Newsgroups: sci.physics

Subject: Re: What is the energy of the universe ? Date: 13 Oct 1995 11:30:26 -0700

In article <45k5mj$aq2@agate.berkeley.edu> ted@physics.berkeley.edu writes:

>In article <813502172.20274@upthorpe.demon.co.uk>,

>Oz wrote:

>>5) (Uninformed speculation) There might be other reasons for

>>existing models to be inaccurate. One is the problem of the

>>high energy density of the vacuum predicted (but largely

>>ignored) by QM. Admittedly it works out to be huge, but I am

>>surprised how it is just swept under the carpet. I know you

>>cannot comment because it's not your field.

>Like I've ever let that stop me before!

>It turns out that "vacuum energy" is precisely what cosmologists and

>general relativists call "the cosmological constant." The idea that

>this quantity might not be zero is actually quite fashionable among

>cosmologists these days. It turns out that throwing a cosmological

>constant into your theory helps a lot in making it fit the data. In

>fact cosmological-constant cold dark matter models work just about as

>well as hot-plus-cold dark matter models.

>Of course, the values for the cosmological constant that are preferred

>by cosmologists are many orders of magnitude smaller than any

>"natural" scale imposed by particle physicists. But that's a problem

>for the particle physicists to work out.

Coming in from left field here, let me add that quantum gravity people are very interested in the cosmological constant. In fact, the cosmological constant seems to serve, at least in some cases so far, as a kind of "regulator" which prevents certain divergences in quantum gravity. So a nonzero cosmological constant would make me very happy. In particular, it would make me very happy because when we quantize the DeSitter universe (no matter, just a cosmological constant), we get a very wonderful thing called the "Chern-Simons state" of quantum gravity, which is closely related to a wonderful knot invariant called the Kauffman bracket. You see, every state of quantum gravity corresponds to an invariant of knots and links, in the loop representation of quantum gravity, and the Kauffman bracket is one of the most charming of all link invariants....

Below I quote from "week56" about that... the stuff about cosmology is mainly near the end.

I don't really take the importance of the cosmological constant *too* seriously, since there are too many ways we could be wrong in cosmology, and too many ways we could be wrong in quantum gravity, to think that an idea that solves some problems of both is necessarily right! But it's amusing, to say the least.

Okay, so naively you might think a state in the *quantum* version of general relativity a la Ashtekar is just a wavefunction psi(A). [Here A is the gravitational vector potential, in Ashtekar's reformulation of gravity as gauge theory... more details in week56!] That's not too far wrong and I won't bother about certain nitpicky technicalities here (again, for the full story try net.tex). But there's one very important catch I can't ignore: general relativity has *constraint* equations, meaning that psi has to satisfy some equations. The first constraint, the Gauss law, just says that we must have

whenever A' is the result of doing a gauge transformation to A. Or at the very least, this should hold up to a phase; the point is that psi is only supposed to record physically significant information about the state of the universe, and two connections are physically equivalent if they differ by a gauge transformation. The second constraint, the diffeomorphism constraint, says we need to have

when A' is the result of applying a diffeomorphism of space, S, to A. Again, the point is that two solutions of general relativity are physically the same if they differ only by a coordinate transformation, or --- *roughly* the same thing --- a diffeomorphism. The third constraint is the real killer. It's meaning is that psi(A) doesn't change when we do a diffeomorphism of spaceTIME to the connection A, but it's usually formulated `infinitesimally' as the Wheeler-DeWitt equation

meaning roughly that the time derivative of psi is zero. Think of it as a screwy quantum gravity version of Schrodinger's equation, where d psi/dt had better be zero!

It's hard to find explicit solutions of these equations. Indeed, it's hard to know what the heck these equations *mean* in a sufficiently precise way to recognize a solution if we found one! However, things were even worse back in the old days. Back in the old days when we thought of states as wavefunctions on the space of metrics, we didn't know ANY solutions of these equations. But nowadays we are very happy, because we know infinitely many times as many solutions! To be precise, we now know ONE solution. This is called the Chern- Simons state, and it was discovered by Kodama:

2) H. Kodama, Holomorphic wavefunction of the universe, Phys. Rev. D42 (1990), 2548-2565.

Now actually people have proposed other explicit solutions, but personally I have certain worries about all those other solutions, and I am not alone in this, whereas everyone seems to agree that, no matter how you slice it, the Chern-Simons state is a solution.

Now there is a slight catch: the Chern-Simons state is a solution of quantum gravity *with cosmological constant*. This is an extra term that Einstein threw into his equations so that they wouldn't make the obviously ridiculous prediction that the universe is expanding. When Hubble took a look and saw galactic redshifts all over, Einstein called this extra term the biggest blunder in his life. That kind of remark, coming from that kind of person, might make you a little bit reluctant to get too excited about having found a state of quantum gravity with this extra term thrown in! Luckily it turns out that you can take the limit as the cosmological constant goes to zero, and get a state of the theory where the cosmological constant is zero. I like to call this the `flat state', because it's zero except where the connection A is flat.

(In fact, if the space S is not simply connected, there are lots of different flat states, because there is what experts call a moduli space of flat connections, i.e., lots of different flat connections modulo gauge transformations. Not many people talk too much about these flat states, but I do so in my paper net.tex and also the harder one grav.tex, both available by anonymous ftp as described at the end of this article.)

Now what is this Chern-Simons state? Well, there is a wonderful thing you can compute from a connection A on a (compact oriented) 3-manifold S, called the Chern-Simons action:

which looks weird when you first see it, but gradually starts seeming very sensible and nice. The reason why folks like it is that it doesn't change when you do a small gauge transformation --- i.e., one you can get to following a continuous path from the identity --- and it changes only by an integral multiple of 8pi^2 if you do a large gauge transformation. Plus, it's diffeomorphism-invariant. It's incredibly hard to write down many functions of A with these properties, so they are precious. There are deeper reasons why it's so nice, but let's leave it at that for now.

So, the Chern-Simons state is

Why does this satisfy the constraints? Well, I just said above that the Chern-Simons action was hand-tailored to have the gauge-invariance and diffeomorphism-invariance we want, so the only big surprise is that we *also* have a solution of the Wheeler-DeWitt equation. Well, we do: a two-line computation shows it.

But clearly nature, or at least the goddess of mathematics, is trying to tell us something if this Chern-Simons state, which has all sorts of wonderful properties relating to *3-dimensional* geometry, is also a solution of the Wheeler-DeWitt equation, which is all about *4-dimensional* geometry, since it expresses the invariance of psi under evolution in TIME. I have been thinking about this for several years now and I think I finally really understand it. There are probably people out there to whom it's perfectly obvious, because it's not really all that complicated, but unfortunately none of these people has ever explained it. Let me briefly say, for those who know about such things, that it all comes down to the fact that the Chern-Simons action was *born* as a 3-dimensional spinoff of a 4-dimensional thing called the 2nd Chern class. (If you want more details, I explained this as well as I could at the time in grav.tex, but now that I understand it better I'll say more about it in a paper which is all about this "3d versus 4d" stuff.)

What is the physical meaning of the Chern-Simons state? As far as I know Kodama's paper hasn't been vastly surpassed in explaining this. He shows that in the classical limit this state becomes something called the anti-deSitter universe, a solution of Einstein's equation describing a (roughly) exponentially expanding universe. If you are wondering what this has to do with Einstein's introduction of the constant to KEEP the universe from expanding, let me just say this. In our current big bang theory the universe is expanding, but the presence of matter, or any sort of positive energy density, tends to pull it back in, and if there is enough matter it'll collapse again. Einstein's stuck in a cosmological constant term to give the vacuum some negative energy density, exactly enough to counteract the positive energy density of matter, so things would neither collapse nor expand, but instead remain in an (unstable, alas) equilibrium. In the deSitter universe there's no matter, just a cosmological constant of the opposite sign, so that the vacuum has positive energy density. In the anti-deSitter universe (invented by deSitter's nemesis anti-deSitter) there's no matter either, but the cosmological constant has the sign giving the vacuum negative energy density, which pushes the universe to keep expanding faster and faster.

Now in addition to this physical interpretation, the Chern-Simons state also has some remarkable relationships to knot theory, which were discovered by Witten and, since then, studied intensively by lots of people. I have written a lot in This Week's Finds about this! But briefly, there should be an invariant of knots and links associated to any state of quantum gravity, and the one associated to the Chern-Simons state is the Kauffman bracket, a close relative of the Jones polynomial, which is distinguished by having a very simple, beautiful definition, and also lots of wonderful relationships to an algebraic structure, the quantum group SU_q(2). I should add that in addition to an invariant of knots and links, a state of quantum gravity should also give an invariant of *spin networks*, and indeed the Kauffman bracket extends to a wonderful invariant of spin networks. One can read about this in many places, but perhaps the most detailed account is Kauffman and Lins' book referred to in "week30".