Sunday, October 27, 2019

What is more incomplete: Copenhagen or many worlds?

In the first stage of the Einstein-Bohr debates, Einstein tried to prove that quantum mechanics was wrong – literally producing predictions that conflicted with observations. When he was persuaded that this goal was just plain stupid around 1930, Einstein switched to the second stage in which he "only" argued that quantum mechanics (as defined in Copenhagen) was "incomplete".

Even today, almost 90 years later, the anti-quantum zealots who are still around – depending on the degree of their stupidity – argue that quantum mechanics is either wrong or incomplete. The typical complaint that "quantum mechanics isn't complete" is formulated as follows:
But the Copenhagen Interpretation fails to tell us what is really going on before we look.
Well, in reality, quantum mechanics tells us everything that is happening before the observation: nothing that could be considered a fact is happening before (or in the absence of) an observation! It is an answer. You may dislike it but it's a lie to say that you weren't given an answer!

Needless to say, the statements are upside down. The Copenhagen Interpretation provides us with a definition which questions are physically meaningful; and with the method to determine the answers to these questions (which must be probabilistic and the axiomatic framework clearly and unambiguously says that no "unambiguous" predictions of the phenomena are possible in general).

Instead, it's the anti-quantum "interpretations" of quantum mechanics such as the Many Worlds Interpretation that are incomplete because
their axioms just don't allow you to determine what you should do if you want to calculate the probability of an outcome of an observation.
In particular, the Many Worlds Interpretation denies that there's any collapse following Born's rule (an axiom) but it is rather obvious that when you omit this only link between quantum mechanics and probabilities, the Many Worlds paradigm will become unable to actually predict these probabilities. You created a hole – (because the building block looked ideologically heretical to him) someone has removed something that was needed (in the Copenhagen paradigm) to complete the argumentation that normally ends with the probabilistic prediction.

This is an actually valid complaint because the primary purpose of science is to explain and predict the results of phenomena. The general purpose of science is not to say "what is out there when no one looks". The assumption that "some facts are true even when no one looks" was compatible with classical physics – and one could usefully make this assumption while improving the theories in classical physics. But the construct "what is out there in the absence of observations" was just a conditional and auxiliary construct that worked when it worked.

In the era of quantum mechanics, it just doesn't work. You must return to the actual purpose of science that is independent of the epochs – and the actual purpose of science is to explain, organize, and predict the outcomes of observations!

Some geography

I found a cute metaphor for the bogus complaint that "Copenhagen is incomplete". Is the City of Copenhagen complete? If you're not Danish, I think you will be stunned by this question. OK, what is the actual Copenhagen Municipality and is it complete?

Well, this is the answer. You should search for Copenhagen on, the wonderful killer of the Google Maps, to see that there is really a nice round hole inside Copenhagen. (On top of that, the water that divides the city to two parts formally doesn't belong to the Copenhagen Municipality, either.)

The hole in Copenhagen is the Frederiksberg Kommune – which, along with the Copenhagen Municipality, makes up the City of Copenhagen. But the Copenhagen Municipality and Frederiksberg have different mayors. Frederiksberg is aggressively independent; it is an affluent hole previously called Tulehøj (the Song Hill) where King Frederik IV built a palace in 1700-1703.

So Copenhagen "isn't complete" in the sense that it has a big hole near the middle which doesn't belong to the Copenhagen Municipality. How could a city with such a hole be complete? Does the Copenhagen Interpretation have an analogous hole? (To avoid confusions, note that the Bohr Institute is outside the hole but not by much.)

I do think that the analogy is great because Frederiksberg, the hole inside Copenhagen, is analogous to all the things that are claimed to be impossible or unphysical by quantum mechanics – and especially by the uncertainty principle – starting with the simultaneously well-defined precise locations and speeds of particles and the factual unambiguous answers to the questions "what exists in the absence of observations" and "what exists independently of the choice of the observer".

You didn't expect that Copenhagen would have a hole in the middle, did you? In the same way, people didn't expect that a better theory would classify some tasks as impossible to solve – and some questions as physically meaningless. But it simply happened because Nature doesn't have the duty to confirm mortals' expectations. The City of Copenhagen is "incomplete" in the sense that it has a hole – something that you could have assumed to be forbidden at the beginning.

In the same way, quantum mechanics – with the correct, Copenhagen Interpretation (or its modern equivalent update) – tells us that some questions must be considered physically meaningless. Relatively to some preconceptions – about simply connected cities and physical theories rooted in classical physics – both Copenhagen and the Copenhagen Interpretation are "incomplete".

However, a sensible person may also see that this "incompleteness" is really just a symptom of our invalid assumptions. There is nothing wrong about cities that are not simply connected (areas with holes in the middle); and there is nothing wrong about theories that just predict probabilities of observations (where one must "insert" the information about what is considered an observation and what isn't in each physical situation). It would be extremely unnatural – especially from the viewpoint of the pompous inhabitants of Frederiksberg – to try to "fill" the hole inside Copenhagen; and it is even more unnatural, and basically mathematically impossible, to complete the "hole" in quantum mechanics (by replacing it with a different theory that fundamentally rejects the uncertainty principle).

The real point is that the map of Copenhagen that I have linked to is complete in the sense of "information". No information is missing there. The topology of the city is unexpected but it's as well-defined and complete as the simply connected shape of Prague or any other boring, classical city. (I mean just topologically boring, of course; Danish youth have used Prague as the place of some fun journeys involving lots of alcohol.)

In the analogy, the Copenhagen axioms of quantum mechanics contain some unexpected features, require you to remove some statements from the list of those that you considered physically meaningful – to create a hole in your previous ideas about which statements physics must address – but when you properly adjust your set of meaningful questions and produce the hole that you're told to produce, everything makes a perfect sense!

The idea that the "holes" are an inconsistency is simply wrong. The inhabitants of Copenhagen may happily live in their city. They may even run around Frederiksberg every day while carefully avoiding it. In the same way, the axioms of quantum mechanics allow you to predict probabilities of the outcomes of all observations that you may actually perceive. You can't really become certain of the existence of any meaningful data that would belong to the "hole" inside quantum mechanics – the statements that contradict the uncertainty principle etc. Any effort to deny the simply disconnected topology of the axiomatic system of quantum mechanics is doomed – it's even much more doomed than attempts to persuade the Frederiksberg folks to finally obey the orders of the Copenhagen Municipality's mayor.

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