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:

H \Psi = i \hbar \frac{d}{dt} \Psi.

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.
    .
  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).
    .
  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.
    .
  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:

H \Psi = 0.

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.A

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]

About Aron Wall

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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2 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

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