Right after the mid 1920s, every physicist who was up to his or her job (OK, let's not be kidding, no woman really understood QM in the 1920s yet) knew that the idea that "the wave function was a real wave, like the electromagnetic wave" was the most naive kind of misconception about the character of quantum mechanics that a layman could have about quantum mechanics. This knowledge continued for many decades. The second quantum mechanical generation – including Feynman and pals – still understood the things perfectly but they already started to express the things in ways that reduced the negative reactions of the listeners.

Now, almost one century later, after a few decades of unlimited proliferation of pop-science books and completely wrong articles, the "unreal character of the wave function" became one of the most misunderstood basic facts about the natural science among the members of the broad public. Almost all the people were not only pushed to buy the completely wrong "the wave function is a real wave" thesis but this delusion has been turned into a moral imperative of its own kind. You should not only parrot such wrong statements: you should morally despise those who dare to point out that these statements are wrong.

Also, the writers who just can't live with the end of classical physics have not only written lots of wrong and confusing stuff about the physics questions themselves. They have also rewritten the history of physics. If a generic person tries to quickly enough find out what the Copenhagen Interpretation was or what Bohr and Heisenberg actually believed about quantum mechanics (and Dirac, Pauli, von Neumann, Wigner, and a few others), they get almost unavoidably drowned in amazing distortions, demagogy, and downright lies. The amount of mess, censorship, and misinformation about these elementary things already trump the chaos and censorship by the Inquisition of the Copernican ideas.

The motivation for almost all these distortions are ideological in character. It may look surprising that such a technical, almost mathematical point may be affected by ideologies – but it simply is affected a lot. In particular, lots of people realize that some kind of Marxism or another unscientific superstition that they hold dear did really assume classical physics which is why classical physics has to be "saved" from the questioning.

Quantum mechanics has completely new foundations that "almost precisely fill" the space that used to be occupied by the foundations of classical physics that had to be thrown away because they had been experimentally ruled out. The rules of quantum mechanics are qualitatively different from the classical ones and almost all critics completely underestimate the mental step that had to be done by the founding fathers of quantum mechanics – and that has to be reproduced by them if they at least want to follow what others could actually discover.

On the other hand, the general purpose of the foundations of quantum mechanics is basically the same as the purpose of the rules of classical physics – to predict and understand the outcomes of experiments in the physical world. And the observations of the physical world may still be phrased in terms of observables – but they are represented very differently in the mathematical formalisms of classical vs quantum physics. In this sense, "quantum mechanics is achieving a similar thing" in a completely unequivalent way than classical physics – a way that is more general, prettier, unavoidably probabilistic, and dependent on some mathematics, at least the complex linear algebra.

All the "wave function is real" people combine the misunderstanding of these two principles – about the differences and analogies between classical and quantum physics – and generally end up assuming that "quantum mechanics cannot be anything else than a new class of theories in classical physics" that deal with a new example of a classical wave, the wave function. But both parts of the statement are wrong separately. Quantum mechanics is really the process of answering the "old kinds of questions" – but using a completely new, non-classical methodology and rules.

Bell got rather famous for no good reason – for finding an explicit example of a situation in which a classical theory and quantum mechanics differ. But this whole observation that "they differ" is about as revolutionary as the observation that creationism and Darwin's theory differ. Or that a donkey differs from a Mercedes car. You can describe the shape of donkey's limbs and the car's wheels in some detail to "rigorously prove" that they're not the same thing. But do you really need it? If your brain is alright, the difference between the donkey and a car is self-evident. You don't need to prove it. And if you outline a specific proof, there is absolutely nothing canonical or important about such a proof. Instead of the round vs linear shape of the wheels and limbs, you could focus on the different greenhouse gases that donkeys and (proper) cars produce. Or look at thousands of other differences.

Among these thousands of differences, one may notice that quantum mechanics allows us to interpolate between a "Yes" answer and a "No" answer really continuously. Let's study a qubit – the electron's spin – and ask whether the spin projection \(J_x\) is positive ("up" along the \(x\)-axis). If it is, the wave function is\[

\ket\psi_{\rm Yes} = \pmatrix{ 1 \\ 1 }

\] and if the answer is No, the spin is "down" along that axis, the wave function is the orthogonal\[

\ket\psi_{\rm No} = \pmatrix{ 1 \\ -1 }.

\] That's great. Yup, the complex overall normalization factor doesn't physically matter. But the relative phase between the two amplitudes does matter and may be changed gradually\[

\ket\psi_{K} = \pmatrix{ 1 \\ \exp(\pi i K/N) }.

\] where \(K=0,1,2,\dots , N\). The relative phase may just change from \(0\) to \(\pi\) continuously, or in small steps \(\pi / N\). The physical interpretation is simple: the spin is "up" along an axis in the \(xy\)-plane that gradually moves from the positive \(x\)-semiaxis to the negative one, through the \(y\)-semiaxis etc.

All the intermediate states are of the same character – they are related by a simple rotation around the \(z\)-axis, after all. You may already feel that the classical probability distributions can't really achieve such a thing easily. In fact, you may see that in classical physics, if the intermediate states may be represented at all, they must be strictly distinguishable for each value of the \(K\).

Just think about the two wave functions for the spin that slightly differ in the relative phase. If the wave function were a classical wave – a part of some information that objectively exists, without any need to discuss observers – then these two states could be in principle experimentally distinguished by a measurement.

Imagine you have some classical field, like the temperature \(T(x,y,z)\). The you may ask whether the temperature is greater at point A or point B. Connect the two points by some tube and if the gas will drift from A to be, then A is hotter than B, or vice versa. Or something like that. If some properties of the classical degrees of freedom objectively exist, there should better exist a way to distinguish them, right?

But the point is that the states \(\ket\psi_K\) and \(\ket\psi_{K+1}\) for a large value of \(N\) are almost completely indistinguishable. If you make any measurement on these two states, you are extremely likely to get the same outcome for both states. Quantum mechanics predicts this conclusion because it may only predict probabilities and those are always continuous functions of the states. If the state – the relative phase – changes just a little bit, so will *all* the probabilities. And nothing else than probabilities of outcomes is implied by the theory. It means that there's no way how the probability of some outcome of a measurement could jump from near 0% to near 100%. The probabilities are really "squared cosines" of some angles and all the relevant angles only change by a tiny amount between \(K\) and \(K+1\).

So if you assume that quantum mechanics is correct, then the nearby non-orthogonal states are "almost the same" for all physical purposes. That's very different from two different classical waves that may always be in principle reliably distinguished with a fine enough apparatus.

OK, quantum mechanics predicts that no such "fine enough apparatus" may exist. Whatever you do in a single measurement of the electron's spin, you can't reliably – and not even "almost reliably" – distinguish the states \(\ket\psi_K\) and \(\ket\psi_{K+1}\) for a large value of \(N\). If you want to prove that quantum mechanics is wrong and the wave function is "real", you should better start building your "fine enough apparatus" that can at least partially reliably distinguish the nearby, non-orthogonal wave functions.

Of course you will fail.

The normal concise proof that the "wave function is not real" looks at some entanglement experiment and describe it from the viewpoint of several inertial frames, as understood in the special theory of relativity. If the wave function is real, the measurement must cause some collapse, and this collapse occurs at "one moment". But relativity indicates that "simultaneously" is only well-defined with respect to a particular inertial frame. If such a preferred frame exists, the Lorentz invariance and special relativity will be broken. If the wave function were real, it would be in principle observable, so it would almost certainly have some observable consequences that betray that relativity doesn't work in Nature. But relativity has passed all tests and the probability that a fundamentally non-relativistic theory makes this prediction is tiny and basically zero. If the non-relativistic fundamental effects were able to show up in principle, they would have almost certainly showed up already.

But even if you don't understand – or, assuming some unjustified self-confidence of yours, don't "agree" with – similar "relativistic" proofs that the "wave function cannot be real", there exists a simpler, philosophically standard way of arguing, namely Occam's razor:

Non sunt multiplicanda entia sine necessitate.Concepts or entities just shouldn't proliferate unless it's necessary. What this principle says is that you should prefer "minimalistic" theories of the Universe – in the sense of theories that need to assume a smaller number of objects that really exist.

If you agree with me that the "fine enough apparatus" that may reliably distinguish between nearby wave functions hasn't been constructed or seen yet, then there's no reason to think that the "objectively real wave function" is one of the measurable entities that objectively exist. And because the existence of this "objectively real wave function" isn't necessary to understand or predict any observations, you shouldn't assume that it exists.

That's why the assumption that the "wave function is just a template to calculate probabilities of some other, actually measurable properties" (like whether the projection \(J_x\) of the spin is positive) is the answer preferred by Occam's razor. To supplement your picture of the world with the "objectively real wave function" not only breaks the rules of relativity. It's also redundant and unnecessary. It's exactly as redundant and unnecessary as if you assume that the vacuum is filled with aether built out of wheels and gears. Or that planets are being pushed by little angels with beautiful bodies who are attached from all sides. (As Feynman said, Newton has figured out that these angels weren't pushing the planets from the back, instead, they were pushing the planets inwards radially LOL.)

Assuming an "objectively real wave function" is on par with the superstitious assumptions that "something that cannot be seen" exists – aether, angels, ghosts, phlogiston, dangerous climate change, or any other superstition you can think of. Until you create a gadget that can distinguish the nearby wave functions with a reliability above 50% (and by this gadget, you would also prove that Einstein was completely wrong about relativity), you should realize it's just plain irrational and unscientific to ascribe the "objective existence" to the wave function.

The wave function cannot be measured (its tiny changes cannot be distinguished by any apparatus that studies the physical system once) which is a good reason to say that "it probably doesn't objectively exist".

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