Hartle and Hawking on quantum cosmology
[John Baez, University of California, Riverside: On the Wave Function of the Universe]
It's incredibly cool. First I'll give a very nontechnical account so that everyone can see just how cool it is, and then I'll fill in a few of the details.
Hartle and Hawking were perhaps the first people to have the nerve to write down a formula for the wavefunction of the universe. They were working in the context of quantum gravity, and their formula made some mind-blowing predictions about the quantum- theoretic aspects of what happens at the big bang and big crunch.
Of course, we don't know for sure whether there will be a big crunch, or whether the universe is finite or infinite in extent. The original Hartle-Hawking formula assumes that the universe is finite in extent, and it predicts that there will be a big crunch. Current conventional wisdom is leaning towards a universe that's infinite in extent, with galaxies moving apart forever - so no big crunch. But let me just explain the original Hartle-Hawking proposal, and not worry about whether it's correct.
In many approaches to quantum gravity, the basic equation is the Wheeler-DeWitt equation
H psi = 0
where H is the "Hamiltonian constraint" and psi is the wavefunction of the universe. The Hamiltonian constraint is a bit like a Hamiltonian - but not quite. Since I want to stay reaonably nontechnical, I'll pretend it's just like a Hamiltonian. Given this simplification, the equation H psi = 0 just says the total energy of the universe is zero - the only sensible value for a closed universe. And solving it is just like looking for an eigenstate of any other Hamiltonian.
Folks who have studied quantum mechanics have probably solved the "time-independent Schrodinger equation" in order to find the bound states of a hydrogen atom. It looks like this:
H psi = E psi
where E is the energy. If we do this, we get a wavefunction psi describing a "probability cloud" of possible positions of the electron. It has a chance of being far away from the nucleus, a chance of being close... and since it's in an energy eigenstate, these probabilities don't change with time.
Similarly, Hartle and Hawking solve the Wheeler-DeWitt equation
H psi = 0
they get a wavefunction defining a "probability cloud" of possible geometries for the universe - or more precisely, geometries for space. Since they are assuming a closed universe, they assume space is the 3-dimensional analogue of a sphere. This sphere has a chance of being very large, a chance of being small... and since the wavefunction is in an energy eigenstate, these probabilities don't change with time. That's why the Wheeler-DeWitt equation is sometimes called the "frozen formalism".
Now, this static description of the universe may seem a bit odd! To see the passage of time, we have to "thaw the frozen formalism". We can do this for the hydrogen atom too, so let's think about that analogy again. When we last left it, the electron was described by a very spread-out wavefunction: an energy eigenstate. But say we observe the electron at some point or other - or more realistically, in some smallish region. Then we can calculate what would happen *given* this observation (i.e., we can work out some conditional probabilities).
In other words, let's assume the wavefunction is a bump centered at some point or other, and see what happens as time passes. Well, the wavefunction spreads out as time passes, but not too fast as long as we didn't make our bump *too* sharply spiked. And the "center of mass" of our bump will follow a roughly classical trajectory, at least as long as it stays far from the nucleus. When it gets close to the nucleus, the wavefunction spreads out faster....
How is all of this analogous to the Hartle-Hawking cosmology? Well, if we want to compare the universe to the hydrogen atom, the analogy works this way: the distance of the electron from the nucleus is analogous to the radius of the universe, so the electron being close to the nucleus is analogous to the universe being very small - close to the big bang or big crunch! The singularity in the electrostatic potential right at the nucleus is analogous to the big bang / big crunch singularity.
So what happens is this. The Hartle-Hawking solution of
H psi = 0
says there is a chance of the universe being very large, or very small, or any size in between... but if we observe the universe to have some particular size and shape, we can calculate what would happen *given* this assumption. In other words, we can assume the wavefunction psi is a bump centered at a particular geometry, and see what happens then. It's a lot like the hydrogen atom: the bump nearly follows a classical trajectory, spreading out gradually, except when it gets really close to the singularity, at which point it spreads out much faster.
In short, if we assume the universe is observed to be very large, the universe will thenceforth act almost like our classical intuition says it should - so long as it stays far from the big bang / big crunch. So if it starts out expanding, it keeps and expanding but then starts to recollapse.... but when it gets close to the big bang / big crunch, this classical picture breaks down: the wavefunction describing the geometry of space "smears out" and quantum effects become very important.
So you see, the cool part is that the big bang and big crunch are not different events - instead, they are just two aspects of the same thing: let me call it the "big bang / big crunch". The "big bang / big crunch" is just a name for the regime where the size of space is very small, where our attempts to model space and time using classical physics break down. Far from the big bang / big crunch, we can approximately pretend that space has a definite geometry which changes with time, at first expanding and then recollapsing. But near the big bang / big crunch, this is revealed for the lie it is.
It's tough to explain this accurately without math....
Now let me fill in some details from stuff taken from "week138", which was about a party in Cambridge in honor of Hartle's 60th birthday: James Hartle and Stephen Hawking, Wavefunction of the universe, Phys. Rev. D28 (1983), 2960.
In quantum mechanics, we often describe the state of a physical system by a wavefunction - a complex-valued function on the classical configuration space. If quantum mechanics applies to the whole universe, this naturally leads to the question: what's the wavefunction of the universe? In the above paper, Hartle and Hawking propose an answer.
Now, it might seem a bit overly ambitious to guess the wavefunction of the entire universe, since we haven't even seen the whole thing yet. And indeed, if someone claims to know the wavefunction of the whole universe, you might think they were claiming to know everything that has happened or will happen. Which naturally led Gell-Mann to ask Hartle: "If you know the wavefunction of the universe, why aren't you rich yet?"
But the funny thing about quantum theory is that, thanks to the uncertainty principle, you can know the wavefunction of the universe, and still be completely clueless as to which horse will win at the races tomorrow, or even how many planets orbit the sun.
That will either make sense to you, or it won't, and I'm not sure anything *short* I might write will help you understand it if you don't already. A full explanation of this business would lead me down paths I don't want to tread just now - right into that morass they call "the interpretation of quantum mechanics".
So instead of worrying too much about what it would *mean* to know the wavefunction of the universe, let me just explain Hartle and Hawking's formula for it. Mind you, this formula may or may not be correct, or even rigorously well-defined - there's been a lot of argument about it in the physics literature. However, it's pretty cool, and definitely worth knowing.
Here things get a wee bit more technical. Suppose that space is a 3-sphere, say X. The classical configuration space of general relativity is the space of metrics on X. The wavefunction of the universe should be some complex-valued function on this classical configuration space. And here's Hartle and Hawking's formula for it:
psi(q) = integral exp(-S(g)/hbar) dg g|X = q
Now you can wow your friends by writing down this formula and saying "Here's the wavefunction of the universe!"
But, what does it mean?
Well, the integral is taken over Riemannian metrics g on a 4-ball whose boundary is X, but we only integrate over metrics that restrict to a given metric q on X - that's what I mean by writing g|X = q. The quantity S(g) is the Einstein-Hilbert action of the metric g - in other words, the integral of the Ricci scalar curvature of g over the 4-ball. Finally, of course, hbar is Planck's constant.
The idea is that, formally at least, this wavefunction is a solution of the Wheeler-DeWitt equation, which is the basic equation of quantum gravity (see "week43").
The measure "dg" is, unfortunately, ill-defined! In other words, one needs to use lots of clever tricks to extract physics from this formula, as usual for path integrals. But one can do it, and Hawking and others have spent a lot of time ever since 1983 doing exactly this. This led to a subject called "quantum cosmology".
I should add that there are lots of ways to soup up the basic Hartle-Hawking formula. If we have other fields around besides gravity, we just throw them into the action in the action in the obvious way and integrate over them too. If our manifold X representing space is not a 3-sphere, we can pick some other 4-manifold having it as boundary. If we can't make up our mind which 4-manifold to use, we can try a "sum over topologies", summing over all 4-manifolds with X as boundary. We can do this even when X is a 3-sphere, actually - but it's a bit controversial whether we should, and also whether the sum converges.
Well, there's a lot more to say, like what the physical interpretation of the Hartle-Hawking formula is, and what predicts. It's actually quite cool - in a sense, it says that the universe tunnelled into being out of nothingness! But that sounds like a bunch of nonsense - the sort of fluff they write on the front of science magazines to sell copies. To really explain it takes quite a bit more work. And unfortunately, it's just about dinner-time, so I want to stop now.