’ll try one last time. What unbounded means: You can never reach the end of the line, but you can’t reach infinity either with finite increments (you could try by divison by zero, though).
You could fix your favorite spacetime by a supplementary condition that says only a finite spacetime interval is physical. You will still have other problems though..
The fact that “time” never can be infinite into the future from the present has very important implications. It means that an infinite amount of time can’t already have passed, even for a physical reality “outside” the BB. In fact all kinds of eternities are eliminated.
You can work out the implications..
Keep in mind that apart from what exits, there is nothing.
If you think that entropy is an important concept:
It requires less effort to specify low entropy initial conditions than high entropy conditions.
In fact zero entropy initial conditions are easiest to arrange.
CarlN, I don’t see how you get the idea that time can’t be “infinite into the future.” That was never supposed to mean, there’d be “a time” at which the clock actually read “infinity” (the same as a “highest integer”, or treading “infinity” like another uber-integer somehow beyond each and every other integer ….) What’s to keep things from simply continuing to behave various ways in an ever-expanding universe?
BTW, it can’t be both the case that time was indefinite into the past and yet constants could change with each iteration of a collapsing universe: the chance of becoming an open universe would always have already happened. (Contradiction: I can’t be even in such an open universe, contemplating “the infinity” of cycles before now, since however tiny the chance, the open cycle must have already happened earlier.)
One thing throwing a monkey wrench into consideration of infinite time, and space, is how we can mathematically compress infinite extents into a finite length. We can for example (“normalize” as is convenient and use x or t), use t’ = ArcTan t, or x’ = x/sqrt(x^2 + 1) and remap an entire infinite extent onto a finite line or space. That possibility challenges a realist, common sense notion of what can coexist “in the same space.” Why? Because I can imagine that my entire “infinite universe” is mapped onto the space defined with ArcTan of the given value of r distance from me (such as it is.) It then has a pseudoboundary. (Not a true boundary, since the limit is not an actual defined value: it’s like the open circle at unity for x < 1.) Mappings are perfectly valid transformations, right, than cannot affect or "be observed" by the inhabitants. And yet, if I do so I can now imagine another entire universe "beyond" the infinity, mapped onto another finite segment in the new x' or t' container space. Even weirder, we can then compress an entire Aleph null of such compressed infinities via a second iteration onto x'' or t'', etc ….
One can even imagine apparent absurdities like moving a point along x' such that it will actually blow past "infinity" of x in a finite time, or outwait the entire infinite time extent in t by slowing down appropriately, etc. (Compare "supertask.") Maybe that mathematical trick is not physically realizable. OTOH compare to the old saw about outsiders never being able to see the an object actually fall into a black hole. Yet the faller finds himself reaching not only the event horizon, but the singularity later in a finite time.
The latter disjunction was the framing of a poignant SF story, "Kyrie" by Poul Anderson. The heroic energy-being type alien falls into the black hole. His telepathic human companion can "hear" his laments forever since for her, the fall never ends. See e.g. http://www.nvcc.edu/home/ataormina/scifi/works/stories/kyrie.htm. I admit to getting a little sniffly, reading the story as a boy (always a sucker for sentimental stuff like "Lassie Come Home.")
Neil B, “What’s to keep things from simply continuing to behave various ways in an ever-expanding universe?”
Exactly. What remains invariant is that the future will always be at a finite (but increasing) “distance” from a given point in the past.
Mathematical transformations won’t help you. You can’t “remap” the infinite future before it has happened. That time will never come.
Neil B’s mathematical example is spot on. He did the work and keyboard tapping I was trying to avoid. The Penrose conformal diagram for the universe can put the t = infinity boundary, maybe a Minkowski spacetime, at a finite distance.
The existence of an accumulation point in Hilbert space permits the continued existence of systems which mark time intervals. Whether they reach the endpoint as a summation of their iterations is besides the point.
Lawrence B. Crowell
That is the other problem with spacetime geometries. People are led to believe that the future actually exists in some 4D (or more) geometry.
As a calculational tool it is OK, but don’t get confused.
This is why we never will see a visitor from the future. The future does not exist 🙂
LC, the point is that the future will ALWAYS remain at a finite “distance” from for example today. This is an invariant fact of reality. Spacetime geometries go in and out of fashion on the other hand 🙂
Come on, is anybody willing to step up and say that there will come a time that is infinite into the future from Dec 1, 2008? I guess at least Sean should do so, since he suggests time going from – to + infinity 🙂
I am reminded of an account of George Bernard Shaw who apparently said he did not think the planets exist at the distances demonstrated by astronomers because nobody could ever walk to them.
Lawrence B. Crowell
Invariant fact of reality is that we will have some great new theories like CN’s popping out now and then.
I don’t have one yet, but I have this question bugging me for some time now: why physics has problems with infinities while mathematics doesn’t?
Roman: of course math has problems with infinities, everything from dividing by zero to the status of infinite sets. BTW, I have yet to find a good rundown of “complex number infinity” descriptions, anyone know? I expect it would be “bifurcated” into two incommensurable limit approaches. One would be along a parallel line (such as 5 + infnty i) as one approached infinity along either the real or complex axes. The other would use r, theta guidelines. It would be in the form r = infinity, theta = whatever. Anyone see this?
CarlN, your intuitive notions and “common sense” reasoning (reminding me of how “analytical philosophers” think) may not really constrain the real world, and they certainly don’t constrain the conceptual world.
My only problem with infinities is getting there 🙂
So if you become eternal you believe that there comes a time when you can say: Gee, look at the calendar. It has passed infinitely many years since 2008.
Either you believe that or you don’t. Or you are not able to make up your mind.