I am perplexed by recent papers by 't Hooft giving an explicit construction for an underlying deterministic theory based on integers that is indistinguishable from quantum mechanics at experimentally accessible scales. Does it mean that it is deterministic complexity masquerading as quantum randomness?



I think at least some readers should have noted by now that many of these arguments, particularly the more pathetic ones, are questions of wording rather than physics. Once you made your model simple enough, you can map anything onto anything. Now this was my starting point: if a system is sufficiently trivial, you can do anything you like. Now how can we subsequently generalize some such very simple results into something more interesting?

This has been the ground rule of my approach. I am not interested at all in "no-go" theorems, I am interested in the question "what can one do instead?" I admit that I cannot solve the problems of the universe, I haven't found the Theory of Everything. Instead of pathetically announcing what you shouldn't do, I try to construct models, step by step.

I now think I have produced some models that are worth being discussed. They may perhaps not yet be big and complicated enough to describe our universe, but it may put our questions concerning the distinctions between quantum mechanics and classical theories in some new perspective. Clearly, if a system is too simple, this distinction disappears. But how far can one go? Remember that cellular automata can become tremendously complex, and quantum mechanical models also. How far can we go relating the two? This is how you should look at my papers. I happen to think that the question is very important, and one can go a lot further in relating quantum models to classical ones than some people want us to believe.

And is a calculation wrong if someone doesn't like the wording?

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  • $\begingroup$ I agree that the models are interesting, and the program is important, but the main problem is that you have not shown that quantum mechanics in a beable basis is equivalent to classical statistics on a CA, and this might not be true (I think it isn't true without further restrictions). I will ask a direct question, as your interesting answers and comments have allowed one to sharpen one's thinking about this. $\endgroup$ – Ron Maimon Aug 14 '12 at 13:21
  • $\begingroup$ here it is physics.stackexchange.com/q/34165/3229 $\endgroup$ – Curious George Aug 14 '12 at 17:29

Current (experimental and theoretical) wisdom on deterministic approaches to quantum nondeterminism just say that any deterministc theory underlying quantum mechanics must be nonlocal. Research then goes on discussing the precise nature of this nonlocalness or ruling out certain versions.

On the other hand, there are those who construct nonlocal deterministic theories that somehow reduce to QM. A lot of work goes into Bohmian mechanics, which however has difficulties to recover realistic quantum field theory.

The paper by t'Hooft pursues a different approach, based on discreteness. However, his results are currently very limited, just reproducing the harmonic oscillator.

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  • $\begingroup$ Your first phrase may be too categorical. For instance, it is my understanding that there has been no loophole-free experimental evidence of nonlocality. On the other hand, nonlocality is an extraordinary claim, so it needs an extraordinary proof. $\endgroup$ – akhmeteli Jul 22 '12 at 1:53
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    $\begingroup$ t'Hooft doesn't "reproduce" anything--- he has a linear space which is a quantum state space, he just truncates the evolution to be deterministic on a special basis. The actual dynamics is still quantum, it isn't really an automaton--- you can't imagine that the states are classical underneath. $\endgroup$ – Ron Maimon Jul 22 '12 at 9:03
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    $\begingroup$ @a25bedc5-3d09-41b8-82fb-ea6c353d75ae: (Please choose a better name for identification) - I haven't looked at the details as my impression about all these attempts is that they are dead ends; lots of technicalities, but in the ned nothing that adds understanding to what we know already. If quantum indeterminism is generated in Nature from an underlying discrete classical dynamics, I bet it is an elegant way, just like quantum mechanics is an elegant variation of classical mechnaics, not a messy one. $\endgroup$ – Arnold Neumaier Jul 22 '12 at 17:44
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    $\begingroup$ @akhmeteli: Particle nonlocality is an ordinary experimental fact (already visible in interference experiments); it is only the precise form it takes in theory that is in doubt. So I don't agree that it needs extraordinary verification. The philosophical problems go away once one gives up the particle picture; then there is nothing counterintuitive left that would need extraordinary attention. $\endgroup$ – Arnold Neumaier Jul 22 '12 at 17:48
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    $\begingroup$ @akhmeteli: Geometric optics is the classical particle view of classical light, and has all the nonlocal features that appear in quantum mechanics, once you go beyond the realm of the validity of the approximation. This tells me (though it is not the mainstream point of view) that the root of quantum nonlocality is treating quantum objects as particles rather than as field excitations. In the context of relativistic quantum field theory, everything is local, showing that the weird aspects are nothing but a poor choice of intuitive visualization. $\endgroup$ – Arnold Neumaier Jul 23 '12 at 10:47

It is certainly possible that QM is based on a deterministic physical mechanism. The no-go theorems like Bell's theorem or the "Free will theorem" of Conway and Kochen are not effective against deterministic hidden variable theories because they require non-determinism as one of their assumptions. There are still many phisicist claiming that determinism has been disproven but they are commiting the logical fallacy . However, it is too early to say if 't Hooft is on the right track.

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    $\begingroup$ -1: This is incorrect. The Bell theorem is effective against deterministic hidden variable theories which are local. The argument from superdeterminism is ridiculous, it is not science, it is conspiracy theory. $\endgroup$ – Ron Maimon Jul 22 '12 at 9:02

t'Hooft's papers are not valid. They make a mistake, which is that they assume that just because the discreate time evolution operator in a quantum system is a permutation in some basis, that the quantum theory is then a classical theory.

t'Hooft considers discrete time quantum systems where the time-evolution in some basis is a discrete permutation. So that if you have a 3 state system you permute 1 to 2 to 3. He then analyzes the space of all superpositions of these three states, and discovers he can recover quantum mechanics. He then declares "quantum mechanics is equivalent to a classical determnistic system".

This is just plain wrong. I suppose t'Hooft is thinking that if you start out in some basis state, you stay in a basis state forever, just permuting the basis state, and therefore this must be a classical deterministic system. But the point is that the state space includes all sorts of quantum superpositions of the basis states, and these other states, the non-basis states, are superpositions not by classical probability, but by quantum amplitudes.

If you have quantum amplitudes, even if the basis states evolve by permutation, the theory can obviously reproduce quantum mechanics, because it is quantum mechanics.

In fact, here is a theorem: Given any finite dimensional quantum mechanical Hamiltonian H, there exists a permutation system which includes this Hamiltonian in an approximation, acting on a subspace of the states.

The proof: diagonalize H to an N by N diagonal matrix with N eigenvalues, and approximate the N energies by rational numbers with enormous prime denominators, $p_i/q_i$ $1<i<N$, and take a unit time-step. Multiply all the q_i's together and call the product Q. Then the exponential of t times the Hamitlonian is periodic with period Q time steps.

Consider now a state-space whose basis is labelled by an N-tuple integers from 1 to Q. Let the permutation Hamiltonian take the basis element (a_1,....a_n) to $a_i\rightarrow a_i + s_i$ where $s_i$ is the product of all the q's except q_i, and the $Z_{q_i}$ multiplicative inverse of $p_i$. This permutation Hamiltonian has to property that it's eigenvalues include a subset with $p_i/q_i$. Project to this subspace, and call this your quantum system.

This process, or anything resembling it, cannot be called a "deterministic system" in any way. There are still states which are superpositions. If you have a true classical system, the state is described by a probability distribution on the unknown starting state, not by probability amplitudes for superpositions of the unknown current state. The moment you describe states by superpositions, you are not getting quantum mechanics out, you are putting it in.

This is the reason t'Hooft is able to derive mathematical results that are quantum mechanical, he is using quantum mechanics, but with a restriction that it reduces to a permutation on one basis. This doesn't explain why we see superpositions of electronic spins in nature, it doesn't produce these superpositions from ignorance of classical values, it puts in the superpositions by hand.

I like t'Hooft's motivations and admire his independent thinking, but this is not valid stuff. It doesn't do what he claims it does. To call the statement that these are classical models misleading is charitable.

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  • $\begingroup$ 't Hooft thinks there is an ontologically preferred basis and superpositions of those basis states are not real. The Bohmian treatment of measurement interactions shows that you can have a preferred basis and still describe other observables correctly - though the Bohmians also still have superpositions in their ontology, as states of the pilot wave (unless they take the "nomological" path, and treat the specific pilot wave of an individual system as a dynamical law rather than a physical state)... $\endgroup$ – Mitchell Porter Jul 22 '12 at 9:59
  • $\begingroup$ @MitchellPorter: I know what he says, but this is ridiculous--- you have to tell me how come a laboratory electron is described by a superposition. It doesn't help to say that there is a basis in which the Hamiltonian is a permutation. If we don't know which basis element out universe is in, we describe that with a probability distribution, not with amplitudes. Then there is no reason that the electron in the lab is described with amplitudes. It is just wrong. $\endgroup$ – Ron Maimon Jul 22 '12 at 10:01
  • $\begingroup$ In Bohmian mechanics with a specific pilot wave, you can substitute the specific pilot wave into the equations of motion for the classical objects of the theory, and you end up with a pseudo-classical theory in which a classical equation of motion is augmented by a nonlocal potential. It must be possible to explain superposition in terms of this nonlocal potential, because the theory is still identical to Bohmian mechanics with a specific pilot wave, but no-one has ever taken this route and exhibited what such explanations look like... $\endgroup$ – Mitchell Porter Jul 22 '12 at 10:19
  • $\begingroup$ In the paper after this one, 't Hooft constructs his alleged mapping from a CA to a QFT. The CA is really simple but the mapping is a little nontrivial; at least, I haven't grasped the essence of it yet. It will be hard to say anything concrete about how or even whether 't Hooft can account for observed superpositions, until someone understands this further stage of his recent work. $\endgroup$ – Mitchell Porter Jul 22 '12 at 10:21
  • $\begingroup$ @MitchellPorter: It took me a long time to understand it, because it is clearly wrong, and I tried to make a true map from CA to QM. What he is doing is what I described--- he takes a QM system and transforms it to a case where it turns into a permutation on a basis, and when he can do this (which is always) he declares he has gotten QM out of a classical automaton. The declaration is false, the method is producing an "t'Hooft quantum automaton" not a classical automaton since it includes superposition states a-priori. $\endgroup$ – Ron Maimon Jul 22 '12 at 19:18

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