Tuesday, July 30, 2019

There's no measurement problem

...just carefully physically define the wave function to see why...

Wikipedia describes the (non-existent) measurement problem as follows:
The measurement problem in quantum mechanics is the problem of how (or whether) wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.
These sentences are very representative of the confusion of the real-world people who repeat the totally incorrect statement that there is something wrong, illogical, inconsistent, incomplete, or scientifically unsatisfactory about the basic postulates of quantum mechanics.

OK, so the first sentence conveys the message that these people don't understand "the collapse of the wave function". On the other hand, the second sentence pretty much answers the would-be question in the first question – the collapse cannot be observed, at least not in detail, as a sequence of steps – and it postulates that people must be split to many "interpretations", like in a multicultural society, and answer all the questions differently depending on their camp.



This is not how science works. Science always looks for the correct, winning theory, as measured by its ability to predict or explain the observations, and when this one is found, the other ones are abandoned. The plan that one should persistently maintain many unequivalent "interpretations" that should be treated on par is already a fundamental violation of the very purpose of science – and the purpose is to discriminate various assertions about Nature.

If a question cannot be answered even in principle, then it's an unscientific question and the scientists shouldn't study it at all. If the question may be answered in principle, then the scientist must define progress as a path towards the answer to this question – or similar questions.



In science, we need to define the increasingly good questions – and provide them with increasingly correct and accurate answers – instead of uncontrollably growing pseudointellectual garbage such as the texts by those who believe that quantum mechanics has a "measurement problem". OK, what are the questions here? As indicated above, the key question of this "measurement problem" is:
What happens during the collapse of the wave function?
To one extent or another, it is this question and only this question that defines the "measurement problem".

That's a good question but we can't automatically assume that it is a "problem" in the sense of "something bad". It is just another question, just like "Is the Earth flat?", which can be answered after some thinking and scientific research – assuming that the question is scientifically meaningful. Now, a trivial observation but a key observation is the following:
When a scientist wants to seriously discuss a question or a proposition, he should have a definition of all the terms that the question or the proposition refers to.
In other words, if you don't know what you're talking about at all, then your talk is a sequence of meaningless sounds and it's too bad – it is surely not good science. Great, what are the words that we should define and understand here? It's primarily:
the wave function
and perhaps: the collapse
I sincerely hope you are still with me. You just need to give me a definition of the wave function for your question about the wave function to have any meaning, right? I suspect that this elementary observation is already a problem for the critics of quantum mechanics – and perhaps almost all critics of anything today, especially the left-wing ones (well, there are no right-wing critics) – because they often seem to assume that when they ask a hostile question, to attack a theory or a victim, it's already the victim's duty to find the definitions of all the words that were used in the critical question.

Well, among rational people, it just isn't so. If you ask a question or articulate a critical proposition but you don't know what the words in this question or proposition mean, then you haven't provided a valid criticism of anything. You have just proven that you are a worthless and obnoxious whiner, babbler, an unproductive heckler, a pile of junk that is naturally ignored by genuine scientists. Sadly, our culture is ceasing to respect these basic rules of the game and critics – including MeToo accusers – are supposed to be "right" even when they talk absolute garbage.

Fine. So let me assume that you agree that we need a definition of the wave function for anything in this discussion to make any sense at all. Approximately speaking, "what is the right definition of the wave function" may be considered equivalent to the "measurement problem" or the choice of the right "interpretation", too.

What is the definition of the wave function? You might say that it is some complex-valued function\[

\psi(x,y,z,t)\in\CC

\] that exists in the mathematical apparatus of the theory and that depends on space and time. More generally, it is an element of a Hilbert space \(\ket{\psi(t)}\). The typical "interpreters" may also add that it is subject to Schrödinger's equation. Fine. We have just heard some description. It is a complex function. But the defect of this "definition" is that
it is just some qualitative talk, and if it can be considered a definition at all, it is just a definition of a mathematical concept.
Instead, we need a physical definition!
I hope that you are still with me – because I just made another small step for a man but a big one for mankind. ;-) If we are talking about physics or natural science in general, we just can't talk merely about mathematical definitions of notions somewhere in the "Platonic realm of mathematical ideas". Mathematical axioms aren't natural science yet, are they? Instead, we need a physical definition. And what I really meant by a definition of \(\psi(x,y,z,t)\) wasn't just "any valid sentence" that may be said about this object. Instead, I meant a procedure to connect the actual value of this complex function with some observations.

Without this quantitative connection, we don't really have a physical definition – and we can't discuss any physics involving this term!



For an easier example of physical definitions, consider a sentence about "voltage" in classical electrodynamics. What is the definition? Well, we may surely use an operational definition of voltage. It's the number that a voltmeter shows when you connect the two wires to the two relevant places. A guide to build your voltmeter may be a part of the definition. Any consumer complaints about the guide may be solved by making it more detailed.

At this moment, with the reference to a voltmeter and with the voltmeter itself, the term "voltage" becomes physically meaningful. If you didn't have any operational definition of this sort, it would be completely legitimate for everyone else to ignore all sentences with the word "voltage" because they would look like some undefined stuff. Well, the concept of voltage could be "real" even in the absence of voltmeters but we wouldn't know that it's real and the complaint involving the word "voltage" would have just a chance of being a problem, not a known problem.

To summarize, it is always legitimate to ignore criticisms – and other sentences – that use terms that seem ill-defined.

Measuring the wave function?

What is the corresponding gadget that shows you the value of \(\psi(x,y,z,t)\) – the wave function of the electron at some spacetime point that you specify to the gadget? That is a really good question. We also know the answer, at least everyone who actually understands quantum mechanics – and the body of evidence that underlies it – knows the answer.

The answer is that \(\psi(x,y,z,t)\) isn't observable. In the clever "Copenhagen" terminology, we may equivalently say that \(\psi(x,y,z,t)\) is not an observable. All observables are expressed by linear Hermitian operators acting on the Hilbert space. But \(\psi(x,y,z,t)\) isn't such an operator. Instead, it is an element of the Hilbert space itself.

So there just can't be any "voltmeter" that would show you \(\psi(x,y,z,t)\) on the display in a single repetition of an experiment involving one electron. How do I know it? As a theorist, I may say that I know it because it directly follows from the axioms (universal postulates) of quantum mechanics. Those say that the data about the physical system may only be obtained by a measurement of observables which are operators on the Hilbert space. Because \(\psi(x,y,z,t)\) isn't an operator, it cannot be measured.

How do I know that the universal postulates of quantum mechanics are right? Well, they agree with all the experiments and the agreement looks highly nontrivial. But at the same moment, in science, we can't never definitively and rigorously prove a theory or statement correct. Instead, we may only be certain when we disprove or rule out a theory or a proposition.

It means that in reality, I know that "\(\psi(x,y,z,t)\) cannot be measured" because I can disprove various proposed methods to measure the wave function. In particular, it is pretty clear that the overall complex phase (or perhaps the scaling) of the wave function is unphysical and therefore unmeasurable. But even the absolute value \(|\psi(x,y,z,t)|^2\) is impossible to measure. Everyone agrees that this number has something to do with some information about the electron's position. But whenever you try to find something accurate about the position, the electron appears at a particular place \((x,y,z)\), and by doing so, it surely doesn't communicate any complex number \(\psi(x,y,z,t)\) or its absolute value to anybody.

However, the value of \(|\psi(x,y,z,t)|^2\) may be measured if you repeat the same experiment with the electron very many times. The repeated measurements of the position end up differently, you draw the histograms, and reconstruct the probability distribution for the position. Similarly, you may measure the momentum many times and reconstruct \(|\tilde\psi(p_x,p_y,p_z,t)|^2\). In combination, these two real non-negative functions are enough to reconstruct the whole \(\psi(x,y,z,t)\) up to an overall phase.

I hope you are still with me because I made another small step for a man, a big one for mankind. The step has produced the insight that
it is unquestionable that any physical or operational definition of the wave function produces the answer that the wave function is a template for the probability distributions describing the odds of various properties of the electron (or a general physical system).
Do we really know it? Again, we can't rigorously prove it because physics isn't mathematics and we can't ever prove anything in physics rigorously. But we can prove it in the same sense as the statement that "the Earth is round". We just carefully look and the Earth looks round. And the wave function looks like a complex template for probability distributions. You may always deny the conclusion and insist that the Earth is flat or the wave function isn't probabilistic. But every single observation of the Earth at the scale of thousands of kilometers or more ends up with the "round Earth" answer; and every single physically successful application of the wave function ultimately confirms or needs to assume that the wave function has a probabilistic meaning.

Great. So the wave function is defined as a set of complex numbers so that various sesquilinear objects of the type \(|\psi(x,y,z,t)|^2\) describe the probabilities or probability densities of various values of assorted properties of the electron (or any physical system).

In general, even in classical physics, probabilities may only be quantitatively measured if you repeat experiments many times. Before each single measurement, we can only guess the answer because it's random – and the correct interpretation of the probabilities such as \(|\psi(x,y,z,t)|^2\) is that of a subjective probability. This fact is an unavoidable property of probabilities in general. It doesn't depend on any specific features of quantum physics or any other physics. Unless you have repeated experiments many times, the value of the probability is Bayesian or subjective.

All these statements – following my small steps – have really been proven to a similar extent to which it has been proven that the Earth isn't flat. It is comparably dumb to question both.

Now, once you appreciate that it is easily established in science that the wave function is a template for probabilities, and (like every probability) it must therefore be subjective or Bayesian within a single repetition of an experiment – because the alternative theory in which the wave function isn't probabilistic or in which the results of individual experiments aren't random is literally incompatible with all experiments – you must realize that we have also answered the original question
What happens during the collapse of the wave function?
The answer is that – because the wave function is a complex template for subjective probabilities – what changes during the wave function collapse is nothing else than someone's knowledge and expectations about the electron (or another physical system). When someone learns any new empirical data, i.e. about the result of a measurement, the wave function – which is his own and has to be his own because different observers know different things in general – gets abruptly modified or "collapses".

That's the answer and it's a complete answer. The statement that something is still "missing" is just demonstrably wrong.

Again: When we try to apply a psi-like mathematical description to phenomena e.g. in the microscopic world, it looks like it can only work if the wave function is basically a complexified probability distribution. So the wave function has to be defined in this way, is therefore subjective in a single repetition of an experiment, and that's why it's unavoidable that it collapses when the observer learns something (the result of a measurement) – because the wave function by definition quantifies all the knowledge of the observer. Because the conclusion follows from the basic axioms of the theory this directly, it's nonsensical to try to "reduce" the collapse to some more elementary steps. There aren't any. A one-step derivation building on the fundamental axioms is as elementary or irreducible as you can get.

The statement that the wave function collapses after the measurement is a trivial tautology – a direct consequence of the physical definition of the wave function. Whoever is incapable of getting this elementary point or understanding my proof is an absolute idiot and any journalist's suggestion that such an absolute idiot is on par with real quantum physicists is another sign of the political correctness that has run amok.

If you wanted to change the answer, i.e. that the reduction is tautologically a mathematical description of the observer's changed knowledge, it would be necessary for you to use a completely different physical definition of the wave function. Any "realist" would probably want to use a wave function that is not probabilistic. In any such picture, a wave function is supposed to be analogous to any "objective" classical observable such as the voltage.

But if that's the type of an alternative answer that you want to find, then I think that it is absolutely scientifically legitimate to demand that you first present your "voltmeter" that shows the values of \(\psi(x,y,z)\) or at least some evidence that such a "voltmeter" could exist at least in principle. Without such a "voltmeter" or its promising design sketch, you haven't even started to construct an alternative answer to mine because the key terms in your sentences, especially the "wave function", remain totally undefined! At most, they are defined as some concepts in a mathematical theory – but not a physical one because physics has to be connected to observations by "something like the voltmeters". You have erased the correct, "Copenhagen" physical definition of the wave function (the probabilistic one) – but you haven't replaced it with any other. So you are using terms that became physically meaningless and you have no idea what you're talking about. It's absolutely ludicrous for you to suggest that quantum mechanics has a problem; it is you who has a problem.

You know, you may imagine that such a hypothetical "voltmeter for the wave function" could count some "worlds of a certain kind" in the many-worlds paradigm; or measure a classical field such as the Bohmian pilot wave or the objectively collapsing Ghirardi-Rimini-Weber "objective wave function". But these are just fantasies. If you don't connect the wave function with observations that can actually be made, at least in principle, in our world, then your ideas have nothing to do with the natural science that studies this world! You haven't connected your mathematical fantasies with physics yet.

Needless to say, I am as certain that no one will be capable of constructing any "voltmeter for the wave functions" as I am certain about the statement that the Earth is round. We have – or at least I have – mastered the basic rules that govern the phenomena in Nature and they clearly imply that such a "voltmeter for the wave function" is exactly as impossible as a perpetuum mobile. So there will obviously never be any alternative definition of the wave function or an alternative viable answer to the question we started with.

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