The n-Category Café

Thanks! I’m hoping that I can do a bit of work on quantum gravity now and then without getting too emotionally involved in it — it’s too much of a rollercoaster ride, and there’s too much fighting between different schools of thought. Math is great because you be quite sure you’re right. Physics is great because you can look up into the sky and see it. I want a balanced diet.

Urs wrote:

My impression is the following, I’d be grateful to hear what is wrong about it.

What happens is a special case of this: start with a differential equation whose solutions diverge at the origin. Replace it with a difference equation on a subset of points that does not include the origin. This will have solutions not showing the singularity.

This may have been true for some early models — there’s been a lot of models between 2001 and now. It’s definitely not true for the currently popular models that Ashtekar was explaining.

It’s true that a differential equation is getting replaced by a difference equation. But the singularity is not being avoided simply by ‘stepping over it’. That would be a cheap trick In the models under study now, there really is an effective ‘force’ that prevents the singularity. And it arises from the mechanism I sketched:

Instead of talking about the curvature of spacetime at infinitesimally small scales, we can only measure curvature by carrying a particle around a finite-sized loop. This has little effect when spacetime is only slightly curved, but when the curvature is big it makes a big difference. This causes an effect very much like a force that prevents the Universe from crushing down to nothing as we follow it backwards in time.

A bit more precisely — but still quite roughly: if the curvature is $x$, in loop quantum gravity we don’t have a quantum operator corresponding to $x$. Instead we have something more like $exp(i x)$. So we use something like $sin(x) = (exp(i x) - exp(-i x))/2i$ as a substitute for $x$ in certain equations. When the curvature is small these are almost the same. When it gets big, they’re different. This gives an effective force that prevents the singularity. This force starts getting big when the density is about 1/100 of the Planck density. When the density hits .41 times the Planck density, it’s enough to create a ‘bounce’.

(The math is actually a bit more complicated than what I said, but it’s similar in spirit.)

In fact, a bunch of calculations have shown that the quantum dynamics are quite nicely approximated by an ‘effective Friedmann equation’ — a differential equation like the usual classical one that describes the Big Bang, but with an extra term.

And, the solutions of the difference equation are very close to the solutions of this differential equation.

I see, thanks for the explanation. I should have a look at a recent review, then.

Christine wrote:

For instance, the smallest nonzero area eigenvalue of LQG is the assumed step size (the so called “area gap”) of the LQC difference equation. For far this can be made more formally justifiable (on either physical and/or mathematical grounds)?

In the work that Ashtekar was summarizing, the step size is not equal to the area gap — but you’re right, it’s related.

This is not an ad hoc assumption, but it’s not the sort of thing I’m particularly expert on, so I suggest you read Quantum nature of the Big Bang: improved dynamics for details. (This may also help Urs with some of his questions.)

I hope it’s clear why loop quantum cosmology is not my thing. Since we don’t really know the dynamics in loop quantum gravity, it takes a mixture of guts, caution, math, physical intuition, and trial and error to guess equations for loop quantum cosmology. This is a job for actual physicists. I’m more of a mathematical physicist. I like spin foams not only because they have a chance of providing loop quantum gravity with some dynamics that reduces to general relativity in a suitable limit, but also because they’re mathematically elegant, with relations to representation theory and category theory. Math like this makes me happy. Quantum cosmology mainly makes me nervous. Of course it’s fun to talk about, since it tackles big issues that everyone can enjoy, like the origin of the universe! But it’s not the sort of thing I’d enjoy working on, so I’m not an expert on it, and don’t plan to become one.

Daniele wrote:

Now that things in spin foams are working nicely, where can you put particles in there?

I wouldn’t say that spin foams are “working nicely”: I just said that a few obstacles have been overcome. As I said:

Of course there are even bigger tests still ahead for this spin foam model. We need to see if it reduces to general relativity in the classical limit. In other words, we need to get Einstein’s equations out of it. And we need to see if it reduces to the usual perturbative theory of quantum gravity in some other limit. In other words, we need to compute, not just graviton propagators (which describe the probability of a lone graviton zipping from here to there on the background of Minkowski spacetime), but graviton scattering amplitudes (which describe the probability of various outcomes when two or more gravitons collide).

Both these tasks are both computationally and conceptually difficult. In other words, it’s not just hard to do the calculations: it’s hard to know what calculations to do! When I said “in some limit” and “in some other limit”, I know what limits these are in a physical sense, but not how to describe them using spin foams. Actually we seem closer to understanding graviton scattering amplitudes, thanks to the work of Rovelli. But it seems miraculous and strange that we can compute graviton propagators (much less scattering amplitudes) using very simple spin foams, as Rovelli and his collaborators have done. Every time I meet him, I ask Rovelli what’s going on here: how we can describe the behavior of a graviton in terms of just a few 4-simplices of spacetime.

So, the road is still long, steep, and fraught with danger.

In particular, I would not be surprised if at some point we needed a spin foam model that includes matter to get everything to work out nicely. But we haven’t reached the point of being able to tell.

As for matter, I hope you know that in 3d quantum gravity, you get matter for free just by carving out ‘tubes’ in spacetime: they automatically act like worldlines of particles! This has been studied intensively by Barrett, Freidel, Baratin and collaborators — I gave a long list of references back in week222, and you really should read them.

In particular, these guys reached the point of being able to compute some Feynman diagrams for ordinary quantum field theory in 3d Minkowski spacetime using the $G \to 0$ limit of the usual spin foam model for 3d quantum gravity!

Baratin and Freidel also know how to compute Feynman diagrams for ordinary quantum field in 4d Minkowski spacetime using a spin foam model. So this is a candidate for the $G \to 0$ limit of a spin foam model containing gravity and matter.

I’m glad someone is paying attention. The ‘downbeat’ Quote of the Week that now graces the end of week280 had formerly been my quote for week279. Since it’s about quantum gravity, I decided it would be more appropriate for week280. It should be read with a tinge of humor.

Originally this week’s Quote of the Week was:

I was sitting in a chair in the patent office in Bern when all of a sudden a thought occurred to me. If a person falls freely, he will not feel his own weight. - Albert Einstein

But I figure this should go on a Week devoted to classical general relativity!

My stock of good math and physics quotes is running low. Anyone know some more?

Of course I was talking about ‘prevention of singularities’ in theories of loop quantum cosmology that are obtained by taking some ideas from loop quantum gravity — itself a speculative theory — and combining them with severe simplifying assumptions — namely, homogeneity and isotropy — in a rather controversial way. So, take everything I say here with a large grain of salt.

But, insofar as we can approximate the formation of a black hole by a process with perfect spherical symmetry, there’s hope that similar calculations can be done here too, and ‘singularity prevention’ will again occur. And indeed, there are calculations supporting this hope.

Also, people are starting to lessen the symmetry assumptions needed to get these results. For example, they’re working with Bianchi I cosmologies, which are anisotropic, and Gowdy spacetimes, which are even more complicated and less symmetrical.

So, Ashtekar is already dreaming of general singularity avoidance results.

Vishal wrote:

After the next ‘bounce’, should we expect from Ashtekar’s model(s) the same kind of fine-tuned universe as the one we are living in now (even if it isn’t the exact universe as ours)?

There probably won’t be a ‘next’ bounce.

It’s long been known that depending on the mass density of the universe and the cosmological constant, the universe may either keep expanding indefinitely or recollapse in a ‘Big Crunch’. Right now all the astronomical evidence points towards a universe that will expand indefinitely. If this is true, there’s just one Big Bounce.

One can adjust the mass density and cosmological constant in loop quantum cosmology and get models where there’s just one bounce. That’s the sort of model I was discussing — because that seems physically realistic. But one can also adjust these parameters to get models where the universe recollapses, and there are many big bounces, as shown in this computer calculation:

In this sort of scenario, it turns out the universe can bounce over $10^{50}$ times before its wavefunction smears out significantly!

An interesting question is how the arrow of time works in this theory. Current research doesn’t answer that, because the model is too simple. But here’s an example of what I mean: in the one-bounce model shown below, does the bounce seem to lie in the past in both branches of the universe… or does it seem to lie in the past of one and the future of the other?

In other words, does this picture show two universes with a Big Bang in their past — or one with a Big Bang in its past and one with a Big Crunch in its future?

Personally I lean towards the former scenario, since a low-entropy state at the bounce is most likely to gain entropy in both directions. But this is the sort of thing where it would be good to do calculations.

Of course, if I’m right, the caption’s phrase “collapse of a preexisting universe” is a misleading way of describing what happened!

Anyway, we can rephrase your question as follows: are both branches of the above universe roughly the same? And if I’m right, the answer is yes.

Vishal wrote:

The “effective force that prevents the singularity”, is that a property of matter or space-time or both?

Slightly tricky question.

It’s mainly the new treatment of spacetime that makes this force arise. The discreteness of spacetime at small distances is what makes loop quantum cosmology different from the standard approach to quantum cosmology. In the standard approach, the singularity is not avoided. In loop quantum cosmology, the discreteness modifies the equations in a way that becomes significant when the curvature of spacetime is big. This gives the effective force.

But, it’s crucial that the model includes matter as well as spacetime! It uses a very simple form of matter: a massless scalar field $\phi$ whose value serves as a ‘clock’. In other words, we use this field to tell what time it is. We need some trick like this because of the famous ‘problem of time’ in quantum gravity.

Very crudely, the ‘problem of time’ is: how do you keep track of time in a world where everything is treated quantum-mechanically, even the geometry of spacetime itself?

Your next question might be: “What is this field $\phi$?” The answer is that we’re just playing around with very simple models here, which are not meant to describe our universe in much detail. But as it happens, the most successful theories of inflation invoke a massless scalar field $\phi$ which has been dubbed the ‘inflaton field’.

Nobody knows what it really is!

Vishal wrote:

… what do the models have to say about the “previous universe” (the one that presumably existed before the ‘bounce’)? In other words, how different was its geometry from the present one?

In the models I’m talking about, there’s almost no difference — nothing you could measure. In the simplest case, there’s no difference at all.