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?
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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?
G. 't Hooft
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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.
Arnold Neumaier
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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.
Ionut Bocan
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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.
