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I've always hit two big stumbling blocks in conceiving of the proof or disproof of hidden variable theories as being even valid idea, let alone an answerable question... I feel I must be misunderstanding some very fundamental things.

  1. Quantum mechanics is deterministic, ignoring the bit where we take our complex number and fudge it into a probability.

  2. We use computers to make calculations to provide the predictions of quantum mechanics - how is this not itself a hidden variable theory?

From what I understand the famous Bell theorem and work around it doesn't disprove hidden variable theories, just one specific flavour of them, and even then I am not convinced. I feel like I must be missing something again, because this result seems to me in no way profound... although it certainly fits into the picture of the EPR 'paradox' and is a good demonstration of the validity of QM.

I guess the crux of my problem is that this 'area' really addresses wave function collapse and the problems it causes, but I don't see the need for some collapse mechanism to begin with... again I feel like I am missing something really fundamental.

Apologies if this is a bad question. I have had a look around and there are similar questions, but I don't feel that either of my points are addressed...

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1) The fact that the complex amplitudes are templates to calculate probabilities isn't a "fudge" in any way. It's not only the reason behind the 1954 physics Nobel prize but also one of the most important, fundamental, and universally valid principles of science. If you pre-decide to add indefensible childish negative adjectives in front of similar paramount principles, you're maneuvering yourself into the inevitable outcome that you will never understand the foundations of modern physics. And that's a great pity because modern physics is amazing. –  Luboš Motl Feb 25 '13 at 15:31
2) We may use computers to simulate the evolution of probability amplitudes but we may still distinguish the reality from the simulation. For example, the simulation explicitly violates the Lorentz symmetry - special relativity - and one only restores the agreement with the exactly Lorentz-invariant reality by fine-tuning infinitely many parameters of the simulation infinitely accurately, by imposing principles that constrain the predictions of our product even though the product fundamentally disagrees with these principles. –  Luboš Motl Feb 25 '13 at 15:33
Quite generally, I am always amazed by the comments e.g. that it's a "fudge" to correctly interpret the wave function. The fact that the wave function must be interpreted as the probability (density), up to the squaring, isn't an optional luxury one could remove. It's a complete essence of the whole quantum framework and it may be directly experimentally verified to hold. If someone gave a different "interpretation" to $\psi$, he would immediately run into conflicts with observations just like if someone interpreted $U$ in $U=RI$ Ohm's law as the number of time a woman has had sex. Just wrong. –  Luboš Motl Feb 25 '13 at 15:37
Don't feel bad if you don't understand these principles right away. They inspired more than a decade of debate among some very bright minds, and are part of what makes QM challenging. While @Lubos is right that negatively labeling these principles does you no favors in understanding them, neither does providing condescending examples of miscalculation do any service to teaching the subject. –  KDN Feb 25 '13 at 15:47
I don't disagree that we haven't learned a lot through ridiculous examples, and even through the condescending guidance of our professors and peers. I am not convinced, however, that because we learned it that way, it is the best way to teach. Calling the asker "childish" and pitiable for following an intuitively appealing line of questioning is probably not helpful. Moreover, the example you give is disenfranchising to women, which does nothing to combat a rather salient problem in science and engineering, and physics in particular. –  KDN Feb 25 '13 at 16:39
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4 Answers

Jherico, I see that you are keen in finding answers to your questions, or putting your views across for a debate, and this is really good. This is what science is all about. I think your questions deserve attention and proper debate.

Here is an effort from my side to help dilute some of the misunderstanding through the comments section of this forum.

(1) Your opening statement “Quantum mechanics is deterministic …”

The deterministic nature of QM is only in relation to the inherent structure of Schrodinger equation, which gives us the ability to predict the *probability for an event to take place. We can only predetermine probabilities, and this does not make QM deterministic in any way. Even if we had the most accurate Hamiltonian or Lagrangian (whatever we want to call it) we would still be calculating probabilities.

The probabilistic nature of QM is related to the fact that nature always has numerous options available to choose from, when she does something and, amazingly she picks just the right amount from each option and does a perfect job out of it!! ADMIRABLE + FASCINATING!!

(2) “We use computers …”

This has been answered by @Lubos Motl very well. I will only add the following: The use of computers to solve the complex problems in the quantum world is not a compromise over the richness of subtleties of QM. The algorithms used are of purely mathematical nature and only help us with the finding solutions to the complex mathematics involved in our problems. The hidden variables you are referring to probably relate to errors propagating and accumulating, and therefore might obscure the accuracy of the answers we obtain. But that is a purely computational problem and has nothing to do with the hidden variables physicists are referring to when they talk about hidden variables.

I hope I have cleared up some of the misunderstanding, and please do keep in touch with physics. You will find it is one of the most fulfilling enterprises undertaken by mankind.

Some discussion on Bell’s inequalities can be found in several places in this forum, but if you wish more detail you could try this book:

Speakable and Unspeakable in Quantum Mechanics (Any new edition)

Cambridge University Press

John S Bell;

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Thanks for the reply. I have continued to do research after being frustrated with the comments etc. I think I am learning something... –  jheriko Feb 25 '13 at 23:48
So, to address the first issue, I like your answer, but I feel it glosses over a point that I fail to explain in enough detail, and perhaps I use bad terminology... You say "We can only predetermine probabilities, and this does not make QM deterministic in any way." - I think that the model is deterministic in terms of the complex valued wave function and that the conversion into probability by taking the norm is the operation which makes it probabilistic, and correspond with our perceptions of the universe. The fact that it is empirically proven and not explained, for me, misses something... –  jheriko Feb 25 '13 at 23:55
Thanks for confirming that my use of 'hidden local variables' is incorrect - I've always construed this to mean something more like something computable in a deterministic fashion. I should stop abusing the comment system now... :) –  jheriko Feb 25 '13 at 23:59
@jheriko Pleasure. Good you keep researching the question. I begin to see the cause of your confusion over probabilities. Probability in QM is not based on the fact that the WF happens to be CMPLX valued, but is not always the case. Many non-QM quantities in physics are represented by CMPLX numbers, but they are not probabilistic. For example, the current and Impedance in an AC electric circuit with R and L or C, are CMPLX numbers. Most important, the WF for a particle in a box is real-valued, but we still calculate probabilities with it! It is the only sensible think to do. Thats nature. –  JKL Feb 26 '13 at 0:58
Cool. I'd like to understand this better... but i'm struggling to find good reference material. Perhaps spending a lot more time calculating things will give me more insight. :) –  jheriko Feb 28 '13 at 16:16
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I answer my own question only because the original answer provided doesn't directly answer it... although it lead me to the correct interpretations.

So to address my points...

  1. We can absolutely replace QM with a deterministic theory and get the same predictions, we don't actually have to stop before we turn our wave functions into probabilities either as the original question suggests...

  2. Local hidden variable theory does not simply mean 'an underlying deterministic theory' - it is meant to imply a concept referred to as 'local realism'. The computer models we use do not include 'local realism' as a constraint - importantly we can show that if they did they would be inaccurate.

What Bell's theorem shows us is that a fully deterministic theory with these properties of 'local realism' does not agree with experiment. This looks like a very special and weird case to consider without context - but previously the idea of 'local realism' was held in high regard.

(Please do correct me further if you can - particularly I can not yet agree with the comment 2) from Lubos Motl - I do not see how merely being a simulation will break Lorentz invariance - more importantly I can construct simulations where I can rotate the universe or make time go backwards without altering the algorithm - making it evolve forwards in time is a combination of choosing a simple integration strategy and providing visual feedback which is intuitive to grasp - without changing the underlying laws I can make it step 'diagonally' along some arbitrary 4-vector to produce some arbitrary foliation and the results are the same up-to approximation errors (rather than fundamental errors))

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Unfortunately I just joined so it seems my low reputation doesn't allow me to simply reply in a comment.

I should also say this answer is a reply to your answer.

I'm somewhat confused by your first point, and I think that it may be claiming something that is false, though I might just be misinterpreting it; namely the claim that QM can be replaced by a "deterministic theory". If what you mean by the former is a model where certain input variables determine the behavior of a system deterministically (for instance, with differential equations relating the input and output variables as in classical mechanics), then you are mistaken. That is exactly what Bell's theorem (inequality) showed. If you haven't already, I would suggest pulling up a copy of Bell's original paper and trying to follow it. I haven't read it myself, but I was told it's quite readable.

To correct the (potential) error you asserted, a clarification: when people say QM is deterministic, what they mean is that the amplitudes (variously called the wave-function, ket vector, state vector) evolve deterministically according to the Schrodinger equation. However, the interpretation of QM states that the information that this deterministically determined amplitude tells us about reality must be interpreted statistically, or probabilistically, and hence non-deterministically.

As a minor addendum, I would guess what Lubos meant is that to make most calculations regarding Lorentz invariant "things", one usually chooses a particular coordinate or inertial system, thus breaking the Lorentz invariance of your quantities.

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"If what you mean by the former is a model where certain input variables determine the behavior of a system deterministically... then you are mistaken." Sorry but you are mistaken. Bohmian mechanics is an explicit counter-example to this common, though incorrect, claim. It is a fully deterministic theory that agrees exactly with ordinary quantum mechanics and therefore agrees with experiment as well. The feature Bohmian mechanics does not have is locality, which is the loophole in Bell's theorem. Bell's theorem is more subtle than you say - I suggest you read the paper yourself. :) –  Michael Brown Mar 12 '13 at 6:14
@MichaelBrown There is what is called an "occams razor " when discussing theories, the simplest one holds. Bohm's QM is a more complicated mathematical way of describing what QM is doing very efficiently. Bohm's QM is an exercise in mathematical complexity, as it does not predict/offer any experimental tests that would validate it and falsify ordinary QM. –  anna v Mar 12 '13 at 7:15
@annav Bohmian mechanics is barely more complicated than ordinary QM, and while it is indistinguishable from QM it suggests ways of modifying QM that could be testable in the future. Also, Ockham's razor is a guideline, not a rule. And what is "simple" to us might be complicated to a future generation and vice versa. I love the chapter of Misner Thorne & Wheeler where they write Newton's gravity in tensor language... which makes GR look positively simple in comparison. So I'm fairly agnostic about what is "simple." –  Michael Brown Mar 12 '13 at 7:31
@annav (cont.) Anyway, I'm hardly a Bohmian partisan. I just use it as a counterexample to the many wrong claims made about QM. For this purpose the fact that it exists is enough. It doesn't matter if it isn't the best interpretation of QM. :) –  Michael Brown Mar 12 '13 at 7:31
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To this date there is no valid argument against the existence of deterministic, local, hidden-variable theories. Bell's theorem and its modifications only deal with non-deterministic theories because they require non-determinism (often encountered as "free-will") as their fundamental assumption.

The possibility of such theories is accepted by John Bell and also by the authors of the so-called "free-will theorem".

Until someone proposes such a theory, or a valid no-go theorem is demonstrated, there is no way to know where the truth is. QM might be fundamental or just a statistical approximation.

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