Insights Struggles With The Continuum - Part 2 - Comments

[john baez]

Hi!

But as far as you know, are there any promising attempts at developing a theory that is fundamentally discrete, at the most basic level? (So that continuum calculations are the approximations to the discrete calculations, rather than the other way around.) Does loop quantum gravity count as one?

Count as fundamentally discrete, or count as promising?

Loop quantum gravity uses real numbers and infinite-dimensional Hilbert spaces all over - in fact, the approach favored by Ashtekar uses Hilbert spaces of uncountable dimension, which are much bigger than the usual infinite-dimensional Hilbert spaces in physics. It also continues to treat space as a continuum.

On the other hand, most versions of loop quantum gravity involve discretization of geometry in the sense that areas and volumes take on a discrete spectrum of allowed values, sort of like energies for the bound states of a hydrogen atom.

In spin foam models, which are a bit different than loop quantum gravity, we tried to remove the concept of a spacetime continuum, and treat it as a quantum superposition of different discrete geometries. However, the real and complex numbers were still deeply involved.

If I considered this line of work highly promising I would still be working on it. I would like to hope it's on roughly the right track, but there's really no solid evidence for that, and at some point that made me decide to quit and work on something that would bear fruit during my own lifetime. I think that was a wise decision.

 

[john baez]

Is the problem with your Newtonian example due to the implied instantaneous-action-at-a-distance in the fundamental equation (which means the result isn't fully conservative)?

I don't know what you mean by "not fully conservative": energy is conserved in this theory, along with everything else that should be conserved.

I said what causes the problem I discussed (namely runaway solutions where particles shoot to infinity in finite time):

Is this is a weakness in the theory, or just the way things go? Clearly it has something to do with three idealizations:

• point particles whose distance can be arbitrarily small,
• potential energies that can be arbitrariy large and negative,
• velocities that can be arbitrarily large.

These are connected: as the distance between point particles approaches zero, their potential energy approaches −∞, and conservation of energy dictates that some velocities approach +∞.

The interesting thing is that when we introduce special relativity and make charged point particles interact via fields, the runaway solutions don't go away. If anything, they get even worse! They don't reach arbitrarily high speeds, but particles that should be attracting tend to shoot away from each other in an ever-accelerating way, approaching the speed of light. I explain this in Part 3.

Of course that's for electromagnetism; I'll talk about general relativity later.

One final note. Your reply to the God-made/Man-made joke was uncomfortable but you have to remember that discussing the continuum is really the physics equivalent to a "religious" question.

I try to avoid "religious" questions: this series of posts is about concrete problems with our favorite theories of physics. Here by "concrete problems" I mean things like runaway solutions and the actual behavior of electrons, which we can study using math and experiment - as opposed to questions like whether the real numbers "really exist" or were "made by Man".

It seems like the only thing we can do is perpetually oscillate between experimental and theoretical advances, never reaching a "ta da, we're done!" moment.

There's so much we don't understand yet about physics, I don't think it's even profitable to think about the "ta da, we're done!' moment. I think we're making good progress, but it may take a few more millennia to understand the fundamental laws of physics. So, I think it's good to be patient and enjoy the process of slowly figuring things out. It is, after all, plenty of fun.

 

[Hendrik Boom]

It doesn't seem to me that anything is ever continuous; continuous is always an approximation, even in classical mechanics. Take a baseball. Suppose you integrate numerically the simple case where there is just gravity, no air resistance.

When you speak of numerical integration you're no longer speaking about a baseball: you're speaking about a computer program. Of course if you do numerical integration on a computer using time steps, there will be time steps.

Space is also always discretized in terms of the characteristic scales of the system.

I have never seen any actual discretization of space, unless you're talking about man-made structures like pixels on the computer screens we're looking at now.

The closest I've encountered about grains of space is in loop quantum gravity where space seems to be broken into tiny pieces, maybe a googol of them in a teaspoon. But as I understand it, the pieces are not arranged in a regular grid; they are constantly rearranging themselves, and all their possible arrangements get quantum-superposed so they are all smudged together into something that feels kind of continuous.

Is this image even approximately a correct view of the theory?

 

[john baez]

Is this image even approximately a correct view of the theory?

Yes, that's a quite good description of what loop quantum gravity is aiming for. The main problem is that there are several different versions of loop quantum gravity and spin foam theories, nobody has been able to show (in any style I find convincing - I'm not hoping for mathematical rigor) that any version reduces to general relativity in a suitable 'classical limit'.

 

[Telemachus]

Hi John. About the LIGO experiment, do you think that there are new perspectives now on the exploration of the nature of space-time? what could be the implications of this measurements of gravitational waves? what about quantum gravity and gravitons? what about string theory? I've heard the announcement by the LIGO team, and they've talked about some cosmological strings. I don't know if this place is the appropriate to talk about this, but this discovery excites me a lot! do you think that with this interferometers we will confirm or discard string theory?

Thanks for your interesting posts.

 

[john baez]

The LIGO experiment is very important, but it says nothing at all about quantum gravity, gravitons or string theory. I explained why in the discussion thread here:

• John Baez, Gravitational waves, 11 February 2016.

If you go there you'll see a long conversation with at least 113 comments. I must have answered at least 40 interesting questions about LIGO and gravitational waves. It was quite fun!

 

[atyy]

The LIGO experiment is very important, but it says nothing at all about quantum gravity, gravitons or string theory. I explained why in the discussion thread here:

• John Baez, Gravitational waves, 11 February 2016.

If you go there you'll see a long conversation with at least 113 comments. I must have answered at least 40 interesting questions about LIGO and gravitational waves. It was quite fun!

But if we take GR to be a low energy effective quantum field theory of a spin 2 particle, then in that sense, wouldn't LIGO say something about gravitons in the same way it says something about GR?

 

[vanhees71]

At least it gives an upper limit of the graviton mass.

 

[john baez]

But if we take GR to be a low energy effective quantum field theory of a spin 2 particle, then in that sense, wouldn't LIGO say something about gravitons in the same way it says something about GR?

Okay, if you insist. What I meant is that we don't know anything more about quantum gravity than we did before LIGO discovered gravitational waves. This was a classical experiment, not a quantum one.

At least it gives an upper limit of the graviton mass.

Okay. If someone thought the graviton had a nonzero mass they might be less convinced of that now. Of course we already knew that either the graviton mass is zero or general relativity is wrong.

I would prefer to say LIGO's first result can help us test a prediction of purely classical general relativity: namely, that gravitational waves don't disperse, at least if they're not too strong and their wavelengths are much shorter than the curvature length scale of the spacetime they're in. You can interpret this in terms of gravitons if you like. But we're no more (or less) sure that gravitons exist now than we were a few weeks ago.

 

[vanhees71]

Sure, perhaps one should say that it gives a limit on a mass of the gravitational field. It's analogous to the measurement of a "photon mass" in the context of electromagnetics. You can test this also by, e.g., high-accuracy measurements of Coulomb's law, i.e., with classical em. field situations.