Download Wave Function of the Universe

Hartle and Hawking on quantum cosmology
*given* this observation (i.e., we can work out some conditional
probabilities).
[John Baez, University of California, Riverside: On the Wave Function
of the Universe]
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....
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
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.
H psi = 0
It's tough to explain this accurately without math....
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 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.
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
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 3sphere, 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 HartleHawking 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.