Does God Play Dice? is not a treatise on religion and gambling, but is instead Gerard ‘t Hooft’s submission to ~~Physics Today~~ Physics World concerning the reconcilliation of quantum theory with general relativity. The most interesting part of this short note is not that ‘t Hooft comes out squarely on the side of hidden variable theories, but instead in his description of an idea for how general relativity might arise in such a theory:

An even more daring proposition is that perhaps also general relativity does not appear in the formalism of the ultimate equations of nature. This journal does not allow me the space to explain in full detail what I have in mind. At the risk of not being understood at all, I’ll summarize my explanation. In making the transition from a deterministic theory to a statistical treatment — read: a quantum mechanical one —, one may find that the quantum description develops much more symmetries than the, deeper lying, deterministic one. If, classically, two different states evolve into the same final state, then quantum mechanically they will be indistinguishable. This induces symmetries not present in the initial laws. General coordinate covariance could be just such a symmetry.

That general coordinate covariance may not be fundamental but is instead a product of our inability to access the beables of a theory seems like quite an interesting idea. It would be interesting to think if this type of hidden variable theory, which is not totally general because it needs to recover the general coordinate covariance, is indeed large enough to be consistent with quantum theory. I.e. in the same way the Bell’s theorem rules out local hidden variable theories, is there a similar theorem ruling out ‘t Hooft’s property? I certainly have no inclination about the answer to this question in either direction.

Of further interest, ‘t Hooft claims as motivation for his perspective the following

Nature provides us with one indication perhaps pointing in this direction: the unnatural, tiny value of the cosmological constant. It indicates that the universe has the propensity of staying flat. Why? No generally invariant theory can explain it. Yet, if an underlying, deterministic description naturally features some preferred flat coordinate frame, the puzzle will cease to perplex us.

Finally, for no reason but to turn some portion of the readers of this blog happy and the other portion of this blog angry, here is ‘t Hooft on string theory:

I am definitely unhappy with the answers that string theory seems to suggest to us. String theory seems to be telling us to believe in “magic”: duality theorems, not properly understood, should allow us to predict amplitudes without proper local or causal structures. In physics, “magic” is synonymous to “deceit”; you rely on magic if you don’t understand what it is that is really going on. This should not be accepted.

I wish I understood what ‘t Hooft means in this critique by “proper local or causal structures.”

This is as bad as Laughlin.

t’Hooft wants something better than quantum information theory, which however he hasn’t studied very deeply. And he wants something better than string theory, which he also hasn’t studied very deeply. It’s just too late for this kind of speculation. Okay, it’s a free world and he can have his opinions. They and a dollar will get you a cup of coffee.

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As usual for this type of proposal, he runs into problems explaining EPR type experiments.

‘t Hooft himself writes[*]: “This is surely the most difficult part to be addressed, and a completely satisfactory response has not been given.”

[*]I cite from his 2001 essay in honor of John Bell.

Perhaps he found a solution meanwhile, but I doubt it. There are some pretty strict no-go theorems on this issue.

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Ah, for the statement: “the unnatural, tiny value of the cosmological constant. It indicates that the universe has the propensity of staying flat.”

Many (even prominent) physicists fail to appreciate that solutions to General Relativity even without matter and cosmological constant can exhibit arbitrarily strong curvature. (e.g. The Schwarzschild solution is a vaccum solution).

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Hm, but calling the Schwarzchild solution a vacuum solution seems a bit strong to me: that damn singularity seems to me to be most exceptionally not the “nothing” of our vacuum.

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Dave,

> calling the Schwarzchild solution a vacuum solution seems a bit strong to me

but this is what it is.

One can use part of it to describe the gravitational collapse of matter or the collapse of gravitational waves (aka gravitons).

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By the way GR really challenges our common sense distinction between matter and information.

What exactly does the metric of a vacuum solution mean, measuring the distance between what?

(The official answer of course is that the metric measures the distances between arbitrarily small test particles; But the field equation does not need them.)

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Maybe I’m missing something but possibly we’re just stuck on terminology. I mean the Coulomb solution to Maxwell’s equations are vacuum solutions to these equations except at the location of the point charge. But I wouldn’t call this a vacuum solution because there is a locale where the charge is nonzero. Similarly, the Schwarzchild solution has stress energy tensor zero everywhere except at the singularity. I think we just have a different definition of vacuum: yours includes any solution which has stress-energy tensor vanishing over some region, whereas mine requires that the solution must have stress-energy tensor nonzero over the entire solution.

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Could you please elaborate on the seeming contradiction between ‘t Hooft’s idea that quantum mechanics can be explained by “hidden-variable” theories and Bell’s theorem which you say forbids it? Thanks!

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Please, everybody, the apostrophe should be before the “t”. Moreover, the apostrophe should be of the closing (right) kind. And, yes, annoyingly most editors automaticllay correct this to the wrong (left) kind. See http://www.phys.uu.nl/~thooft/ap.html.

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“tâ€™Hooft wants something better than quantum information theory, which however he hasnâ€™t studied very deeply.”

I don’t think ‘t Hooft says this, at least in this essay. He certainly acknowledges Bell’s theorem and understands that he needs a way around this, but certainly I don’t see how ‘t Hooft’s idea would end up destroying quantum information theory. Just because a theory is realistic doesn’t mean that it can’t give the power of quantum computers, right?

“And he wants something better than string theory, which he also hasnâ€™t studied very deeply.”

Well maybe. But not knowing the guy, I can’t say this is true. Also, I’ll note: these lecture notes for a course ‘t Hooft taught on string theory. Plus his work on 2+1 dimensional quantum gravity is pretty awesome, so, even if you think he doesn’t understand string theory you’ve got to at least acknowledge the guy knows something about quantum gravity (and yes, I know 2+1 is very different from 3+1, but still!)

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Dave,

There’s a typo in your link to ‘t Hooft’s article. (“hre” should be “href”). And sorry, but I can’t resist commenting on:

“he wants something better than string theory, which he also hasnâ€™t studied very deeply. Itâ€™s just too late for this kind of speculation.”

Isn’t string fanaticism a wonderful thing? Let’s see, ‘t Hooft is a Nobel prize winning particle theorist, generally acknowledged to be among the very few active geniuses in the field, and has taught a graduate course in string theory. Kuperberg knows virtually nothing about the subject (see his comments on other postings here and elsewhere), and yet feels quite comfortable publicly attacking ‘t Hooft’s professional competence and announcing that it is “too late” for alternatives to string theory. Absolutely amazing….

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Dave: You are right, I overstated it. Fair enough, ‘t Hooft gave a course on string theory and he has at least studied it more than I have. My comments were too ad hominem.

But still, I can only read his essay as saying that he wants something better than quantum information thery and something better than string theory. Not necessarily something that contradicts quantum information theory, but maybe something that refines it. Whether he wants to contradict string theory or refine it is less clear.

Somehow this opinion piece in Physics Today seems impatient to me. Whatever real wisdom ‘t Hooft himself has on this subject — which is a great deal if you count quantum field theory as related to both string theory and QIP — he’s not really using it in this essay. If someone wants to improve quantum information theory (with or without contradicting it), I would rather see a quantum information theory narrative that identifies some logical departure point to something else. Same for string theory.

For example, he might have wanted quaternionic quantum information theory. That idea has been tried and seems not to work, but at least it is a tangible proposal.

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Dave: Again, if I can step back from my initially ad hominem tone, I have at least studied 2+1-dimensional gravity some too. I don’t have any professional contributions and I don’t doubt that what ‘t Hooft has done here is very good. However, the real lesson of gravity is just how different it is in different dimensions. It’s not just that there are big differences; ultimately the differences are the thing to learn.

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> I mean the Coulomb solution to Maxwellâ€™s equations are vacuum solutions to these equations except at the location of the point charge.

The point charge lives along a timelike world line, the Schwarzschild singularity is spacelike.

If you imagine the singularity as some sort of ‘matter’, it would have some unusual properties.

The conventional point of view is that T=0 everywhere in the Schwarzschild solution and the singularity is strictly speaking not part of it.

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“If you imagine the singularity as some sort of â€˜matterâ€™, it would have some unusual properties”

I thought that Einstein, Infield, and Hoffman showed that the singularity behaves, at lest to lowest order, like a massive particle, i.e. the singularity would follow a geodesic.

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“Could you please elaborate on the seeming contradiction between â€˜t Hooftâ€™s idea that quantum mechanics can be explained by â€œhidden-variableâ€ theories and Bellâ€™s theorem which you say forbids it? Thanks! ”

Bell’s theorem only rules out a particularly nice set of hidden variable theories, so-called local hidden variable theories. Certainly ‘t Hooft will inevitably be led to messing around with something which is, at least at our effect energy scales, not local. There is nothing a priori which disallows us from thinking about such theories and certain models exist which do exactly this (from toy models to more serious replacements for quantum theory) but, in my opinion, all these results are not very satisfying (because they are either experimentally indistinguishable from quantum theory, or seem ad hoc and do nothing to help shed light on theory, by for example, helping us understand how to reconcile quantum theory with gravity. This later point is why ‘t Hooft’s idea is interesting, at least to me.)

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In a certain sense and approximation black holes move along geodesics. But the singualrity is spacelike (and does not appear in the EIH approximation).

By the way, I wrote something as a follow-up to our discussion here:

http://wbtsm.blogspot.com/2005/11/information.html

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In my little brain, the 2+1 dimensional quantum gravity work is among the coolest stuff I’ve learned about in the last few years, and I wish I had more time to think about it. But you’re right Greg: What a difference a dimension can make!

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I think I have a more or less clear idea what ‘tHooft is complaining about. The basic quantities one knows how to compute in string theory are S-matrix elements: you send some disturbance from infinity, wait a long time and calculate the probability for all possible eventual outcomes. This does not allow you to tell a local story involving particle trajectories, people falling into black holes and what not.

Some people take the attitude that string theory has to be extended (e.g using string field theory) and others take the attitude that local stories are to be given up, the same way that definite trajectories were given up in QM. To get a local description one has to have system of measuring device, heavy if they are to behave classically, so then they disturb the metric a lot…more mathematically, local quantities are not gauge invariant. Note that this argument refers only to QM and GR, not to string theory specifically.

So from this viewpoint this is similar to a classical physicist complaint: you guys have all these fancy Hilbert spaces, but you still cannot answer the simple question: which slit did the electron go through? this “magic” of the wavefunction is “deceit” etc etc.

‘tHooft of course knows all that, and he realizes that elementary considerations in GR and QM will prevent him from knowing “what is going on” to his satisfaction, so he is trying to modify both…mere mortals like myself, on the other hand, try to stick to the well-defined questions, to avoid getting really confused…

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The main point is that the answer to “Does God play dice?” has been given, and it is: yes! He plays them with Einstein!

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