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Socks and electrons are more analogous than Bellists pretend

On aspects that are different and especially those that are the same in classical and quantum physics

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George Musser identified himself as the latest promoter of the delusion started by John Bell, the delusion saying that the world has to be "non-local" but the objective reality independent of any observers (i.e. the information about the right point in a phase space) is surely something that is gonna be with us forever.

The truth is just the opposite one, of course. Locality works perfectly – at least in non-gravitational context. As understood since 1905, the influences or signals (they're exactly the same thing in discussions about causality: a signal is nothing else than an influence that was just considered helpful to send information by someone who cares but what's happening in a signal and a generic influence is exactly the same thing) cannot propagate faster than light. This consequence of special relativity is manifest in quantum field theory, too. In non-relativistic quantum theories, locality may be violated and signals may be superluminal but it's still true that they play absolutely no role in "explaining" the correlations coming from entangled states etc.

On the other hand, quantum mechanics has taught us – and nothing changes about these basic principles in quantum field theory, a special subclass of quantum mechanical theories – that all facts about Nature must be determined through observations. That means that they always depend on the choice of an observer, i.e. on the identification which events are classified as observations (interactions resulting in a change of the observer's knowledge about Nature). This classification is unavoidably subjective in principle.




A popular metaphor started by anti-quantum zealot John Bell in the 1980s involves the Bertlmann's socks. A guy named Bertlmann wears socks. If one of them is green, the other has to be red, and vice versa. Many of you must have thought that it was just a thought experiment or some mythology.




Not at all. Reinhold Bertlmann is not only real but this university professor in Vienna is alive and well – despite the fact that these Bell's confusions about quantum mechanics are about 50 years old. He was born in 1945.



Here you have Reinhold Bertlmann (right) along with a relatively sane physicist, David Mermin (left). You may verify that Bertlmann's left sock is red while the right sock is green. Since his teenage years, he's been wearing colorful socks in order to protest against the world which hadn't been turned into a communist paradise yet. Ironically, for decades, I have loved motley colors as symbols of creativity of freedom and capitalism. Sadly, Herr Bertlmann has never matured so now, at the age of 71, he still has similar ideas about politics and he still wears incoherent socks.



On his Viennese web page, you may also find this 1980 picture of a Bertlmann-Bell love scene involving tea taken at Bell's home. OK, these people were always nuts but this is not the main problem here.

The main problem is the question whether the predictions of entangled electrons' spins should be viewed as an "analogous problem" – with analogous reasons of correlations – to the problem of correlations between Bertlmann's socks. The sect of anti-quantum zealots whom I will refer to as Bellists in this blog post loves to parrot the claim by John Bell that these two situations are totally different and the reasons for the correlations are completely different, too.

The analogy

But it's just rubbish. The classical and quantum methods or algorithms or formulae to predict the particular values of the probabilities and the amount of correlation are different. But the meaning of the probabilities, the meaning of the correlations, and the location of the cause of the correlations as well as the observation of the correlations are fully analogous.

First, let's take comrade Bertlmann. We've been told – we observed – how this device works inside. When you leave him in the changing room, he will install a green sock on the left foot and a red one on the right foot or vice versa. This is what allows you to predict the colors of both socks once you see at least one sock. Note that this works even for sane people who normally take on two identical socks – a perfect anticorrelation is replaced by a perfect correlation – but sane people are less entertaining.

(Concerning changing rooms, the May 2016 video showing a fat Yankee urinating on the belongings of the Russian ice-hockey captain turned out to be a fake video – created by Czech top prankster Kazma using a perfect copy of the part of the changing room. See the previous prank. They finally pocketed the one million roubles that the insulted Russian news outlets offered to catch the U.S. criminal.)

Let's also assume that comrade Bertlmann randomly decides whether the left sock is green and the right sock is red – or vice versa, with equal odds. So whenever we see Bertlmann but before we see his socks, we know that the probabilities are\[

(0,1/2,1/2,0)

\] for the possibilities "left red, right red; left red, right green; left green, right red; left green, right green". A perfect anticorrelation, OK?

Now, take a spin \(J=0\) particle decaying to two identical spin-1/2 fermions. We may measure the spin up/down of each particle at the end. The probabilities of the four options are\[

(0,1/2,1/2,0)

\] for "left up, right up; left up, right down; left down, right up; left down, right down". Here, "left" and "right" refer to the two electrons at the end, according to their location. Clearly, the situations are completely isomorphic: I just replaced the words "red" and "green" by "up" and "down", respectively (the association follows from the traffic lights!).

The probabilities mean the same thing. We may prepare comrade Bertlmann; or the pair of fermions repeatedly, and "measure" the probabilities by the frequentist formula \(N_i / N_{\rm total}\).

Also, in both cases, the measurements are done at a later time and allow us to deduce the color/spin of the other sock/electron.

Finally, most importantly, the perfect anticorrelation was caused or created by an event at the analogous moment, at the very beginning. The perfect anticorrelation of the two socks' colors was created when comrade Bertlmann was taking on the socks in his changing room; the perfect anticorrelation of the two electrons' spins was created/caused when a spinless particle decayed to the two fermions.

Now, we may assume that we don't know the individual spins/colors before we measure them in both cases. This is analogous, too.

A difference is that something is telling you that even if you don't know the two individual colors of Bertlmann's socks, they objectively exist even before you observe the colors. This can't be assumed in the case of spins. In quantum mechanics, the results (values of the two individual spins) are really uncertain, even in principle, up to the moment of the measurement.

But I want to emphasize that this difference has no impact on your ability to make predictions and the character of the predictions themselves. You don't know the individual colors of the two socks before you observe at least one, so the fact that the two colors are "knowable" or "objectively exist" is absolutely useless from a predictive viewpoint – from a physical or scientific viewpoint!

On the other hand, in both cases, electrons and socks, you do know about the perfect anticorrelation of the two spins/colors from the very beginning. Nothing is changing about this perfect anticorrelation at the moment of the observation of the first color or the first spin, so no signal or no influence has to be sent anywhere, let alone a superluminal signal.

Differences

All the differences between classical physics and quantum mechanics are consequences of the nonzero commutators in quantum mechanics i.e. the uncertainty principle. There are absolutely no other differences between classical physics and quantum mechanics. That fact also means that whenever the commutators between the relevant quantities are zero or negligible, the difference between classical physics and quantum mechanics becomes zero or negligible, too.

The uncertainty principle is the actual reason why it's inconsistent in quantum mechanics to assume that the observables have their values before they're actually observed. The reason is that we may observe different observables, like \(J_{1z}\) and \(J_{1x}\), but these two don't commute with each other, so they simply can't be equal to classical numbers because classical numbers do commute with each other! That's it.

The uncertainty principle is also the reason why the 50-50 split between the two possibilities may be guaranteed in quantum mechanics – even though there is no good reason why it should be the case in classical physics. I have mentioned that the probabilities of the four spin/color arrangements are\[

(0,1/2,1/2,0).

\] Both nonzero entries are equal to one another i.e. equal to \(1/2\). This may be derived from a \(\ZZ_2\) symmetry but there's really no unavoidable reason for such a symmetry in classical physics. Comrade Bertlmann could place the red sock on the left foot more often (this 2007 picture confirms the same pattern) – because the Left is red, after all (Republicans in the U.S. will surely forgive me that this is the dominant convention). The greens belong to the Left as well, but let's not get too distracted. ;-)

On the other hand, in the case of the spins, there is a reason why the two nonzero probabilities have to be equal. In quantum mechanics, the probabilities aren't really fundamental; they are the squared absolute values of complex probability amplitudes. So the complex probability amplitudes are\[

(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0).

\] in quantum mechanics. The equal magnitude of the two nonzero amplitudes – as well as their particular relative phase (the minus sign, in this case) – is fully determined e.g. by the condition that the state \(\ket\psi\) of the two electrons is annihilated by \(J_{x, \rm total}\).

Just to be sure, the state\[

\ket \psi = \frac{\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow}}{\sqrt{2}}

\] is annihilated by \(J_{z,\rm total}=J_{z1}+J_{z2}\) because only states with the opposite spins are included with nonzero coefficients: \(J_{z1}=+1/2\) is combined with \(J_{z2}=-1/2\) or vice versa. So the state \(\ket \psi\) "has" \(J_{z,\rm total}=0\).

However, you may verify that the state \(\ket\psi\) is also annihilated by \(J_{x,\rm total}=J_{x1}+J_{x2}\). Note that \(J_{xj}\) acts like \(\hbar/2\) times the off-diagonal \(\sigma_x\) Pauli matrix (which has the number \(1\) on the two off-diagonal places). The action of \(J_{x,\rm total}\) on the two terms in \(\ket\psi\) produces four terms proportional to either \(\ket{\uparrow\uparrow}\) or \(\ket{\downarrow\downarrow}\) and they cancel in pairs assuming that the ratio of nonzero coefficients in \(\ket\psi\) is exactly \(-1\). You should verify that.

So our "maximal" (50-50) uncertainty about the individual spin (up/down) of the left electron (and similarly for the right electron) is actually a consequence of our certainty about the component \(J_{x,\rm total}\) of the angular momentum of the whole state! It would work with \(J_{y,\rm total}\), too. Note that when \(\ket\psi\) is annihilated both by \(J_{x,\rm total}\) and \(J_{z,\rm total}\), then it is unavoidably annihilated by \(J_{y,\rm total}\) as well because the third operator is proportional to the commutator of the first two.

This kind of magic is characteristically quantum mechanical.

In classical physics, the uncertainty – all probabilities that differ from 0% as well as 100% – are just artifacts of our failure. A better, more informed observer may always eliminate all this uncertainty, we are allowed to assume (even if this assumption is totally useless for us). On the other hand, the uncertainty principle of quantum mechanics bans any observer who could say that all these things are perfectly known or well-defined. Observables don't commute with each other – from anyone's viewpoint – so they simply can't have \(c\)-number values at the same moment! When some observables are known with certainty, almost all others are uncertain (because they have nonzero commutators with the known ones).

So the uncertainty – probabilities larger than 0% but smaller than 100% – are absolutely unavoidable in quantum mechanics. This is what the uncertainty principle says; this is how the most important consequence of the nonzero commutators should be interpreted physically. As I showed you, the uncertainty about something, like the individual electrons' spins, may be derived from the certainty about the value of some other observables.

Both in the practical case of socks and electrons, the individual spins/colors are unknown. But it's still true that the correlation (perfect anticorrelation in our two cases) is known and it is determined from the very beginning. Because the value of the "perfect anticorrelation" is "yes" from the very beginning in both cases, no observation of an individual spin/color ever influences this value, and that's why there is no influence – let alone a superluminal influence – anywhere.

On the other hand, in quantum mechanics, we must admit that the first spin (up/down) of an individual electron that we measure is created in the measurement itself. The reality – a particular \(c\)-number associated with the observable – wouldn't exist without the observation. The spin of the distant electron is therefore "determined" by our local measurement, too.

But as the sock analogy makes manifest, this "determination" doesn't require any superluminal or instantaneous influence because the spin of the remote electron wasn't strictly changed by the measurement of the spin of our local electron. Both individual spins were uncertain, just like both colors of the individual socks, and the measurement of the first spin just eliminated our uncertainty which may be thought of as residing in our head – just like in the case of the two socks. So only our knowledge about the individual spins was changed, not the spins themselves. And this change of knowledge may take place immediately, regardless of the separation of both electrons: it takes place in our head.

Once we make the particular measurement and turn e.g. \(J_{z1}\) into a \(c\)-number, we may reinterpret the events right before this measurement and assume that the individual spins (just like sock colors) had these values, either \(\uparrow\downarrow\) or \(\downarrow\uparrow\), already right before the first measurement of an individual spin. But in quantum mechanics, due to the uncertainty principle, we mustn't do the same thing in the absence of an actual measurement. We're only allowed to assign sharp \(c\)-number values to observables that are actually measured and once they are measured.

Again, let me repeat, the cause of the perfect anticorrelation resides at the very beginning – in Bertlmann's changing room or in the event when the original spinless particle decays. The anticorrelation is guaranteed from that moment on and nothing ever changes about the certainty of this anticorrelation which is why it's completely wrong to say that "someone affected it later".

Bellists seem obsessed by the "commissars" that do the work to "guarantee" the perfect anticorrelation when it's measured – and they want to believe that such "commissars" have to influence things and do so superluminally or instantaneously. But these Bellists never ask whether the "commissars" actually exist and whether they are actually needed. Also, they never ask what is the actual cause of the anticorrelation – or the reason why they believe it will be confirmed at all. Instead, they mindlessly adopt it and focus on the question how to enforce it. If they approached these questions rationally, they would know that the anticorrelation is known and guaranteed from the beginning. Entanglement or correlations simply can't exist without the previous interaction or contact of the two subsystems. Because physics is about predictions and this aspect of the prediction (perfect anticorrelation) never changes, there are no "commissars" who would affect this aspect later.

In quantum mechanics, the whole knowledge about the physical system can't be determined just by the set of probabilities for a particular basis, e.g. the \(J_{1z},J_{2z}\) simultaneous eigenstates. Instead, we need the complex probability amplitudes and all the relative phases do matter because they affect the probabilities of all other observables not commuting with \(J_{1z}\) or with \(J_{2z}\). This is why the "generic" assumption that a random observable "already exists" before the measurement, whether or not the measurement is actually made, is just forbidden in quantum mechanics. To assume that one basis vector may be objectively chosen in the absence of a particular measurement is equivalent to using regular probabilities instead of the complex probability amplitudes for the basis vectors – i.e. to forgetting about all the relative phases and neglecting all the quantum interference.

But once we know what we measure, the uncertainty about that observable may always be viewed as our idiosyncratic ignorance, just like our ignorance about the individual sock colors. When we learn about the individual spins, we are not remotely changing the two spins. We are just changing our knowledge from the ignorance (of individual spins/colors) to the knowledge, and because our brain is small (especially Bellists' brains), this change may take place very quickly without violating any constraints from relativity.

Because the results of the individual spins' measurements are random even in principle, there is no way for us to get the result we "want" and that's also why there's no way how to use the (anti)correlation to influence remote objects or send signals. If we only study the probabilities of various results of the remote measurement, these probabilities may be shown to be independent of all events or decisions that take place here. This independence of probabilities in the "right region" on all events/decisions in the "left region" is what quantitatively proves the locality of the theory.

The (anti)correlation may only be experimentally proven if you observe both electrons, socks, regions, or subsystems. And when you are observing both, you can't pretend that you only deal with one of them and influencing the other remotely: you are touching both so of course, you affect both.

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