# Did the Universe Begin? IV: Quantum Eternity Theorem

Having pointed out that the BVG theorem presupposes the existence of a classical spacetime, Carroll goes on to cite some evidence that the universe did not have a beginning, based on quantum mechanics (QM):

If you need to invoke a theorem, because that’s what you like to do rather than building models, I would suggest the quantum eternity theorem. If you have a universe that obeys the conventional rules of quantum mechanics, has a non-zero energy, and the individual laws of physics are themselves not changing with time, that universe is necessarily eternal. The time parameter in Schrödinger’s equation, telling you how the universe evolves, goes from minus infinity to infinity. Now this might not be the definitive answer to the real world because you could always violate the assumptions of the theorem but because it takes quantum mechanics seriously it’s a much more likely starting point for analyzing the history of the universe. But again, I will keep reiterating that what matters are the models, not the abstract principles.

First of all, some background.  In QM, there's a gizmo called the wavefunction $\Psi$.  This is the thing that tells you what are the probabilities for any particular thing to be happening, at any given moment.  It involves specifying a complex number $z = a + bi$ for each possible configuration of the universe.  Complex numbers are two-dimensional, so they have both an absolute value (or magnitude) $|z| = \sqrt{a^2 + b^2}$ and a phase (or direction) in the two dimensional plane.  The square of the absolute value $|z|^2$ gives you the probability to be in that state, while the phase (or direction) of the complex number is an additional weird extra piece of information which is special to QM.  (There's some deep conceptual issues about what the wavefunction "really" means, but let's not get into that here.)

In ordinary QM, the wavefunction of the universe changes with time.  If you want to work out how it changes with time, you need to know the formula for the total energy of the universe, written out as a function of the positions and momenta of all the particles or fields.  Once you know what $H$ is, you can then use Schrödinger's equation:

This equation tells you that if your state $\Psi$ is in a state with a specific energy $H \Psi = E \Psi$ (this is called an energy eigenstate), then its phase just spins around and around, at a rate proportional to the energy $E$ divided by Planck's constant $\hbar$.  That's rather boring, since it would mean that none of the probabilities actually change at all.  On the other hand, if you have a state where the energy has quantum uncertainty (meaning that it is actually a superposition of states with definite energy) then more interesting things can happen due to interference patterns between the different energy eigenstates.

So, if you know what H is (that specifies the dynamics of your theory) and you know what the wavefunction $\Psi$ is at some specific time $t_1$, and if you assume that this theory is valid at all moments of time, then you can work out what $\Psi$ is at any other moment of time, past or future.  And it particular, you know what it would have been at a time $t_2$ which is arbitrarily earlier than $t_1$ is.  Hence—so Carroll's argument goes—the universe cannot have had a beginning.

That's all the Quantum Eternity Theorem (QET) says.  It's a little bombastic for Carroll to even refer to this as a "theorem", since it's just an elementary restatement of one of the most basic principles of QM.   As Carroll said in his post-debate reflections:

For convenience I quoted my own paper as a reference, although I’m surely not the first to figure it out; it’s a fairly trivial result once you think about it.

You could still imagine that God miraculously created the universe at a given moment of time $t = 0$, and that the laws of physics only apply after that moment of time.  Then physics as such would have nothing to say about the actual Beginning, but only what happens after that.  There's no logical contradiction in saying that, but it might make some people uncomfortable if—so far as we can tell from Science—the universe has to have lasted forever.  In some ways, this is the position Christians were in prior to Modern Science, when the study of the heavens seemed to indicate that the universe just kept going on and on, like a clock that never needs winding up.  Back then, Christians mostly believed there was a Beginning for philosophical reasons, or else because it said so in the Bible.  We now know that the Universe developed from a simpler form, and that it has only existed in its currently observable form for about 13.8 billion years.  The scientific case for a Beginning is certainly much more conclusive now than it was then, since back then there wasn't much of a scientific case at all!

But if Carroll's QET does apply, then no matter how many fireworks there were at the "Big Bang", it could only really have been the universe passing from one form to another.  So is he right?

Probably not.  Carroll himself states the important loophole in his reasoning, although he does it in a kind of a cryptic way so that only another physicist like me knows what it really means.  Let's have it again:

If you have a universe that obeys the conventional rules of quantum mechanics, has a non-zero energy, and the individual laws of physics are themselves not changing with time, that universe is necessarily eternal.

What Carroll neglected to say during the debate, is that there's very good reason to believe that the energy of the universe is zero (if it is defined at all).

It's actually rather tricky to make precise the concept of "energy" in General Relativity.  The reason is that energy is defined with respect to how things change with time, and time is a rather slippery concept in GR.  There isn't just one notion of time, but rather any choice of "t" coordinate you might choose is equally valid.  If there's no well-defined concept of time, then there's also no well-defined concept of energy, and the QET won't apply.

So when people do refer to energy in GR, they need to be some type of special situation that allows them to invoke the concept.  Here are the cases people talk about most often:

1. If we zoom in close to one point, we can adopt a particular local inertial reference frame and define the energy of an object using that local coordinate system.  But Special Relativity tells us there are several equally good notions of time , and even those are only good in the neighborhood of a single point, so this won't work for the QET.
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2. If you have a spacetime which is approximately unchanging with respect to some special time coordinate "t", you can define the energy of objects with respect to that time coordinate, as long as their gravitational field is small.  This is called the Killing energy, but this is also inapplicable in cosmology since the universe is not anywhere close to static (it is expanding).
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3. If you have a system of objects sitting by themselves inside an otherwise empty infinite space, then you can use the notion of time defined by a clock which is very far away from the system.  This is called the ADM energy, and it tells you the effective gravitational mass of the system as measured from far away.  But this is also inapplicable to cosmological settings, since so far as we know the universe is not a clump of matter in an empty space.
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4. Finally, if you have a closed universe (one with no boundary) then there is an unambiguous notion of energy associated with the gravitational Hamiltonian $H$.  However, it is exactly zero for all physically allowed states: $H = 0$!

The conventional view of researchers in quantum gravity—with, apparently, the exception of Carroll himself—is that the same thing is likely to be true in quantum gravity.  That is, instead of the usual Schrödinger's equation, the dynamics of the theory are encoded in the Wheeler-DeWitt equation:

Now since $H$ tells us how $\Psi$ changes with time, the Wheeler-DeWitt equation tells us that the quantum state does not change with time at all!  That's weird, since we all know that things do change with time.

Does that mean that Zeno was right and time is an illusion?  Well, we have to be very careful with interpretation here.  The real reason why this happens in gravitational theories is because the choice of spacetime coordinates is arbitrary—you are free to label your spacetime points with any $(x,y,z,t)$ coordinates you like: there is not one "best" way to do it.  (Although I've been focusing on General Relativity, physicists expect similar issues to pop up in almost any decent theory of gravity.  So long as it does not reintroduce a notion of absolute Newtonian time, there will necessarily be a "Hamiltonian constraint" saying that the only physically allowed states of a closed universe are those for which $H = 0$.)

So when we say that the wavefunction doesn't change with time, what this really means is that the choice of time coordinate is arbitrary.  "Time" needs to be measured relative to some physical clock.  There is no absolute "t" coordinate relative to which everything else moves,  So, I think I would say that in this case, the QET "applies", but in a totally trivial way, and when you unpack its real meaning, it doesn't tell us anything about whether or not there was any time before the Big Bang.  Thus the formalism of ordinary QM is not applicable.

To summarize, in a closed cosmology, the energy is zero, and in an open cosmology it might not even be defined.  Thus Carroll's appeal to the QET probably doesn't make sense.

As I said to Carroll in the comments to his post-debate reflections:

Regarding the QET, to my mind the most conservative belief about quantum gravity is that it is—like GR—governed by a Hamiltonian constraint rather than an ordinary Hamiltonian (as in standard QM). In this setup, it’s not obvious that the QET applies.

What’s more, since there is no “absolute time” in GR, there are lots of different, inequivalent ways to evolve space forwards in time. As Wheeler put it, time is many fingered. This concept of time evolution will be much more subtle to quantize, and it’s far from obvious (to me, at any rate) that it’s forbidden for time to begin or end. In any case, this is quantum gravity, so none of us really know what we’re talking about!

And he replied:

Aron– That’s certainly a respectable point of view. It’s basically choosing the option that the energy is zero, which is definitely a possibility. And if that does turn out to be the case, time can certainly “end,” but in a very funny sense, since “time” was only emergent to begin with.

But the other option, that the energy is not zero and the ordinary time-dependent Schrödinger equation applies, is at the very least equally reasonable (perhaps more so). Our best-understood example of quantum gravity is the AdS/CFT correspondence, where the theory is most carefully defined in terms of the Hamiltonian of the boundary theory — in which perfectly conventional Schrödinger evolution applies. My suspicion is that quantum gravity will work similarly in other cases as well. But as you say, it’s quantum gravity, so we’re allowed to speculate but not allowed to act like we know the answer.

AdS/CFT is a famous duality between an ordinary QM theory (the CFT) and a gravitational (string) theory with a negative cosmological constant.  In this case, there is a well-defined nonzero $H$, but that is because you have a bunch of matter sitting in an otherwise empty AdS space, so you can use the ADM definition of the Hamiltonian.  (This duality tells us very interesting things about general aspects of quantum gravity, but it probably doesn't apply directly to our own universe, which has a positive cosmological constant, among other considerations.)

GR predicts (A) that $H \ne 0$ for matter sitting in empty AdS space, and (B) that $H = 0$ for closed universes.  It doesn't make any sense to me to say that because string theory agrees with GR about (A), it probably disagrees with GR about (B).  To me, the most conservative thing to say is that both of these facts continue to be true.  Furthermore, case (B) is far more likely to describe the real universe than (A) is.

Although, as we both said to each other, no one really knows for sure how the correct theory of quantum gravity is going to be formulated.  Of course, there is nothing wrong with Carroll putting forward his personal opinion in the debate—I can hardly complain about that after Craig put forward my opinions!  But I think he could have been more clear that it was his personal opinion, and that, given more "conventional" beliefs about quantum gravity, the QET probably can't be applied in cosmological settings.

[9/22/14: a few minor wording changes—AW]

I am a postdoctoral researcher studying quantum gravity and black hole thermodynamics at the Institute for Advanced Study in Princeton. Before that, I read Great Books at St. John's College (Santa Fe), got my physics Ph.D. from U Maryland, and did my first postdoc at UC Santa Barbara.
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### 16 Responses to Did the Universe Begin? IV: Quantum Eternity Theorem

1. TY says:

Aron,
Well done again, Sir.

Just like to make a few observations and ask a few question.

In the Carroll-Craig debate, Carroll said in his opening speech . “I want to argue that naturalism is far and away the winner when it comes to cosmological explanation. And it comes down to three points. First, naturalism works—it accounts for the data we see. Second, the evidence is against theism. Third, theism is not well defined.” Having read your Blogs and your responses on to readers’ questions, I have to say that the totality is a devastating refutation of Naturalism. Isn’t it interesting that energy isn’t such an easy thing to define after all? Which brings me to Carroll’s special pleading of non-zero energy of the universe to make the argument of an eternal universe work.

Aron, Don’t we need a model that explains why the energy is non-zero instead of assuming the condition exists? Is this assumption necessary to make the mathematics "manageable".
It seems that until some new theory supplants the Big Bang and is supported by observations, I think the notion of a beginning of the universe and time remains unrefuted.

Zero energy (the positive energy from matter exactly balanced by negative energy from gravity) is taken to mean the universe consists of nothing. In physics “nothing” is “something” as your blog, “A Universe from Nothing? Posted on December 7, 2012, “A Universe from Nothing”, and David Albert’s review of the book in the New York Times, March 23, 2012.).

Why do so many smart physicists continue to write this fallacy?

2. Kevin V says:

Good stuff! Would love to hear more about some pre-Big Bang models that are aimed at getting around the BGV such as Emergent Models and Aguirre-Gratton, and whether we know anything that falsifies them.

Thanks

3. Dennis Jensen says:

Craig responded in part to Carroll’s argument from the QET for an eternal past by saying that the QET tells us that if there is a prior moment then we can describe that moment, it does not tell us whether that prior moment existed (http://www.reasonablefaith.org/does-quantum-mechanics-indicate-an-eternal-universe). Is this correct? (Forgive me if you have answered this in your essay above and I just missed it.)

4. Aron Wall says:

Dennis,
It's really an ambiguous question. Suppose that the universe began at a particular moment of time; call it $t = 0$. So only times with $t \ge 0$ exist.

Now the basic premise of Carroll's QET is the Schrodinger equation, which I wrote in the post above. If we say it's a "law of nature" that the Schrodinger equation has to be true for all values of $t$, then there must exist times $t < 0$ for it to apply to. On the other hand, if we phrase the "law of nature" so that it is only valid at moments when there is time, then there is no contradiction with it only applying at times $t > 0$.

I think I addressed this issue in the paragraph beginning "You could still imagine that God miraculously created the universe at a given moment of time t=0, and that the laws of physics only apply after that moment of time."

So I agree with St. Craig that there's a way to phrase things where a beginning of time is consistent with the Schrodinger equation, but I don't quite agree with his phrasing when he says that "in order to know whether there is such a[n earlier] moment we must look to empirical evidence." If the premises of the QET are true (though I think they probably aren't) then there is at least an answer to the question of what would have occured before $t = 0$ if one continued to apply the Schrodinger equation to those earlier times, and I see no way that purely empirical physics considerations could rule out those times actually existing.

5. GS says:

Aron,

I have just been listening to the Craig Carroll debate where Craig stated that you have formulated, " a new singularity theorem". Is it the case the you have done so and could you direct me to the journal where it is published. Was the publication peer reviewed?

Thank you

6. Aron Wall says:

Dear GS,

Yes, it's true. The article in question is published in Classical and Quantum Gravity, which is a peer-reviewed, high quality journal. (However, you might be surprised how often wrong stuff makes it past peer-review! It's not a magic bullet by any means.) It was also highlighted by one of the CQG editors.

You can find the article online at the journal webpage or on the arxiv.

My article extends the Penrose singularity theorem to situations involving quantum fields, which can have negative energy densities. The main assumption is that the Generalized Second Law of thermodynamics holds for all causal horizons, for which there is significant theoretical evidence. There's also some other technical assumptions discussed in the article. I also have a section speculating that these assumptions might be applicable to full quantum gravity, but this can't be proven since nobody understands quantum gravity.

I've also described aspects of the article in parts II and VI of this series, and also in a Scientific American blog post.

7. Jonathan Gelfusco says:

Aron,

Since we are discussing Quantum Physics, I would really love to hear about your views on the orthodox or copenhagen interpretation of Quantum Mechanics.

I'm not sure if you have heard, but there is a you-tube channel called "InspiringPhilosophy" and he is an idealist Christian who uses some quantum mechanical findings to infer the existence of a mind (God).

I know I may be asking for too much, but please allow me to beg you to watch some of the videos in this playlist of his and give me your own personal opinions on his work? I really would love the input of someone with your expertise on this!

One video you may especially like is the one called "the introspective argument" or "Quantum Physics debunks materialism".

I really hope you view these!

8. Ed Slattery says:

In Section III, you mentioned a contracting and then expanding (but, I'm assuming, non-cyclic) universe as an alternative of infinite age to the one hypothesized as expanding from a beginning. A long thread on the Physics Forum mentioned that such a "one shot" contraction to a bounce into expansion was characteristic of those cosmological alternatives that rely on quantum mechanics rather than relativity. I find any beginning to everything inconceivable, but, to me, it would be simpler to say that the expansion arose from a state lacking any averaged tendency toward either. Why the takeoff roll through a contracting phase?

9. Scott Church says:

I'm out of my league here, so any needed correction/elaboration to the following would be appreciated...

If I'm reading this right, Carroll's argument for QET is heavily dependent on AdS/CFT, which as Aron pointed out, probably doesn't apply to our universe anyway. If so, it's interesting that AdS/CFT has been running afoul of real-world data as well. Over at Backreaction Sabine Hossenfelder discusses recent attempts to model LHC heavy ion collision data with it (specifically the impact of quark gluon plasma viscosity on the hadronized jets emitted by such collisions). As luck would have it, the AdS/CFT predictions for these runs do not match the data either quantitatively or qualitatively. Add to that the fact that LHC Run 1 has taken us well into the regime where we should've found ample evidence of Minimal SuSY superpartners by now, and so far it would appear the real-world hasn't been kind to AdS/CFT or its M-theory underpinnings. I would imagine this doesn't explicitly disprove Carroll's version of QET, but in the very least it ought to put the burden of proof squarely on his shoulders.

Of course, explorations of M-theory and AdS/CFT are still in their infancy, and I don't know how relevant any of this is to cosmological questions. And although M-theory does require some form of SuSY it need not be the Minimal version. But it seems to me that these things don't bode well for AdS/CFT, M-theory, QET or M-theory-based multiverse scenarios as viable answers to fine tuning. Unfortunately, one of the problems with M-theory is that it's pliable enough to accommodate virtually any dataset thrown at it, rendering it virtually untestable for the foreseeable future. (No doubt, this is why Carroll and others are crusading for the elimination of real-world testability as a requirement for science). Is there anything I'm missing here...?

In any event, LHC Run 2 is now under way and is slated to push into the 13-14 TeV collision range. If it fails to turn up a gluino or other SuSY superpartner it'll be interesting to see how Carroll and other M-theory true believers respond. 50 bucks says that within a month of run completion there'll be multiple papers at Arxiv predicting that Run 3 will turn up the desired evidence, and they'll be even more convinced that QET and the M-theory multiverse are on a more solid footing than anything real-world-facts folks could ever come up with... especially if they believe in God. ;-)

10. Aron Wall says:

Ed,
You could try to construct a universe which remains in a static configuration for a long time, and then starts to expand, but that raises some troublesome questions. For example, if the static universe is really unchanging with time until it decays to the expanding phase, and if it has some finite probability to decay at any time, then it is impossible for it to have been in the static phase for an infinite amount of time. (Alex Vilenkin has some talks and papers discussing models like these.)

Also, it's generally very hard to stabilize such universes. For example, Einstein constructed a static universe where he balanced the attraction due to matter's gravity with repulsion due to a cosmological constant, but this turns out to be unstable. If it shrinks a little bit, there is a runaway contraction to a singularity, and if it grows a bit, there is a runaway expansion.

Whereas de Sitter space is a very natural solution to GR which contracts and then expands.

Scott,
I think it's important to sharply distinguish between:
1) string theory or AdS/CFT as an effective phenomenological model for the strong interactions
2) string theory as a candidate theory of quantum gravity at the Planck scale,
and
3) string theory or AdS/CFT as a (probably) mathematically consistent structure, serving as a toy model for quantum gravity ideas, some of which may be useful even if the real world is described by a different theory of quantum gravity.

There's lots of good (mathematical) evidence for the validity of (3), which is enough I think to motivate a nonzero Hamiltonian in asymptotically AdS spacetimes. But in that respect AdS/CFT is no different from classical GR, and classical GR predicts $H = 0$ in a closed universe.

That string theory is valid at the Planck scale is a plausible hypothesis, supported by the great difficulty in constructing other good theories of quantum gravity, but it is not supported by any kind of direct experimental evidence. And as you say, the LHC evidence for things like low energy supersymmetry is weak. (Supersymmetry could be broken at higher energy scales, which would be fine so far as string theory is concerned, but makes it no longer useful for explaining the fine tuning of the Higgs mass.) In any case, this would require finding solutions to string theory with a positive cosmological constant, so AdS/CFT would not directly apply although it may still give clues about various questions like e.g. do black holes lose information?

Whether or not AdS/CFT is useful for describing the phenomenology of quark-gluon plasmas is a completely separate question, which stands or falls independently of the other things. (It uses the "CFT" side of the duality, not the "AdS quantum gravity" side.) Some of the CFT's which have useful holographic duals are qualitatively similar to QCD in certain respects, but there are also a lot of important differences (lots of supersymmetry instead of none, a very large number of colors instead of 3, the quarks interact with the gluons differently than in QCD, etc.) So there was no good reason to expect anything more than qualitative predictions for heavy ion collision physics, and if it turns out that the fit is terrible, I don't think that's particularly surprising, or evidence against the consistency of AdS/CFT as a mathematical structure.

11. Scott Church says:

Aron, thanks... this spoke directly to my questions. As you might have guessed from my comments, while the AdS/CFT issues with modeling QCD interactions seem significant to me, I wasn't sure to what extent they're relevant to quantum gravity, cosmology and M-theory in general, if at all. To be honest, 3) never occurred to me, although it should have since AdS/CFT is useful for making many otherwise intractable problems solvable and toy models are a lot more useful than many folks realize. I didn't realize that M-theory requires Minimal SuSY to explain the fine tuning of the Higgs mass either.

As always, great stuff Sir! Since last fall I honestly I think I've learned more theologically relevant physics and philosophy from this blog than I had in my entire life before then. And speaking of M-theory and the Higgs, I'm anxiously awaiting your series on fine tuning and the multiverse. I'm planning on an essay or two of my own covering some of that as well. If I finish mine before you do yours I may hit you up for a sanity check. Thanks again, and keep up the stellar work! :-)

12. Scott Church says:

Btw, for the record I agree that string theory (or more specifically M-theory) is very plausible, and perhaps even as they say, "the best game in town" for a potential quantum gravity theory. The framework is elegant and in theory at least predicts things that might physically exist (e.g. extra dimensions, compactified Calabi-Yau spaces, and the associated vacua). But it's so fluid that it can be made to agree with/predict virtually anything, and I don't see how we're ever going to get beyond that without probing on distance scales comparable to those of its compactifications. Last time I saw any estimates of such, this would require an LHC roughly the diameter of the solar system, which needless to say ain't happening any time soon. If there's another way to get useful predictions out of any string theory I have no idea what they would be, and as near as I can tell no one else does either. And much as I love the inflationary paradigm and hate to admit it, in the absence of any useful candidates for an inflaton (beyond some generic scalar field) the same can be said for it as well (although to be fair, BICEP2 + Planck does appear to have ruled out a lot of inflationary models, specifically those involving higher scalar-to-tensor ratios).

My real complaint isn't with M-theory per se, but with folks like Sean "Can-I-Build-A-Model?" Carroll who insist that none of this matters. As long as we can dream up any old scenario that can be stretched to fit whatever we see (or even can't), anyone who thinks twice before jumping on the M-theory bandwagon risks being labeled as an anti-science obscurantist. It also annoys me that they unquestioningly assume that Inflation = Eternal Inflation + String Landscape + Anthropic Multiverse. While it is true that most inflationary scenarios are eternal, not all are... and in fact, most of those that survived BICEP2 + Planck aren't (e.g. the Starobinsky model). And whatever the final quantum gravity theory turns out to be, unless it actually takes us beyond the Standard Model (which, I might add, virtually nothing observationally supportable to date has), even eternal inflation will only yield universes similar to ours, which won't do much to address all those fine-tuning issues they so quickly side-step. It seems to me that whether Carroll et al. admit it or not, to them the real appeal of all this is less scientific than "a-theological."

But now I'm off topic. I'll save further such thoughts for the much-anticipated Fine-Tuning/Multiverse series... coming soon to a blog near you! :-)

13. David says:

This seems to be as good a place as any to ask a question about closed universes.

See, in a lot of popular science books, they teach you that an "open" universe is one where space is infinite, saddle-shaped, and keeps expanding forever; a "flat" universe is infinite, plane-shaped, and the rate of expansion eventually peters out to zero; and a "closed" universe is finite, sphere-shaped, and eventually contracts in a big crunch. They then talk about the cosmological constant and "dark energy," which make our universe expand at an accelerating rate, something that doesn't fit the taxonomy of possibilities for the universe's topology, and which they do not relate back to that taxonomy in any way.

Can a universe with lots of dark energy be a closed universe? Will a closed universe with dark energy keep on expanding and accelerating, or will it eventually collapse in a big crunch like a "normal" closed universe? Is the three-type Taxonomy only relevant given certain energy conditions? (Strong/weak/null)

14. David says:

Oh, and I almost forgot:

are there any good reasons to think that the universe is closed in the first place, other than Kalam-esqe arguments against actual infinities?

15. Aron Wall says:

David,
I responded to your question here.