8.2 In­stan­ta­neous In­ter­ac­tions

Spe­cial rel­a­tiv­ity has shown that we hu­mans can­not trans­mit in­for­ma­tion at more than the speed of light. How­ever, ac­cord­ing to the or­tho­dox in­ter­pre­ta­tion, na­ture does not limit it­self to the same silly re­stric­tions that it puts on us. This sec­tion dis­cusses why not.

Con­sider again the H$_2^+$-ion, with the sin­gle elec­tron equally shared by the two pro­tons. If you pull the pro­tons apart, main­tain­ing the sym­me­try, you get a wave func­tion that looks like fig­ure 8.1.

Fig­ure 8.1: Sep­a­rat­ing the hy­dro­gen ion.
\begin{figure}\centering
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\epsffile{h2-symf.eps}
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You might send one pro­ton off to your ob­server on Mars, the other to your ob­server on Venus. Where is the elec­tron, on Mars or on Venus?

Ac­cord­ing to the or­tho­dox in­ter­pre­ta­tion, the an­swer is: nei­ther. A po­si­tion for the elec­tron does not ex­ist. The elec­tron is not on Mars. It is not on Venus. Only when ei­ther ob­server makes a mea­sure­ment to see whether the elec­tron is there, na­ture throws its dice, and based on the re­sult, might put the elec­tron on Venus and zero the wave func­tion on Mars. But re­gard­less of the dis­tance, it could just as well have put the elec­tron on Mars, if the dice would have come up dif­fer­ently.

You might think that na­ture cheats, that when you take the pro­tons apart, na­ture al­ready de­cides where the elec­tron is go­ing to be. That the Venus pro­ton se­cretly hides the elec­tron “in its sleeve”, ready to make it ap­pear if an ob­ser­va­tion is made. John Bell de­vised a clever test to force na­ture to re­veal whether it has some­thing hid­den in its sleeve dur­ing a sim­i­lar sort of trick.

The test case Bell used was a gen­er­al­iza­tion of an ex­per­i­ment pro­posed by Bohm. It in­volves spin mea­sure­ments on an elec­tron/positron pair, cre­ated by the de­cay of an $\pi$-​me­son. Their com­bined spins are in the sin­glet state be­cause the me­son has no net spin. In par­tic­u­lar, if you mea­sure the spins of the elec­tron and positron in any given di­rec­tion, there is a 50/50% chance for each that it turns out to be pos­i­tive or neg­a­tive. How­ever, if one is pos­i­tive, the other must be neg­a­tive. So there are only two dif­fer­ent pos­si­bil­i­ties:

1.
elec­tron pos­i­tive and positron neg­a­tive,
2.
elec­tron neg­a­tive and positron pos­i­tive.

Now sup­pose Earth hap­pens to be al­most the same dis­tance from Mars and Venus, and you shoot the positron out to Venus, and the elec­tron to Mars, as shown at the left in the fig­ure be­low:

Fig­ure 8.2: The Bohm ex­per­i­ment be­fore the Venus mea­sure­ment (left), and im­me­di­ately af­ter it (right).
\begin{figure}\centering
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% -125 left aligns the...
...b]{Earth}}
\put(70.7,17){\makebox(0,0)[b]{Mars}}
}
\end{picture}
\end{figure}

You have ob­servers on both plan­ets wait­ing for the par­ti­cles. Ac­cord­ing to quan­tum me­chan­ics, the trav­el­ing elec­tron and positron are both in an in­de­ter­mi­nate state.

The positron reaches Venus a frac­tion of a sec­ond ear­lier, and the ob­server there mea­sures its spin in the di­rec­tion up from the eclip­tic plane. Ac­cord­ing to the or­tho­dox in­ter­pre­ta­tion, na­ture now makes a ran­dom se­lec­tion be­tween the two pos­si­bil­i­ties, and as­sume it se­lects the pos­i­tive spin value for the positron, cor­re­spond­ing to a spin that is up from the eclip­tic plane, as shown in fig­ure 8.2. Im­me­di­ately, then, the spin state of the elec­tron on Mars must also have col­lapsed; the ob­server on Mars is guar­an­teed to now mea­sure neg­a­tive spin, or spin down, for the elec­tron.

The funny thing is, if you be­lieve the or­tho­dox in­ter­pre­ta­tion, the in­for­ma­tion about the mea­sure­ment of the positron has to reach the elec­tron in­stan­ta­neously, much faster than light can travel. This ap­par­ent prob­lem in the or­tho­dox in­ter­pre­ta­tion was dis­cov­ered by Ein­stein, Podol­ski, and Rosen. They doubted it could be true, and ar­gued that it in­di­cated that some­thing must be miss­ing in quan­tum me­chan­ics.

In fact, in­stead of su­per­lu­mi­nal ef­fects, it seems much more rea­son­able to as­sume that ear­lier on earth, when the par­ti­cles were sent on their way, na­ture at­tached a se­cret lit­tle note of some kind to the positron, say­ing the equiv­a­lent of “If your spin up is mea­sured, give the pos­i­tive value”, and that it at­tached a lit­tle note to the elec­tron If your spin up is mea­sured, give the neg­a­tive value. The re­sults of the mea­sure­ments are still the same, and the lit­tle notes travel along with the par­ti­cles, well be­low the speed of light, so all seems now fine. Of course, these would not be true notes, but some kind of ad­di­tional in­for­ma­tion be­yond the nor­mal quan­tum me­chan­ics. Such pos­tu­lated ad­di­tional in­for­ma­tion sources are called “hid­den vari­ables.”

Bell saw that there was a fun­da­men­tal flaw in this idea if you do a large num­ber of such mea­sure­ments and you al­low the ob­servers to se­lect from more than one mea­sure­ment di­rec­tion at ran­dom. He de­rived a neat lit­tle gen­eral for­mula, but the dis­cus­sion here will just show the con­tra­dic­tion in a sin­gle case. In par­tic­u­lar, the ob­servers on Venus and Mars will be al­lowed to se­lect ran­domly one of three mea­sure­ment di­rec­tions $\vec{a}$, $\vec{b}$, and $\vec{c}$ sep­a­rated by 120 de­grees:

Fig­ure 8.3: Spin mea­sure­ment di­rec­tions.
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...ec b$}}
\put(-14,-5){\makebox(0,0)[b]{$\vec c$}}
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Let’s see what the lit­tle notes at­tached to the elec­trons might say. They might say, for ex­am­ple, “Give the $+$ value if $\vec{a}$ is mea­sured, give the $\vphantom0\raisebox{1.5pt}{$-$}$ value if $\vec{b}$ is mea­sured, give the $+$ value if $\vec{c}$ is mea­sured.” The rel­a­tive frac­tions of the var­i­ous pos­si­ble notes gen­er­ated for the elec­trons will be called $f_1,f_2,\ldots$. There are 8 dif­fer­ent pos­si­ble notes:

\begin{displaymath}
\begin{array}{r\vert c\vert c\vert c\vert c\vert c\vert c\v...
... - & - \\
\vec c & + & - & + & - & + & - & + & -
\end{array}\end{displaymath}

The sum of the frac­tions $f_1$ through $f_8$ must be one. In fact, be­cause of sym­me­try, each note will prob­a­bly on av­er­age be gen­er­ated for $\frac18$ of the elec­trons sent, but this will not be needed.

Of course, each note at­tached to the positron must al­ways be just the op­po­site of the one at­tached to the elec­tron, since the positron must mea­sure $+$ in a di­rec­tion when the elec­tron mea­sures $\vphantom0\raisebox{1.5pt}{$-$}$in that di­rec­tion and vice-versa.

Now con­sider those mea­sure­ments in which the Venus ob­server mea­sures di­rec­tion $\vec{a}$ and the Mars ob­server mea­sures di­rec­tion $\vec{b}$. In par­tic­u­lar, the ques­tion is in what frac­tion of such mea­sure­ments the Venus ob­server mea­sures the op­po­site sign from the Mars ob­server; call it $f_{ab,\mbox{\scriptsize {opposite}}}$. This is not that hard to fig­ure out. First con­sider the case that Venus mea­sures $\vphantom0\raisebox{1.5pt}{$-$}$and Mars $+$. If the Venus ob­server mea­sures the $\vphantom0\raisebox{1.5pt}{$-$}$value for the positron, then the note at­tached to the elec­tron must say “mea­sure $+$ for $\vec{a}$; fur­ther, if the Mars ob­server mea­sures the $+$ value for $\vec{b}$, that one should say “mea­sure $+$ too. So, look­ing at the ta­ble, the rel­a­tive frac­tion where Venus mea­sures $\vphantom0\raisebox{1.5pt}{$-$}$and Mars mea­sures $+$ is where the elec­tron's note has a $+$ for both $\vec{a}$ and $\vec{b}$: $f_1+f_2$.

Sim­i­larly, the frac­tion of cases where Venus finds $+$ and Mars $\vphantom0\raisebox{1.5pt}{$-$}$is $f_7+f_8$, and you get in to­tal:

\begin{displaymath}
f_{ab,\mbox{\scriptsize opposite}} = f_1 + f_2 + f_7 + f_8
= 0.25
\end{displaymath}

The value 0.25 is what quan­tum me­chan­ics pre­dicts; the de­riva­tion will be skipped here, but it has been ver­i­fied in the ex­per­i­ments done af­ter Bell's work. Those ex­per­i­ments also made sure that na­ture did not get the chance to do sublu­mi­nal com­mu­ni­ca­tion. The same way you get

\begin{displaymath}
f_{ac,\mbox{\scriptsize opposite}} = f_1 + f_3 + f_6 + f_8
= 0.25
\end{displaymath}

and

\begin{displaymath}
f_{bc,\mbox{\scriptsize opposite}} = f_1 + f_4 + f_5 + f_8
= 0.25
\end{displaymath}

Now there is a prob­lem, be­cause the num­bers add up to 0.75, but the frac­tions add up to at least 1: the sum of $f_1$ through $f_8$ is one.

A seem­ingly per­fectly log­i­cal and plau­si­ble ex­pla­na­tion by great minds is tripped up by some num­bers that just do not want to match up. They only leave the al­ter­na­tive no­body re­ally wanted to be­lieve.

At­tach­ing notes does not work. In­for­ma­tion on what the ob­server on Venus de­cided to mea­sure, the one thing that could not be put in the notes, must have been com­mu­ni­cated in­stantly to the elec­tron on Mars re­gard­less of the dis­tance.

It can also safely be con­cluded that we hu­mans will never be able to see in­side the ac­tual ma­chin­ery of quan­tum me­chan­ics. For, sup­pose the ob­server on Mars could see the wave func­tion of the elec­tron col­lapse. Then the ob­server on Venus could send her Morse sig­nals faster than the speed of light by ei­ther mea­sur­ing or not mea­sur­ing the spin of the positron. Spe­cial rel­a­tiv­ity would then al­low sig­nals to be sent into the past, and that leads to log­i­cal con­tra­dic­tions such as the Venus ob­server pre­vent­ing her mother from hav­ing her.

While the re­sults of the spin mea­sure­ments can be ob­served, they do not al­low su­per­lu­mi­nal com­mu­ni­ca­tion. While the ob­server on Venus af­fects the re­sults of the mea­sure­ments of the ob­server on Mars, they will look com­pletely ran­dom to that ob­server. Only when the ob­server on Venus sends over the re­sults of her mea­sure­ments, at a speed less than the speed of light, and the two sets of re­sults are com­pared, do mean­ing­ful pat­terns how up.

The Bell ex­per­i­ments are of­ten used to ar­gue that Na­ture must re­ally make the col­lapse de­ci­sion us­ing a true ran­dom num­ber gen­er­a­tor, but that is of course crap. The ex­per­i­ments in­di­cate that Na­ture in­stan­ta­neously trans­mits the col­lapse de­ci­sion on Venus to Mars, but say noth­ing about how that de­ci­sion was reached.

Su­per­lu­mi­nal ef­fects still cause para­doxes, of course. The left of fig­ure 8.4 shows how a Bohm ex­per­i­ment ap­pears to an ob­server on earth.

Fig­ure 8.4: Earth’s view of events (left), and that of a mov­ing ob­server (right).
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...[b]{Venus}}
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The spins re­main un­de­cided un­til the mea­sure­ment by the Venus ob­server causes both the positron and the elec­tron spins to col­lapse.

How­ever, for a mov­ing ob­server, things would look very dif­fer­ent. As­sum­ing that the ob­server and the par­ti­cles are all mov­ing at speeds com­pa­ra­ble to the speed of light, the same sit­u­a­tion may look like the right of fig­ure 8.4, chap­ter 1.1.4. In this case, the ob­server on Mars causes the wave func­tion to col­lapse at a time that the positron has only just started mov­ing to­wards Venus!

So the or­tho­dox in­ter­pre­ta­tion is not quite ac­cu­rate. It should re­ally have said that the mea­sure­ment on Venus causes a con­ver­gence of the wave func­tion, not an ab­solute col­lapse. What the ob­server of Venus re­ally achieves in the or­tho­dox in­ter­pre­ta­tion is that af­ter her mea­sure­ment, all ob­servers agree that the positron wave func­tion is col­lapsed. Be­fore that time, some ob­servers are per­fectly cor­rect in say­ing that the wave func­tion is al­ready col­lapsed, and that the Mars ob­server did it.

It should be noted that when the equa­tions of quan­tum me­chan­ics are cor­rectly ap­plied, the col­lapse and su­per­lu­mi­nal ef­fects dis­ap­pear. That is ex­plained in sec­tion 8.6. But, due to the fact that there are lim­its to our ob­ser­va­tional ca­pa­bil­i­ties, as far as our own hu­man ex­pe­ri­ences are con­cerned, the para­doxes re­main real.

To be per­fectly hon­est, it should be noted that the ex­am­ple above is not quite the one of Bell. Bell re­ally used the in­equal­ity:

\begin{displaymath}
\vert 2(f_3+f_4+f_5+f_6) - 2(f_2+f_4+f_5+f_7)\vert \mathrel{\raisebox{-.7pt}{$\leqslant$}}2(f_2+f_3+f_6+f_7)
\end{displaymath}

So the dis­cus­sion cheated. And Bell al­lowed gen­eral di­rec­tions of mea­sure­ment not just 120 de­gree ones. See [25, pp. 423-426]. The above dis­cus­sion seems a lot less messy, even though not his­tor­i­cally ac­cu­rate.