8.6 The Many-Worlds In­ter­pre­ta­tion

The Schrö­din­ger equa­tion has been enor­mously suc­cess­ful, but it de­scribes the wave func­tion as al­ways smoothly evolv­ing in time, in ap­par­ent con­tra­dic­tion to its pos­tu­lated col­lapse in the or­tho­dox in­ter­pre­ta­tion. So, it would seem to be ex­tremely in­ter­est­ing to ex­am­ine the so­lu­tion of the Schrö­din­ger equa­tion for mea­sure­ment processes more closely, to see whether and how a col­lapse might oc­cur.

Of course, if a true so­lu­tion for a sin­gle ar­senic atom al­ready presents an un­sur­mount­able prob­lem, it may seem in­sane to try to an­a­lyze an en­tire macro­scopic sys­tem such as a mea­sure­ment ap­pa­ra­tus. But in a bril­liant Ph.D. the­sis with Wheeler at Prince­ton, Hugh Everett, III did ex­actly that. He showed that the wave func­tion does not col­lapse. How­ever it seems to us hu­mans that it does, so we are cor­rect in ap­ply­ing the rules of the or­tho­dox in­ter­pre­ta­tion any­way. This sub­sec­tion ex­plains briefly how this works.

Let’s re­turn to the ex­per­i­ment of sec­tion 8.2, where a positron is sent to Venus and an en­tan­gled elec­tron to Mars, as in fig­ure 8.6.

Fig­ure 8.6: Bohm’s ver­sion of the Ein­stein, Podol­ski, Rosen Para­dox.

The spin states are un­cer­tain when the two are sent from Earth, but when Venus mea­sures the spin of the positron, it mirac­u­lously causes the spin state of the elec­tron on Mars to col­lapse too. For ex­am­ple, if the Venus positron col­lapses to the spin-up state in the mea­sure­ment, the Mars elec­tron must col­lapse to the spin-down state. The prob­lem, how­ever, is that there is noth­ing in the Schrö­din­ger equa­tion to de­scribe such a col­lapse, nor the su­per­lu­mi­nal com­mu­ni­ca­tion be­tween Venus and Mars it im­plies.

The rea­son that the col­lapse and su­per­lu­mi­nal com­mu­ni­ca­tion are needed is that the two par­ti­cles are en­tan­gled in the sin­glet spin state of chap­ter 5.5.6. This is a 50% / 50% prob­a­bil­ity state of (elec­tron up and positron down) / (elec­tron down and positron up).

It would be easy if the positron would just be spin up and the elec­tron spin down, as in fig­ure 8.7.

Fig­ure 8.7: Nonen­tan­gled positron and elec­tron spins; up and down.

You would still not want to write down the su­per­colos­sal wave func­tion of every­thing, the par­ti­cles along with the ob­servers and their equip­ment for this case. But there is no doubt what it de­scribes. It will sim­ply de­scribe that the ob­server on Venus mea­sures spin up, and the one on Mars, spin down. There is no am­bi­gu­ity.

The same way, there is no ques­tion about the op­po­site case, fig­ure 8.8.

Fig­ure 8.8: Nonen­tan­gled positron and elec­tron spins; down and up.

It will pro­duce a wave func­tion of every­thing de­scrib­ing that the ob­server on Venus mea­sures spin down, and the one on Mars, spin up.

Everett, III rec­og­nized that the so­lu­tion for the en­tan­gled case is blind­ingly sim­ple. Since the Schrö­din­ger equa­tion is lin­ear, the wave func­tion for the en­tan­gled case must sim­ply be the sum of the two nonen­tan­gled ones above, as shown in fig­ure 8.9.

Fig­ure 8.9: The wave func­tions of two uni­verses com­bined

If the wave func­tion in each nonen­tan­gled case de­scribes a uni­verse in which a par­tic­u­lar state is solidly es­tab­lished for the spins, then the con­clu­sion is un­de­ni­able: the wave func­tion in the en­tan­gled case de­scribes two uni­verses, each of which solidly es­tab­lishes states for the spins, but which end up with op­po­site re­sults.

This ex­plains the re­sult of the or­tho­dox in­ter­pre­ta­tion that only eigen­val­ues are mea­sur­able. The lin­ear­ity of the Schrö­din­ger equa­tion leaves no other op­tion:

As­sume that any mea­sure­ment de­vice at all is con­structed that for a spin-up positron re­sults in a uni­verse that has ab­solutely no doubt that the spin is up, and for a spin-down positron re­sults in a uni­verse that has ab­solutely no doubt that the spin is down. In that case a com­bi­na­tion of spin up and spin down states must un­avoid­ably re­sult in a com­bi­na­tion of two uni­verses, one in which there is ab­solutely no doubt that the spin is up, and one in which there is ab­solutely no doubt that it is down.

Note that this ob­ser­va­tion does not de­pend on the de­tails of the Schrö­din­ger equa­tion, just on its lin­ear­ity. For that rea­son it stays true even in­clud­ing rel­a­tiv­ity.

The two uni­verses are com­pletely un­aware of each other. It is the very na­ture of lin­ear­ity that if two so­lu­tions are com­bined, they do not af­fect each other at all: nei­ther uni­verse would change in the least whether the other uni­verse is there or not. For each uni­verse, the other uni­verse ex­ists only in the sense that the Schrö­din­ger equa­tion must have cre­ated it given the ini­tial en­tan­gled state.

Non­lin­ear­ity would be needed to al­low the so­lu­tions of the two uni­verses to cou­ple to­gether to pro­duce a sin­gle uni­verse with a com­bi­na­tion of the two eigen­val­ues, and there is none. A uni­verse mea­sur­ing a com­bi­na­tion of eigen­val­ues is made im­pos­si­ble by lin­ear­ity.

While the wave func­tion has not col­lapsed, what has changed is the most mean­ing­ful way to de­scribe it. The wave func­tion still by its very na­ture as­signs a value to every pos­si­ble con­fig­u­ra­tion of the uni­verse, in other words, to every pos­si­ble uni­verse. That has never been a mat­ter of much con­tro­versy. And af­ter the mea­sure­ment it is still per­fectly cor­rect to say that the Venus ob­server has marked down in her note­book that the positron was up and down, and has trans­mit­ted a mes­sage to earth that the positron was up and down, and earth has marked on in its com­puter disks and in the brains of the as­sis­tants that the positron was found to be up and down, etcetera.

But it is much more pre­cise to say that af­ter the mea­sure­ment there are two uni­verses, one in which the Venus ob­server has ob­served the positron to be up, has trans­mit­ted to earth that the positron was up, and in which earth has marked down on its com­puter disks and in the brains of the as­sis­tants that the positron was up, etcetera; and a sec­ond uni­verse in which the same hap­pened, but with the positron every­where down in­stead of up. This de­scrip­tion is much more pre­cise since it notes that up al­ways goes with up, and down with down. As noted be­fore, this more pre­cise way of de­scrib­ing what hap­pens is called the “rel­a­tive state for­mu­la­tion.”

Note that in each uni­verse, it ap­pears that the wave func­tion has col­lapsed. Both uni­verses agree on the fact that the de­cay of the $\pi$-​me­son cre­ates an elec­tron/positron pair in a sin­glet state, but af­ter the mea­sure­ment, the note­book, ra­dio waves, com­puter disks, brains in one uni­verse all say that the positron is up, and in the other, all down. Only the un­ob­serv­able full wave func­tion knows that the positron is still both up and down.

And there is no longer a spooky su­per­lu­mi­nal ac­tion: in the first uni­verse, the elec­tron was al­ready down when sent from earth. In the other uni­verse, it was sent out as up. Sim­i­larly, for the case of the last sub­sec­tion, where half the wave func­tion of an elec­tron was sent to Venus, the Schrö­din­ger equa­tion does not fail. There is still half a chance of the elec­tron to be on Venus; it just gets de­com­posed into one uni­verse with one elec­tron, and a sec­ond one with zero elec­tron. In the first uni­verse, earth sent the elec­tron to Venus, in the sec­ond to Mars. The con­tra­dic­tions of quan­tum me­chan­ics dis­ap­pear when the com­plete so­lu­tion of the Schrö­din­ger equa­tion is ex­am­ined.

Next, let’s ex­am­ine why the re­sults would seem to be cov­ered by rules of chance, even though the Schrö­din­ger equa­tion is fully de­ter­min­is­tic. To do so, as­sume earth keeps on send­ing en­tan­gled positron and elec­tron pairs. When the third pair is on its way, the sit­u­a­tion looks as shown in the third col­umn of fig­ure 8.10.

Fig­ure 8.10: The Bohm ex­per­i­ment re­peated.

The wave func­tion now de­scribes 8 uni­verses. Note that in most uni­verses the ob­server starts see­ing an ap­par­ently ran­dom se­quence of up and down spins. When re­peated enough times, the se­quences ap­pear ran­dom in prac­ti­cally speak­ing every uni­verse. Un­able to see the other uni­verses, the ob­server in each uni­verse has no choice but to call her re­sults ran­dom. Only the full wave func­tion knows bet­ter.

Everett, III also de­rived that the sta­tis­tics of the ap­par­ently ran­dom se­quences are pro­por­tional to the ab­solute squares of the eigen­func­tion ex­pan­sion co­ef­fi­cients, as the or­tho­dox in­ter­pre­ta­tion says.

How about the un­cer­tainty re­la­tion­ship? For spins, the rel­e­vant un­cer­tainty re­la­tion­ship states that it is im­pos­si­ble for the spin in the up/down di­rec­tions and in the front/back di­rec­tions to be cer­tain at the same time. Mea­sur­ing the spin in the front/back di­rec­tion will make the up/down spin un­cer­tain. But if the spin was al­ways up, how can it change?

This is a bit more tricky. Let’s have the Mars ob­server do a cou­ple of ad­di­tional ex­per­i­ments on one of her elec­trons, first one front/back, and then an­other again up/down, to see what hap­pens. To be more pre­cise, let’s also ask her to write the re­sult of each mea­sure­ment on a black­board, so that there is a good record of what was found. Fig­ure 8.11 shows what hap­pens.

Fig­ure 8.11: Re­peated ex­per­i­ments on the same elec­tron.

When the elec­tron is sent from Earth, two uni­verses can be dis­tin­guished, one in which the elec­tron is up, and an­other in which it is down. In the first one, the Mars ob­server mea­sures the spin to be up and marks so on the black­board. In the sec­ond, she mea­sures and marks the spin to be down.

Next the ob­server in each of the two uni­verses mea­sures the spin front/back. Now it can be shown that the spin-up state in the first uni­verse is a lin­ear com­bi­na­tion of equal amounts of spin-front and spin-back. So the sec­ond mea­sure­ment splits the wave func­tion de­scrib­ing the first uni­verse into two, one with spin-front and one with spin-back.

Sim­i­larly, the spin-down state in the sec­ond uni­verse is equiv­a­lent to equal amounts of spin-front and spin-back, but in this case with op­po­site sign. Ei­ther way, the wave func­tion of the sec­ond uni­verse still splits into a uni­verse with spin front and one with spin back.

Now the ob­server in each uni­verse does her third mea­sure­ment. The front elec­tron con­sists of equal amounts of spin up and spin down elec­trons, and so does the back elec­tron, just with dif­fer­ent sign. So, as the last col­umn in fig­ure 8.11 shows, in the third mea­sure­ment, as much as half the eight uni­verses mea­sure the ver­ti­cal spin to be the op­po­site of the one they got in the first mea­sure­ment!

The full wave func­tion knows that if the first four of the fi­nal eight uni­verses are summed to­gether, the net spin is still down (the two down spins have equal and op­po­site am­pli­tude). But the ob­servers have only their black­board (and what is recorded in their brains, etcetera) to guide them. And that in­for­ma­tion seems to tell them un­am­bigu­ously that the front-back mea­sure­ment de­stroyed the ver­ti­cal spin of the elec­tron. (The four ob­servers that mea­sured the spin to be un­changed can re­peat the ex­per­i­ment a few more times and are sure to even­tu­ally find that the ver­ti­cal spin does change.)

The un­avoid­able con­clu­sion is that the Schrö­din­ger equa­tion does not fail. It de­scribes the ob­ser­va­tions ex­actly, in full agree­ment with the or­tho­dox in­ter­pre­ta­tion, with­out any col­lapse. The ap­pear­ance of a col­lapse is ac­tu­ally just a lim­i­ta­tion of our hu­man ob­ser­va­tional ca­pa­bil­i­ties.

Of course, in other cases than the spin ex­am­ple above, there are more than just two sym­met­ric states, and it be­comes much less self-ev­i­dent what the proper par­tial so­lu­tions are. How­ever, it does not seem hard to make some con­jec­tures. For Schrö­din­ger’s cat, you might model the ra­dioac­tive de­cay that gives rise to the Geiger counter go­ing off as due to a nu­cleus with a neu­tron wave packet rat­tling around in it, try­ing to es­cape. As chap­ter 7.12.1 showed, in quan­tum me­chan­ics each rat­tle will fall apart into a trans­mit­ted and a re­flected wave. The trans­mit­ted wave would de­scribe the for­ma­tion of a uni­verse where the neu­tron es­capes at that time to set off the Geiger counter which kills the cat, and the re­flected wave a uni­verse where the neu­tron is still con­tained.

For the stan­dard quan­tum me­chan­ics ex­am­ple of an ex­cited atom emit­ting a pho­ton, a model would be that the ini­tial ex­cited atom is per­turbed by the am­bi­ent elec­tro­mag­netic field. The per­tur­ba­tions will turn the atom into a lin­ear com­bi­na­tion of the ex­cited state with a bit of a lower en­ergy state thrown in, sur­rounded by a per­turbed elec­tro­mag­netic field. Pre­sum­ably this sit­u­a­tion can be taken apart in a uni­verse with the atom still in the ex­cited state, and the en­ergy in the elec­tro­mag­netic field still the same, and an­other uni­verse with the atom in the lower en­ergy state with a pho­ton es­cap­ing in ad­di­tion to the en­ergy in the orig­i­nal elec­tro­mag­netic field. Of course, the process would re­peat for the first uni­verse, pro­duc­ing an even­tual se­ries of uni­verses in al­most all of which the atom has emit­ted a pho­ton and thus tran­si­tioned to a lower en­ergy state.

So this is where we end up. The equa­tions of quan­tum me­chan­ics de­scribe the physics that we ob­serve per­fectly well. Yet they have forced us to the un­com­fort­able con­clu­sion that, math­e­mat­i­cally speak­ing, we are not at all unique. Be­yond our uni­verse, the math­e­mat­ics of quan­tum me­chan­ics re­quires an in­fin­ity of un­ob­serv­able other uni­verses that are non­triv­ially dif­fer­ent from us.

Note that the ex­is­tence of an in­fin­ity of uni­verses is not the is­sue. They are al­ready re­quired by the very for­mu­la­tion of quan­tum me­chan­ics. The wave func­tion of say an ar­senic atom al­ready as­signs a nonzero prob­a­bil­ity to every pos­si­ble con­fig­u­ra­tion of the po­si­tions of the elec­trons. Sim­i­larly, a wave func­tion of the uni­verse will as­sign a nonzero prob­a­bil­ity to every pos­si­ble con­fig­u­ra­tion of the uni­verse, in other words, to every pos­si­ble uni­verse. The ex­is­tence of an in­fin­ity of uni­verses is there­fore not some­thing that should be as­cribed to Everett, III {N.15}.

How­ever, when quan­tum me­chan­ics was first for­mu­lated, peo­ple quite ob­vi­ously be­lieved that, prac­ti­cally speak­ing, there would be just one uni­verse, the one we ob­serve. No se­ri­ous physi­cist would deny that the mon­i­tor on which you may be read­ing this has un­cer­tainty in its po­si­tion, yet the un­cer­tainty you are deal­ing with here is so as­tro­nom­i­cally small that it can be ig­nored. Sim­i­larly it might ap­pear that all the other sub­stan­tially dif­fer­ent uni­verses should have such small prob­a­bil­i­ties that they can be ig­nored. The ac­tual con­tri­bu­tion of Everett, III was to show that this idea is not ten­able. Non­triv­ial uni­verses must de­velop that are sub­stan­tially dif­fer­ent.

For­mu­lated in 1957 and then largely ig­nored, Everett's work rep­re­sents with­out doubt one of the hu­man race's great­est ac­com­plish­ments; a stun­ning dis­cov­ery of what we are and what is our place in the uni­verse.