# The Born Probabilities

Pre­vi­ously in se­ries: De­co­her­ence is Pointless
Fol­lowup to: Where Ex­pe­rience Con­fuses Physicists

One se­ri­ous mys­tery of de­co­her­ence is where the Born prob­a­bil­ities come from, or even what they are prob­a­bil­ities of. What does the in­te­gral over the squared mod­u­lus of the am­pli­tude den­sity have to do with any­thing?

This was dis­cussed by anal­ogy in “Where Ex­pe­rience Con­fuses Physi­cists”, and I won’t re­peat ar­gu­ments already cov­ered there. I will, how­ever, try to con­vey ex­actly what the puz­zle is, in the real frame­work of quan­tum me­chan­ics.

A pro­fes­sor teach­ing un­der­grad­u­ates might say: “The prob­a­bil­ity of find­ing a par­ti­cle in a par­tic­u­lar po­si­tion is given by the squared mod­u­lus of the am­pli­tude at that po­si­tion.”

This is over­sim­plified in sev­eral ways.

First, for con­tin­u­ous vari­ables like po­si­tion, am­pli­tude is a den­sity, not a point mass. You in­te­grate over it. The in­te­gral over a sin­gle point is zero.

(His­tor­i­cal note: If “ob­serv­ing a par­ti­cle’s po­si­tion” in­voked a mys­te­ri­ous event that squeezed the am­pli­tude dis­tri­bu­tion down to a delta point, or flat­tened it in one sub­space, this would give us a differ­ent fu­ture am­pli­tude dis­tri­bu­tion from what de­co­her­ence would pre­dict. All in­ter­pre­ta­tions of QM that in­volve quan­tum sys­tems jump­ing into a point/​flat state, which are both testable and have been tested, have been falsified. The uni­verse does not have a “clas­si­cal mode” to jump into; it’s all am­pli­tudes, all the time.)

Se­cond, a sin­gle ob­served par­ti­cle doesn’t have an am­pli­tude dis­tri­bu­tion. Rather the sys­tem con­tain­ing your­self, plus the par­ti­cle, plus the rest of the uni­verse, may ap­prox­i­mately fac­tor into the mul­ti­plica­tive product of (1) a sub-dis­tri­bu­tion over the par­ti­cle po­si­tion and (2) a sub-dis­tri­bu­tion over the rest of the uni­verse. Or rather, the par­tic­u­lar blob of am­pli­tude that you hap­pen to be in, can fac­tor that way.

So what could it mean, to as­so­ci­ate a “sub­jec­tive prob­a­bil­ity” with a com­po­nent of one fac­tor of a com­bined am­pli­tude dis­tri­bu­tion that hap­pens to fac­tor­ize?

Re­call the physics for:

(Hu­man-BLANK * Sen­sor-BLANK) * (Atom-LEFT + Atom-RIGHT)
=>
(Hu­man-LEFT * Sen­sor-LEFT * Atom-LEFT) + (Hu­man-RIGHT * Sen­sor-RIGHT * Atom-RIGHT)

Think of the whole pro­cess as re­flect­ing the good-old-fash­ioned dis­tribu­tive rule of alge­bra. The ini­tial state can be de­com­posed—note that this is an iden­tity, not an evolu­tion—into:

(Hu­man-BLANK * Sen­sor-BLANK) * (Atom-LEFT + Atom-RIGHT)
=
(Hu­man-BLANK * Sen­sor-BLANK * Atom-LEFT) + (Hu­man-BLANK * Sen­sor-BLANK * Atom-RIGHT)

We as­sume that the dis­tri­bu­tion fac­tor­izes. It fol­lows that the term on the left, and the term on the right, ini­tially differ only by a mul­ti­plica­tive fac­tor of Atom-LEFT vs. Atom-RIGHT.

If you were to im­me­di­ately take the multi-di­men­sional in­te­gral over the squared mod­u­lus of the am­pli­tude den­sity of that whole sys­tem,

Then the ra­tio of the all-di­men­sional in­te­gral of the squared mod­u­lus over the left-side term, to the all-di­men­sional in­te­gral over the squared mod­u­lus of the right-side term,

Would equal the ra­tio of the lower-di­men­sional in­te­gral over the squared mod­u­lus of the Atom-LEFT, to the lower-di­men­sional in­te­gral over the squared mod­u­lus of Atom-RIGHT,

For es­sen­tially the same rea­son that if you’ve got (2 * 3) * (5 + 7), the ra­tio of (2 * 3 * 5) to (2 * 3 * 7) is the same as the ra­tio of 5 to 7.

Do­ing an in­te­gral over the squared mod­u­lus of a com­plex am­pli­tude dis­tri­bu­tion in N di­men­sions doesn’t change that.

There’s also a rule called “uni­tary evolu­tion” in quan­tum me­chan­ics, which says that quan­tum evolu­tion never changes the to­tal in­te­gral over the squared mod­u­lus of the am­pli­tude den­sity.

So if you as­sume that the ini­tial left term and the ini­tial right term evolve, with­out over­lap­ping each other, into the fi­nal LEFT term and the fi­nal RIGHT term, they’ll have the same ra­tio of in­te­grals over etcetera as be­fore.

What all this says is that,

If some roughly in­de­pen­dent Atom has got a blob of am­pli­tude on the left of its fac­tor, and a blob of am­pli­tude on the right,

Then, af­ter the Sen­sor senses the atom, and you look at the Sen­sor,

The in­te­grated squared mod­u­lus of the whole LEFT blob, and the in­te­grated squared mod­u­lus of the whole RIGHT blob,

Will have the same ra­tio,

As the ra­tio of the squared mod­uli of the origi­nal Atom-LEFT and Atom-RIGHT com­po­nents.

This is why it’s im­por­tant to re­mem­ber that ap­par­ently in­di­vi­d­ual par­ti­cles have am­pli­tude dis­tri­bu­tions that are mul­ti­plica­tive fac­tors within the to­tal joint dis­tri­bu­tion over all the par­ti­cles.

If a whole gi­gan­tic hu­man ex­per­i­menter made up of quin­til­lions of par­ti­cles,

In­ter­acts with one teensy lit­tle atom whose am­pli­tude fac­tor has a big bulge on the left and a small bulge on the right,

Then the re­sult­ing am­pli­tude dis­tri­bu­tion, in the joint con­figu­ra­tion space,

Has a big am­pli­tude blob for “hu­man sees atom on the left”, and a small am­pli­tude blob of “hu­man sees atom on the right”.

And what that means, is that the Born prob­a­bil­ities seem to be about find­ing your­self in a par­tic­u­lar blob, not the par­ti­cle be­ing in a par­tic­u­lar place.

But what does the in­te­gral over squared mod­uli have to do with any­thing? On a straight read­ing of the data, you would always find your­self in both blobs, ev­ery time. How can you find your­self in one blob with greater prob­a­bil­ity? What are the Born prob­a­bil­ities, prob­a­bil­ities of? Here’s the map—where’s the ter­ri­tory?

I don’t know. It’s an open prob­lem. Try not to go funny in the head about it.

This prob­lem is even worse than it looks, be­cause the squared-mod­u­lus busi­ness is the only non-lin­ear rule in all of quan­tum me­chan­ics. Every­thing else—ev­ery­thing else—obeys the lin­ear rule that the evolu­tion of am­pli­tude dis­tri­bu­tion A, plus the evolu­tion of the am­pli­tude dis­tri­bu­tion B, equals the evolu­tion of the am­pli­tude dis­tri­bu­tion A + B.

When you think about the weather in terms of clouds and flap­ping but­terflies, it may not look lin­ear on that higher level. But the am­pli­tude dis­tri­bu­tion for weather (plus the rest of the uni­verse) is lin­ear on the only level that’s fun­da­men­tally real.

Does this mean that the squared-mod­u­lus busi­ness must re­quire ad­di­tional physics be­yond the lin­ear laws we know—that it’s nec­es­sar­ily fu­tile to try to de­rive it on any higher level of or­ga­ni­za­tion?

But even this doesn’t fol­low.

Let’s say I have a com­puter pro­gram which com­putes a se­quence of pos­i­tive in­te­gers that en­code the suc­ces­sive states of a sen­tient be­ing. For ex­am­ple, the pos­i­tive in­te­gers might de­scribe a Con­way’s-Game-of-Life uni­verse con­tain­ing sen­tient be­ings (Life is Tur­ing-com­plete) or some other cel­lu­lar au­toma­ton.

Re­gard­less, this se­quence of pos­i­tive in­te­gers rep­re­sents the time se­ries of a dis­crete uni­verse con­tain­ing con­scious en­tities. Call this se­quence Sen­tient(n).

Now con­sider an­other com­puter pro­gram, which com­putes the nega­tive of the first se­quence: -Sen­tient(n). If the com­puter run­ning Sen­tient(n) in­stan­ti­ates con­scious en­tities, then so too should a pro­gram that com­putes Sen­tient(n) and then negates the out­put.

Now I write a com­puter pro­gram that com­putes the se­quence {0, 0, 0...} in the ob­vi­ous fash­ion.

This se­quence hap­pens to be equal to the se­quence Sen­tient(n) + -Sen­tient(n).

So does a pro­gram that com­putes {0, 0, 0...} nec­es­sar­ily in­stan­ti­ate as many con­scious be­ings as both Sen­tient pro­grams put to­gether?

Ad­mit­tedly, this isn’t an ex­act anal­ogy for “two uni­verses add lin­early and can­cel out”. For that, you would have to talk about a uni­verse with lin­ear physics, which ex­cludes Con­way’s Life. And then in this lin­ear uni­verse, two states of the world both con­tain­ing con­scious ob­servers—world-states equal but for their op­po­site sign—would have to can­cel out.

It doesn’t work in Con­way’s Life, but it works in our own uni­verse! Two quan­tum am­pli­tude dis­tri­bu­tions can con­tain com­po­nents that can­cel each other out, and this demon­strates that the num­ber of con­scious ob­servers in the sum of two dis­tri­bu­tions, need not equal the sum of con­scious ob­servers in each dis­tri­bu­tion sep­a­rately.

So it ac­tu­ally is pos­si­ble that we could pawn off the only non-lin­ear phe­nomenon in all of quan­tum physics onto a bet­ter un­der­stand­ing of con­scious­ness. The ques­tion “How many con­scious ob­servers are con­tained in an evolv­ing am­pli­tude dis­tri­bu­tion?” has ob­vi­ous rea­sons to be non-lin­ear.

(!)

Robin Han­son has made a sug­ges­tion along these lines.

(!!)

De­co­her­ence is a phys­i­cally con­tin­u­ous pro­cess, and the in­ter­ac­tion be­tween LEFT and RIGHT blobs may never ac­tu­ally be­come zero.

So, Robin sug­gests, any blob of am­pli­tude which gets small enough, be­comes dom­i­nated by stray flows of am­pli­tude from many larger wor­lds.

A blob which gets too small, can­not sus­tain co­her­ent in­ner in­ter­ac­tions—an in­ter­nally driven chain of cause and effect—be­cause the am­pli­tude flows are dom­i­nated from out­side. Too-small wor­lds fail to sup­port com­pu­ta­tion and con­scious­ness, or are ground up into chaos, or merge into larger wor­lds.

Hence Robin’s cheery phrase, “man­gled wor­lds”.

The cut­off point will be a func­tion of the squared mod­u­lus, be­cause uni­tary physics pre­serves the squared mod­u­lus un­der evolu­tion; if a blob has a cer­tain to­tal squared mod­u­lus, fu­ture evolu­tion will pre­serve that in­te­grated squared mod­u­lus so long as the blob doesn’t split fur­ther. You can think of the squared mod­u­lus as the amount of am­pli­tude available to in­ter­nal flows of causal­ity, as op­posed to out­side im­po­si­tions.

The se­duc­tive as­pect of Robin’s the­ory is that quan­tum physics wouldn’t need in­ter­pret­ing. You wouldn’t have to stand off beside the math­e­mat­i­cal struc­ture of the uni­verse, and say, “Okay, now that you’re finished com­put­ing all the mere num­bers, I’m fur­ther­more tel­ling you that the squared mod­u­lus is the ‘de­gree of ex­is­tence’.” In­stead, when you run any pro­gram that com­putes the mere num­bers, the pro­gram au­to­mat­i­cally con­tains peo­ple who ex­pe­rience the same physics we do, with the same prob­a­bil­ities.

A ma­jor prob­lem with Robin’s the­ory is that it seems to pre­dict things like, “We should find our­selves in a uni­verse in which lots of very few de­co­her­ence events have already taken place,” which ten­dency does not seem es­pe­cially ap­par­ent.

The main thing that would sup­port Robin’s the­ory would be if you could show from first prin­ci­ples that man­gling does hap­pen; and that the cut­off point is some­where around the me­dian am­pli­tude den­sity (the point where half the to­tal am­pli­tude den­sity is in wor­lds above the point, and half be­neath it), which is ap­par­ently what it takes to re­pro­duce the Born prob­a­bil­ities in any par­tic­u­lar ex­per­i­ment.

What’s the prob­a­bil­ity that Han­son’s sug­ges­tion is right? I’d put it un­der fifty per­cent, which I don’t think Han­son would dis­agree with. It would be much lower if I knew of a sin­gle al­ter­na­tive that seemed equally… re­duc­tion­ist.

But even if Han­son is wrong about what causes the Born prob­a­bil­ities, I would guess that the fi­nal an­swer still comes out equally non-mys­te­ri­ous. Which would make me feel very silly, if I’d em­braced a more mys­te­ri­ous-seem­ing “an­swer” up un­til then. As a gen­eral rule, it is ques­tions that are mys­te­ri­ous, not an­swers.

When I be­gan read­ing Han­son’s pa­per, my ini­tial thought was: The math isn’t beau­tiful enough to be true.

By the time I finished pro­cess­ing the pa­per, I was think­ing: I don’t know if this is the real an­swer, but the real an­swer has got to be at least this nor­mal.

This is still my po­si­tion to­day.

Next post: “De­co­her­ence as Pro­jec­tion

Pre­vi­ous post: “De­co­her­ent Essences

• Psy-Kosh, the am­pli­tudes of ev­ery­thing ev­ery­where could be chang­ing by a con­stant mod­u­lus and phase, with­out it be­ing no­ticed. But if it were pos­si­ble for you to carry out some phys­i­cal pro­cess that changed the squared mod­u­lus of the LEFT blob as a whole, with­out split­ting it and with­out chang­ing the squared mod­u­lus of the RIGHT blob, then you would be able to use this phys­i­cal pro­cess to change the ra­tio of the squared mod­uli of LEFT and RIGHT, hence con­trol the out­come of ar­bi­trary quan­tum ex­per­i­ments by in­vok­ing it se­lec­tively.

It would be an Out­come Pump.

Con­trol­lable uni­tar­ity vi­o­la­tion wouldn’t just let you win the lot­tery, it would let you com­mu­ni­cate faster than light, by forc­ing a par­tic­u­lar out­come in a quan­tum en­tan­gle­ment, Bell’s Inequal­ity type situ­a­tion.

• Roland: yes, at least one. Where did you give up and why?

• My guess is that the Born’s Rule is re­lated to the Solomonoff Prior. Con­sider a pro­gram P that takes 4 in­puts:

• bound­ary con­di­tions for a wavefunction

• a time co­or­di­nate T

• a spa­tial re­gion R

• a ran­dom string

What P does is take the bound­ary con­di­tions, use Schröd­inger’s equa­tion to com­pute the wave­func­tion at time T, then sam­ple the wave­func­tion us­ing the Born prob­a­bil­ities and the ran­dom in­put string, and fi­nally out­put the par­ti­cles in the re­gion R and their rel­a­tive po­si­tions.

Sup­pose this pro­gram, along with the in­puts that cause it to out­put the de­scrip­tion of a given hu­man brain, is what makes the largest con­tri­bu­tion to the prob­a­bil­ity mass of the bit­string rep­re­sent­ing that brain in the Solomonoff Prior. This seems like a plau­si­ble con­jec­ture (putting aside the fact that quan­tum me­chan­ics isn’t ac­tu­ally the TOE of this uni­verse).

(Does any­one think this is not true, or if it is true, has noth­ing to do with the an­swer to the mys­tery of “why squared am­pli­tudes”?)

This idea seems fairly ob­vi­ous, but I don’t re­call see­ing it pro­posed by any­one yet. One pos­si­ble di­rec­tion to ex­plore is to try to prove that any mod­ifi­ca­tion to Born’s rule would cause a dras­tic de­crease in the prob­a­bil­ity that P, given ran­dom in­puts, would out­put the de­scrip­tion of a sen­tient be­ing. But I have no idea how to go about do­ing this. I’m also not sure how to de­velop this ob­ser­va­tion/​con­jec­ture into a full an­swer of the mys­tery.

• I just read in Scott Aaron­son’s Quan­tum Com­put­ing, Post­s­e­lec­tion, and Prob­a­bil­is­tic Polyno­mial-Time that if the ex­po­nent in the prob­a­bil­ity rule was any­thing other than 2, then we’d be able to do post­s­e­lec­tion with­out quan­tum suicide and solve prob­lems in PP. (See Page 6, The­o­rem 6.) The same is true if quan­tum me­chan­ics was non-lin­ear.

Given that, my con­jec­ture is im­plied by one that says “sen­tience is un­likely to evolve in a world where prob­lems in PP (which is prob­a­bly strictly harder than PH, which is prob­a­bly strictly harder than NP) can be eas­ily solved” (pre­sum­ably be­cause in­tel­li­gence wouldn’t be use­ful in such a world).

• In­ter­est­ing. What would such a world look like? I imag­ine in­stead of a se­lec­tion pres­sure for in­tel­li­gence there would be a se­lec­tion pres­sure for raw mem­ory, so that you could perfectly model any crea­ture with less mem­ory than your­self. It seems that this would be a very in­tense pres­sure, since the up­per hand is es­sen­tially guaran­teed su­pe­ri­or­ity, and you would ul­ti­mately wind up with galaxy sized com­put­ers run­ning through all pos­si­ble simu­la­tions of other galaxy sized com­put­ers.

I never put much stock in the simu­la­tion hy­poth­e­sis, be­cause I couldn’t see why an en­tity ca­pa­ble of simu­lat­ing our uni­verse would de­rive any value from do­ing it. This sce­nario makes me re­think that a lit­tle.

In any case, while this is an­other po­ten­tial rea­son why the rule must be 2 in our uni­verse, it still doesn’t shed any light on the mechanism by which our sub­jec­tive ex­pe­rience fol­lows this rule.

• What would such a world look like?

I don’t know. I don’t have a very good un­der­stand­ing of reg­u­lar quan­tum com­put­ing, much less the non-Born “fan­tasy” quan­tum com­put­ers that Aaron­son used in his pa­per. But I’m go­ing to guess that your spec­u­la­tion is prob­a­bly wrong, un­less you hap­pen to be an ex­pert in this area. Th­ese things tend not to be very in­tu­itive at all.

• I hon­estly can’t imag­ine my evolu­tion story is right. It just seemed like an im­mensely fun op­por­tu­nity for spec­u­la­tion.

• Sup­pose this pro­gram, along with the in­puts that cause it to out­put the de­scrip­tion of a given hu­man brain, is what makes the largest con­tri­bu­tion to the prob­a­bil­ity mass of the bit­string rep­re­sent­ing that brain in the Solomonoff Prior.

More speci­fi­cally, to re­place my pre­vi­ous sum­mary com­ment: the above state­ment sounds kind-a re­deemable, but it’s so vague and com­mon-sen­su­ally ab­surd that I think it makes a nega­tive con­tri­bu­tion. Things like this need to be said clearly, or not at all. It in­vites all sorts of kook­ery, not just with the for­mat of pre­sen­ta­tion, but in own mind as well.

• Huh, that’s a sur­pris­ing re­sponse. I thought that at least the in­tended mean­ing would be ob­vi­ous for some­one fa­mil­iar with the Solomonoff Prior. I guess “vague” I can ad­dress by mak­ing my claim math­e­mat­i­cally pre­cise, but why “com­mon-sen­su­ally ab­surd”?

• Re ab­surd: It’s not clear why you would say some­thing like the quote.

• I was hop­ing that it would trig­ger an in­sight in some­one who might solve this mys­tery for me. As I said, I’m not sure how to de­velop it into a full an­swer my­self (but it might be re­lated to this other vague/​pos­si­bly-ab­surd idea).

Per­haps I’m abus­ing this com­mu­nity by pre­sent­ing ideas that are half-formed and “epistem­i­cally un­hy­gienic”, but I ex­pect that’s not a se­ri­ous dan­ger. It seems like a promis­ing di­rec­tion to ex­plore, that I don’t see any­one else ex­plor­ing (kind of like UDT un­til re­cently). I have too many ques­tions I’d like to see an­swered, and not enough time and abil­ity to an­swer them all my­self.

• Hello Wei Dai. Your paradigm is a bit opaque to me. There’s a cos­mol­ogy here which in­volves pro­grams, pro­gram out­puts, and prob­a­bil­ity dis­tri­bu­tions over each, but I can’t tell what’s sup­posed to ex­ist. Just the pro­gram out­puts? The pro­gram out­puts and the pro­grams? Does the pro­gram cor­re­spond to “ba­sic phys­i­cal law”, and pro­gram out­put to “the phys­i­cal world”?

If I try to ab­stract away from the meta­phys­i­cal idiosyn­crasies, the idea seems to be that Born’s rule is true be­cause the wor­lds which func­tion ac­cord­ing to Born’s rule are the ma­jor­ity of the wor­lds in which sen­tient be­ings show up. Well, it could be true. But here’s an in­ter­est­ing Bohmian fact: if you start out with an en­sem­ble of Bohmian wor­lds de­vi­at­ing from the Born dis­tri­bu­tion, they will ac­tu­ally con­verge on it, solely due to Bohmian dy­nam­ics. (See quant-ph/​0403034.) So some­thing like the Bohmian equa­tion of mo­tion may ac­tu­ally be the more fun­da­men­tal fact.

• In gen­eral, I think what ex­ists are math­e­mat­i­cal struc­tures, which in­clude com­pu­ta­tions as a sub­class.

But here’s an in­ter­est­ing Bohmian fact: if you start out with an en­sem­ble of Bohmian wor­lds de­vi­at­ing from the Born dis­tri­bu­tion, they will ac­tu­ally con­verge on it, solely due to Bohmian dy­nam­ics.

Thanks for the link. That looks in­ter­est­ing, and I have a cou­ple of ques­tions that maybe you help me with.

1. Why do they con­verge to the Born dis­tri­bu­tion? The au­thors make an anal­ogy with ther­mal re­lax­ation, but there is a stan­dard ex­pla­na­tion of the sec­ond law of ther­mo­dy­nam­ics in terms of sizes of macrostates in con­figu­ra­tion space, and I don’t see what the equiv­a­lent ex­pla­na­tion is for Bohmian re­lax­ation.

2. What about de­co­her­ence? Sup­pose you have a wave­func­tion that has de­co­hered into two ap­prox­i­mately non-in­ter­act­ing branches oc­cu­py­ing differ­ent parts of con­figu­ra­tion space. If you start with a Bohmian world that be­longs to one branch, then in all like­li­hood its fu­ture evolu­tion will stay within that branch, right? Now if you take an en­sem­ble of Bohmian wor­lds that all be­long to that branch, how will it con­verge to the Born dis­tri­bu­tion, which oc­cu­pies both branches?

3. This is more of an ob­jec­tion to the Bohmian on­tol­ogy than a ques­tion. If you look at Bohmian Me­chan­ics as a com­pu­ta­tion, it con­sists of two parts: (1) evolu­tion of the wave­func­tion, and (2) evolu­tion of a point in con­figu­ra­tion space, guided by the wave­func­tion. But it seems like all of the real work is be­ing done in part 1. If you wanted to simu­late a quan­tum sys­tem, for ex­am­ple, it seems suffi­cient to just do part 1, and then sam­ple the re­sult­ing wave­func­tion ac­cord­ing to Born’s rule, and part 2 adds more com­plex­ity and com­pu­ta­tional bur­den with­out any ap­par­ent benefit.

• Why do they con­verge to the Born dis­tri­bu­tion?”

Let’s dis­t­in­guish two ver­sions of this ques­tion. First ver­sion: why does a generic non-Born en­sem­ble of Bohmian wor­lds tend to be­come Born-like? I think the tech­ni­cal an­swer is to be found in foot­note 9 and the dis­cus­sion around equa­tion 20. But ul­ti­mately I think it will come back to a Liou­ville the­o­rem in the space of dis­tri­bu­tions. There is some nat­u­ral met­ric un­der which the Born-like dis­tri­bu­tions are the ma­jor­ity. (Or per­haps it is that non-Born re­gions are tra­versed rel­a­tively quickly.)

Se­cond ver­sion: why does an in­di­vi­d­ual Bohmian world con­tain a Born dis­tri­bu­tion of out­comes? This fol­lows from the first part. An in­di­vi­d­ual Bohmian world con­sists of a uni­ver­sal wave­func­tion and a qua­si­clas­si­cal tra­jec­tory. If you pick just a few of the clas­si­cal vari­ables, you can con­struct a cor­re­spond­ing re­duced den­sity ma­trix in the usual fash­ion, and a re­duced Bohmian equa­tion of mo­tion in which the evolu­tion of those vari­ables de­pends on that den­sity ma­trix and on in­fluences com­ing from all the de­grees of free­dom that were traced over. So when you look at all the in­stances, within a sin­gle Bohmian his­tory, of a par­tic­u­lar phys­i­cal pro­cess, you are look­ing at an en­sem­ble of noisy Bohmian micro­his­to­ries. The ar­gu­ment above sug­gests that even if this starts as a non-Born en­sem­ble, it will evolve into a Born-like en­sem­ble. The only com­pli­ca­tion is the noise fac­tor. But it is at least plau­si­ble that in the ma­jor­ity of Bohmian wor­lds, this non­lo­cal noise is just noise and does not in­tro­duce an anti-Born ten­dency.

From an all-wor­lds-ex­ist per­spec­tive, which we both fa­vor, I would sum­ma­rize as fol­lows: (1) the Born dis­tri­bu­tion is the nat­u­ral mea­sure on the sub­set of wor­lds con­sist­ing of the Bohmian wor­lds (2) most Bohmian wor­lds will ex­hibit an in­ter­nal Born dis­tri­bu­tion of phys­i­cal out­comes. At pre­sent these are con­jec­tures rather than the­o­rems, but I would con­sider them plau­si­ble con­jec­tures in the light of Valen­tini’s work.

As we’ve just dis­cussed, Bohmian dy­nam­ics both pre­serves ex­act Born dis­tri­bu­tions and evolves non-Born dis­tri­bu­tions to­wards Born-like dis­tri­bu­tions (and this is true for sub­sys­tems of a Bohmian world as well as for the whole). So the sub-en­sem­bles in the de­co­hered branches will pre­serve or evolve to­wards Born.

“part 2 adds more com­plex­ity and com­pu­ta­tional bur­den with­out any ap­par­ent benefit”

This is a com­pli­cated mat­ter to dis­cuss, not least be­cause there is an in­ter­pre­ta­tion of Bohmian me­chan­ics, the nomolog­i­cal in­ter­pre­ta­tion, ac­cord­ing to which the “wave­func­tion” is a law of mo­tion and not a thing. In nomolog­i­cal Bohmian me­chan­ics, the con­figu­ra­tion is all that ex­ists, evolv­ing ac­cord­ing to a non­lo­cal po­ten­tial.

• Is it pos­si­ble (I’m not sure it makes sense to ask about easy) un­der our physics to build an in­tel­li­gence that op­ti­mizes (or at least a struc­ture that prop­a­gates it­self) ac­cord­ing to some met­ric other than the Born Rule? If not, then it should be an­throp­i­cally un­sur­pris­ing that we per­ceive prob­a­bil­ity as squared am­pli­tude, even if there is no law of physics to that effect. Otoh if it is pos­si­ble, then you could have a TOE from which you can’t de­rive how to com­pute prob­a­bil­ity, and there’s noth­ing wrong with that, be­cause then there re­ally is an­other way to in­ter­pret prob­a­bil­ity that other peo­ple in the uni­verse (though of course not in our Everett branch) may be us­ing.

Fair rephras­ing?

• The Solomonoff prior de­pends on the en­cod­ing of al­gorithms, the Born rule doesn’t. Or am I miss­ing any­thing?

• That seems like a gen­eral ar­gu­ment against the whole Solomonoff In­duc­tion ap­proach. I’d be happy to see the de­pen­dence on an en­cod­ing of al­gorithms re­moved, but un­til some­one finds a way to do so, it doesn’t seem to be a deal-breaker. I think my claim should ap­ply to any en­cod­ing of al­gorithms one might use that isn’t con­trived speci­fi­cally to make it false.

• Thanks to Eliezer’s QM se­ries, I’m start­ing to have enough back­ground to un­der­stand Robin’s pa­per (kind of, maybe). And now that I do (kind of, maybe), it seems to me that Robin’s point is com­pletely de­mol­ished by Wal­lace’s points about de­co­her­ence be­ing con­tin­u­ous rather than dis­crete and there­fore there be­ing no such thing as a num­ber of dis­crete wor­lds to count.

There seems to be noth­ing to re­solve be­tween the prob­a­bil­ities given by mea­sure and the prob­a­bil­ities im­plied by world count if you sim­ply say that mea­sure is prob­a­bil­ity.

Eliezer ob­jects. We’re in­ter­pret­ing. We’re adding some­thing out­side the math­e­mat­ics.

I fail to see the prob­lem.

If we’re to ac­cept that par­ti­cles mov­ing like billiard balls are an illu­sion, and con­figu­ra­tion space is real, and blobs of am­pli­tude are real, and time evolu­tion of am­pli­tude within con­figu­ra­tion space ac­cord­ing to the wave equa­tions is real, and that con­figu­ra­tions and am­pli­tude and wave equa­tions are fun­da­men­tal parts of re­al­ity, be­cause that’s the best model we’ve come up with that agrees with ex­per­i­men­tal ob­ser­va­tion… why not ac­cept that the mod­u­lus-squared law is real and fun­da­men­tal, too?

It cer­tainly agrees with ex­per­i­men­tal ob­ser­va­tions, and doesn’t seem any less de­sir­able a part of our model of re­al­ity than con­figu­ra­tions, am­pli­tude blobs, and wave equa­tions.

I wish some­one would ex­plain the prob­lem more clearly, al­though if Eliezer’s ex­pla­na­tions so far haven’t cleared it up for me yet, per­haps noth­ing will.

• why not ac­cept that the mod­u­lus-squared law is real and fun­da­men­tal, too?

Read­ing through this, and Han­son’s quick overview page of man­gled wor­lds, I was won­der­ing the same thing my­self. For some rea­son though, see­ing you ask the ques­tion I hadn’t quite ver­bal­ized put the an­swer right on the tip of my tongue: for the same rea­son Ein­stein was so sure of Gen­eral Rel­a­tivity. The mod­u­lus squared law con­flicts with a reg­u­lar­ity in the form that the fun­da­men­tal laws seem to take, speci­fi­cally their lin­ear evolu­tion, and Eliezer puts stock in that reg­u­lar­ity. In fact, he does so suffi­ciently to let him ele­vate any the­ory which ac­counts for the data while hold­ing the reg­u­lar­ity far above those that don’t, similar to how Ein­stein picked GR out of hy­poth­e­sis space.

The benefit of the man­gled wor­lds in­ter­pre­ta­tion is that while the uni­verse-am­pli­tude-blobs do have mea­sure (a non-lin­ear el­e­ment), it is ir­rele­vant to what ac­tu­ally hap­pens. It re­ally only comes into play when try­ing to un­der­stand the in­ter­ac­tion be­tween the uni­verse-am­pli­tude-blobs, but it doesn’t play a part in ac­tu­ally de­scribing that in­ter­ac­tion. For ex­am­ple, the pos­si­ble man­gling of a world of small mea­sure would be de­scribed by nor­mal lin­ear quan­tum evolu­tion, but since the calcu­la­tions are not very nice, we can de­ter­mine whether it would be man­gled us­ing that mea­sure. Thus, we are us­ing the mea­sure as a math­e­mat­i­cal short­cut to de­ter­mine gen­er­al­ized be­hav­ior, but all evolu­tion is lin­ear, and ob­ser­va­tions can be ex­plained with­out the ex­tra hy­poth­e­sis that “mea­sure is prob­a­bil­ity”.

• You wouldn’t have to stand off beside the math­e­mat­i­cal struc­ture of the uni­verse, and say, “Okay, now that you’re finished com­put­ing all the mere num­bers, I’m fur­ther­more tel­ling you that the squared mod­u­lus is the ‘de­gree of ex­is­tence’.”

In­stead, you’d have to stand off beside the math­e­mat­i­cal struc­ture of the uni­verse, and say, “Okay, now that you’re finished com­put­ing all the mere num­bers, I’m fur­ther­more tel­ling you that the world count is the ‘de­gree of ex­is­tence’.”

• I’m a bit puz­zled by the prob­lem here. What’s wrong with the in­ter­pre­ta­tion that the Born prob­a­bil­ities just are the limit­ing fre­quen­cies in in­finite in­de­pen­dent rep­e­ti­tions of the same ex­per­i­ment? Fur­ther, that these limit­ing fre­quen­cies re­ally are defined be­cause the uni­verse re­ally is spa­tially in­finite, with in­finitely many causally iso­lated re­gions. There is noth­ing hy­po­thet­i­cal at all about the in­finite rep­e­ti­tion—it ac­tu­ally hap­pens.

My un­der­stand­ing is that in such a uni­verse model, the Everett-Wheeler ver­sion of quan­tum the­ory makes a pre­cise pre­dic­tion: the limit­ing fre­quen­cies will with cer­tainty cor­re­spond to the Born prob­a­bil­ities be­cause the am­pli­tude van­ishes com­pletely over the sub­space of Hilbert space where they don’t. More for­mally, the wave func­tion of the uni­verse is in an eigen­state of the rel­a­tive fre­quency op­er­a­tor with the eigen­value equal to the Born prob­a­bil­ity. Job done, surely?

Is the ob­jec­tion here just that we don’t want to be­lieve that the uni­verse is spa­tially in­finite?

Well why on Earth(s) would a MWI fan have any prob­lem with that at all? Is it re­ally any harder to be­lieve that each branch of the wave func­tion de­scribes a strictly in­finite uni­verse (but that these in­finite uni­verses are all es­sen­tially iden­ti­cal, be­cause they all have the cor­rect fre­quency limits) than to be­lieve that each branch de­scribes a finite uni­verse, and that while some of the branches get the fre­quency limits right, most of them don’t?

• That gave me, if I am not mis­taken, the last piece of the puz­zle. Let’s just take the naive defi­ni­tion of prob­a­bil­ity—the rel­a­tive fre­quency of out­comes as N goes to in­finity. Now pre­pare N sys­tems in­de­pen­dently in the state a|0>+b|1>. Now mea­sure one af­ter an­other—cou­ple the mea­sure­ment de­vice to the sys­tem. At first we have (a|0>+b|1>)^N |0>. Now the first one is mea­sured: (a|0>+b|1>)^(N-1) (a|0,0>+b|1,1>) where the num­ber af­ter the comma de­notes the state of the mea­sur­ing de­vice, which just counts the num­ber of mea­sured ones. After the sec­ond mea­sure­ment we have (a|0>+b|1>)^(N-2) (a²|00,0>+ab|01,1>+ab|10,1>+b²|11,2>) Since the two states ab|01,1> and ab|10,1> are not dis­t­in­guished by the mea­sure­ment, the ba­sis should be changed—and this is the cru­cial point: |01>+|10> has a length of sqrt(2), so if we change the ba­sis to |+>=(|01>+|10>)/​sqrt(2), we have (a|0>+b|1>)^(N-2) (a²|00,0>+ab­sqrt(2)|+,1>+b²|11,2>).

The co­effi­ci­ants are like in the bino­mial the­o­rem, but note the sqare root!

Con­tin­u­ing, we will get some­thing similar to a bino­mial dis­tri­bu­tion:

sum(k=0..N: sqrt(N!/​(k!(N-k)!))a^k b^(N-k) |...,k>).

Now it re­mains to prove that for j/​N not equal to a² the am­pli­tudes go to zero as N goes to in­finity. This is equiv­a­lent to the square of the am­pli­tude go­ing to zero (this is just to make the calcu­la­tion eas­ier, it does not have any­thing to do with the Born rule). It is, for |...,k>,

ck² = N!/​(k!(N-k)!) a²^k b²^(N-k)

which be­comes a Gaus­sian dis­tri­bu­tion for large N, with mean at k=Na² and width Na²b². So at k/​N=a²+d it has a value pro­por­tional to exp(-(Nd)²/​(2Na²b²))=exp(-Nd²/​(2a²b²)) --> 0 as N --> inf.

So a time cap­sule where the records in­di­cate that some quan­tum ex­per­i­ment has been performed a great num­ber of times and the Born rule is bro­ken will have an am­pli­tude that goes to zero (yeah, I just read Bar­bour’s book).

• Yes, this is called the Finkel­stein-Har­tle the­o­rem (D. Finkel­stein, Trans­ac­tions of the New York Academy of Sciences 25, 621 (1963); J. B. Har­tle, Am. J. Phys. 36, 704 (1968)).

This the­o­rem is the ba­sis for con­struct­ing a limit op­er­a­tor for the rel­a­tive fre­quency when there are in­finitely many in­de­pen­dent rep­e­ti­tions of a mea­sure­ment, and show­ing that the product wave-func­tion is an ex­act eigen­state of the rel­a­tive fre­quency op­er­a­tor. Un­for­tu­nately, it seems that Har­tle’s con­struc­tion of the fre­quency op­er­a­tor wasn’t quite right, and needed to be gen­er­al­ized. (E. Farhi, J. Gold­stone, and S. Gut­mann, Ann. Phys. 192, 368 (1989)).

Even so, the crit­ics are still picky about the con­struc­tion. There is a line of crit­i­cism that in­finite fre­quency op­er­a­tors can be con­structed ar­bi­trar­ily as func­tions over Hilbert space, and un­less you already know the Born rule, you won’t know how to con­struct one sen­si­bly (so that the Har­tle deriva­tion is cir­cu­lar). How­ever this seems un­fair, be­cause if you want the rel­a­tive fre­quency op­er­a­tor to obey the Kol­mogorov ax­ioms of prob­a­bil­ity then it has to co­in­cide with the Born rule, some­thing which is an­other long-stand­ing re­sult called Glea­son’s the­o­rem. (The squared mod­u­lus of the am­pli­tude is the only func­tion of the mea­sure which fol­lows the ax­ioms of prob­a­bil­ity.) Hence the full deriva­tion is:

1) (Pos­tu­late) If the wave­func­tion is in an eigen­state of a mea­sure­ment op­er­a­tor, then the mea­sure­ment will with cer­tainty have the cor­re­spond­ing eigen­value.

2) (Pos­tu­late) Prob­a­bil­ity is rel­a­tive fre­quency over in­finitely many in­de­pen­dent rep­e­ti­tions.

3) (Pos­tu­late) Rel­a­tive fre­quency fol­lows the Kol­mogorov ax­ioms of prob­a­bil­ity.

4) (Glea­son’s the­o­rem) Rel­a­tive fre­quency must con­verge to the Born rule (squared mod­u­lus of am­pli­tude) over in­finitely many rep­e­ti­tions, or it won’t be able to fol­low the Kol­mogorov ax­ioms.

5) (Har­tle’s the­o­rem, as strength­ened by Farhi et al) There is a unique defi­ni­tion of the rel­a­tive fre­quency op­er­a­tor over in­finite rep­e­ti­tions, and such that the in­finite product state is an eigen­state of the rel­a­tive fre­quency op­er­a­tor.

6) (Con­clu­sion) The rel­a­tive fre­quency over in­finitely many mea­sure­ments is with cer­tainty the Born prob­a­bil­ity.

It seem pretty clean to me.

• A ma­jor prob­lem with Robin’s the­ory is that it seems to pre­dict things like, We should find our­selves in a uni­verse in which lots of de­co­her­ence events have already taken place,” which ten­dency does not seem es­pe­cially ap­par­ent.

Ac­tu­ally the the­ory sug­gests we should find our­selves in a state with near the least fea­si­ble num­ber of past de­co­her­ence events

I don’t un­der­stand this—doesn’t de­co­her­ence oc­cur all the time, in ev­ery quan­tum in­ter­ac­tion be­tween all am­pli­tudes all the time? So, like for ev­ery amptli­tude sep­a­rate enough to be a “par­ti­cle” in the uni­verse (=fac­tor) ev­ery planck time it will de­co­here with other fac­tors?

Or did I mi­s­un­der­stand some­thing big time here?

Cheers, Peter

• I’d also love to know the an­swer to Peter’s ques­tion… A similar ques­tion is whether we should ex­pect all wor­lds to even­tu­ally be­come man­gled (as­sum­ing the “man­gled wor­lds” model). I un­der­stand “world” to mean “some­what iso­lated blob of am­pli­tude in an am­pli­tude dis­tri­bu­tion”—is that right?

• The an­swer to Peter’s ques­tion is: no, de­co­her­ence doesn’t hap­pen with a con­stant rate and it cer­tainly doesn’t hap­pen on the Planck time scale.

The an­swer to your ques­tion is that “man­a­gled wor­lds” is a col­lapse the­ory: some wor­lds get man­a­gled and go away, leav­ing other wor­lds.

• Then I’m still un­clear about what a world is. Care to ex­plain?

• Eliezer gave a sim­pler an­swer to my ques­tion: “yes”. (I’m still not sure what yours means.)

Back to Peter’s ques­tion. What makes you say de­co­her­ence doesn’t hap­pen on the Planck time scale? Can you ex­plain that fur­ther?

• Any given in­stance of de­co­her­ence is an in­ter­ac­tion be­tween two or more par­ti­cles. And all known in­ter­ac­tions take rather longer than Planck time.

There prob­a­bly are enough de­co­her­ence events in the uni­verse that at least one oc­curs some­where in each Plank time­u­nit. But that doesn’t in­stantly de­co­here ev­ery­thing. Other ob­jects re­main co­her­ent un­til they in­ter­act with the de­co­hered sys­tem, which is limited by the rate at which in­for­ma­tion prop­a­gates (both la­tency and band­width) (un­less of course they de­co­here on their own). i.e. af­ter a blob of am­pli­tude has split, the sub-blobs are only sep­a­rated along some di­men­sions of con­figu­ra­tion space, and re­tain the same cross-sec­tion along the rest of the di­men­sions (hence “fac­tors”).

• Okay, given one sub-de­co­her­ence event per planck time, some­where in the uni­verse, prop­a­gat­ing through­out it at some rate less than or equal to the speed of light...we ei­ther have con­stant (one per planck time or less) full de­co­her­ence events af­ter some fixed time as each finishes prop­a­gat­ing suffi­ciently, or we have no full de­co­her­ence events at all as the sub-de­co­her­ences fail to de­co­here the whole suffi­ciently.

The lat­ter seems more re­al­is­tic, es­pe­cially given the light speed limit, as the ex­pan­sion of space can com­pletely causally iso­late two parts of the uni­verse pre­vent­ing the prop­a­ga­tion of the de­co­her­ence.

So, with this un­der­stood, we’re left to de­ter­mine how large a por­tion of the uni­verse has to be de­co­hered to qual­ify as a “de­co­her­ence event” in terms of the many wor­lds the­o­ries which rely on the term. I hon­estly doubt that, once a suit­able de­ter­mi­na­tion has been made, the events will be in­fre­quent in al­most any sense of the word. It re­ally does seem, given the mas­sive quan­tities of in­ter­ac­tions in our uni­verse(even just the causally linked sub­space of it we in­habit), that the fre­quency of de­co­her­ence events should be ridicu­lously high. And given some ba­sic unifor­mity as­sump­tions, the rate should be quite reg­u­lar too.

• “I will point out, though, that the ques­tion of how con­scious­ness is bound to a par­tic­u­lar branch (and thus why the Born rule works like it does) doesn’t seem that much differ­ent from how con­scious­ness is tied to a par­tic­u­lar point in time or to a par­tic­u­lar brain when the Spaghetti Mon­ster can see all brains in all times and would have to be given ex­tra in­for­ma­tion to know that my con­scious­ness seems to be liv­ing in this par­tic­u­lar brain at this par­tic­u­lar time.”

Agreed!

More gen­er­ally, it seems to me that many ob­jec­tions peo­ple raise about the foun­da­tions of QM ap­ply equally well to clas­si­cal physics when you re­ally think about it.

How­ever, I think Eli’s ob­jec­tion to the Born rule is differ­ent. The spe­cial weird thing about quan­tum me­chan­ics as cur­rently un­der­stood is that Born’s rule seems to sug­gest that the bind­ing of qualia is a sep­a­rate rule in fun­da­men­tal physics.

• None of the con­fu­sion over du­pli­ca­tion and quan­tum mea­sures seems unique to be­ings with qualia; any Bayesian sys­tem ca­pa­ble of an­thropic rea­son­ing, it would seem, should be sur­prised the uni­verse is or­derly. So maybe ei­ther the con­fu­sion is sep­a­rate from and deeper than ex­pe­rience, or AIXItl has qualia.

• Ed­die,

My un­der­stand­ing of Eli’s beef with the Born rule is this (he can cor­rect me if I’m wrong): the Born rule ap­pears to be a bridg­ing rule in fun­da­men­tal physics that di­rectly tells us some­thing about how qualia bind to the uni­verse. This seems odd. Fur­ther­more, if the bind­ing of qualia to the uni­verse is given by a sep­a­rate fun­da­men­tal bridg­ing rule in­de­pen­dent of the other laws of physics, then the zom­bie world re­ally is log­i­cally pos­si­ble, or in other words epiphe­nom­e­nal­ism is true. (Just pos­tu­late a uni­verse with all the laws of physics ex­cept Born’s bridg­ing rule. Such a uni­verse is, as far as we know, log­i­cally con­sis­tent.) Eli ar­gues against epiphe­nom­e­nal­ism on the grounds that if epiphe­nom­e­nal­ism is true, then the cor­re­la­tion be­tween be­liefs (which are qualia) with our state­ments and ac­tions (which are phys­i­cal pro­cesses) is just a mirac­u­lous co­in­ci­dence.

What fol­lows are my own com­ments as op­posed to a sum­mary of what I be­lieve Eli thinks:

Why can’t the cor­re­la­tion be­tween phys­i­cal states and be­liefs arise by an ar­row of cau­sa­tion that goes from the phys­i­cal states to the be­liefs? In this case epiphe­nom­e­nal­ism would be true (since qualia have no effect on the phys­i­cal world), but the cor­re­la­tion would not be a co­in­ci­dence (since the phys­i­cal world di­rectly causes qualia). I think the ob­jec­tion to this is that if there re­ally is a bridg­ing law, then the co­in­ci­dence re­mains that it is such a rea­son­able bridg­ing law. That is, what we say we ex­pe­rience and phys­i­cally act as though we ex­pe­rience ac­tu­ally matches (usu­ally) what we do ex­pe­rience, as op­posed to re­lat­ing to what we do ex­pe­rience in some ar­bi­trar­ily scram­bled way. If qualia bind to some higher emer­gent level hav­ing to do with in­for­ma­tion pro­cess­ing, then it seems non-co­in­ci­den­tal that the bridg­ing law is rea­son­able. (Be­cause the things it is map­ping be­tween seem to have a close and clear re­la­tion­ship.) How­ever, the Born rule seems to sug­gest that the bridg­ing rule is at the level of fun­da­men­tal physics.

Maybe if we could de­rive the Born rule as a prop­erty of the in­for­ma­tion pro­cess­ing performed by a quan­tum uni­verse the mys­tery would go away.

• “Eli ar­gues against epiphe­nom­e­nal­ism on the grounds that if epiphe­nom­e­nal­ism is true, then the cor­re­la­tion be­tween be­liefs (which are qualia) with our state­ments and ac­tions (which are phys­i­cal pro­cesses) is just a mirac­u­lous co­in­ci­dence.”

Sup­pos­ing he does, I must point out that it is false to say that be­liefs are qualia. In fact, be­liefs are part of the in­ten­tional stance. That is well worked out in Den­nett’s book by the same name.

The in­ten­tional level can be ac­counted for in phys­i­cal terms (See for in­stance “Kinds of Minds” by Den­nett to see how in­ten­tion­al­ity un­folds from genes to amoe­bas to Karl Pop­per.

One could in­sist on be­ing a phe­nom­e­nal re­al­ist, and say that be­liefs are both an in­ten­tional in­ter­pre­ta­tion of a phys­i­cal sys­tem that can be ac­counted for with­out the aid of qualia, and fur­ther­more that there was an­other as­pect of be­liefs that is the ex­pe­ri­en­tial as­pect, the qualia-ness of them.

Even hold­ing such a po­si­tion, one needs only to ex­plain our be­liefs as long as they are phys­i­cally causally effec­tive upon the world (for in­stance caus­ing us to talk about qualia, be­liefs, etc..).

So if there are be­liefs as in­ten­tional de­scrip­tions of or­ganisms, AND in ad­di­tion be­liefs as qualia, the sec­ond kind is UTTERLY un­ex­plain­able by its very na­ture.

There is no need to ac­count for them, be­cause we have no rea­son to be­lieve they ex­ist, since if they did, they would not figure in our the­o­ries, be­ing causally in­nefi­cient.

• A ma­jor prob­lem with Robin’s the­ory is that it seems to pre­dict things like, “We should find our­selves in a uni­verse in which lots of de­co­her­ence events have already taken place,” which ten­dency does not seem es­pe­cially ap­par­ent.

Ac­tu­ally the the­ory sug­gests we should find our­selves in a state with near the least fea­si­ble num­ber of past de­co­her­ence events. Yes, it is not clear if this in fact holds, and yes I’d put the chance of some­thing like man­gled wor­lds be­ing right as more like 14 or 13.

• Hi,

I haven’t com­mented on a while. I’m just cu­ri­ous, are there any non-physi­cists who are able to fol­low this whole quan­tum-se­ries? I’ve given up some posts ago.

Peace!

• I guess I was too quick to as­sume that man­gled wor­lds in­volved some ad­di­tional pro­cess. Oops.

• Sup­pose that the prob­a­bil­ity of an ob­server-mo­ment is de­ter­mined by its com­plex­ity, in­stead of the prob­a­bil­ity of a uni­verse be­ing de­ter­mined by its com­plex­ity and the prob­a­bil­ity of an ob­ser­va­tion within that uni­verse be­ing de­scribed by some differ­ent an­thropic se­lec­tion.

You can spec­ify a par­tic­u­lar hu­man’s brain by de­scribing the uni­ver­sal wave func­tion and then point­ing to a brain within that wave func­tion. Now the mere “phys­i­cal ex­is­tence” of the brain is not rele­vant to ex­pe­rience; it is nec­es­sary to de­scribe pre­cisely how to ex­tract a de­scrip­tion of their thoughts from the uni­ver­sal wave func­tion. The sig­nifi­cance of the ob­server mo­ment de­pends on the com­plex­ity of this speci­fi­ca­tion.

How might you spec­ify a brain within the uni­ver­sal wave­func­tion? The de­tails are slightly tech­ni­cal, but in­tu­itively: de­scribe the uni­verse, spec­ify a ran­dom seed to an al­gorithm which sam­ples clas­si­cal con­figu­ra­tions with prob­a­bil­ity pro­por­tional to the am­pli­tude squared, and then point to the brain within the re­sult­ing con­figu­ra­tion.

Of course, you could also write down the al­gorithm which sam­ples clas­si­cal con­figu­ra­tions with prob­a­bil­ity pro­por­tional to the am­pli­tude, or the am­pli­tude cubed, etc. and I would have to pre­dict that all of the ob­server-mo­ments gen­er­ated in this way also ex­ist. In the same sense, I would have to pre­dict that all of the ob­server-mo­ments gen­er­ated by other laws of physics also ex­ist, with prob­a­bil­ity de­cay­ing ex­po­nen­tially with the com­plex­ity of those laws (and no­tice that ob­server mo­ments gen­er­ated ac­cord­ing to QM with non-Born prob­a­bil­ities are just as for­eign as ob­server mo­ments gen­er­ated with wildly differ­ent phys­i­cal the­o­ries).

Why do we ex­pect the Born rules to hold when we perform an ex­per­i­ment to­day? The same rea­son we ex­pect the same laws of physics that cre­ated our uni­verse to con­tinue to ap­ply in our labs. More pre­cisely:

In or­der to find the blob of am­pli­tude which cor­re­sponds to Earth as we know it, you have to use the Born prob­a­bil­ities to sam­ple. If you use some sig­nifi­cantly differ­ent dis­tri­bu­tion then physics looks com­pletely differ­ent. There are prob­a­bly no stars be­hav­ing like we ex­pect stars to be­have, atoms don’t be­have rea­son­ably, etc. So in or­der to pick out our Earth you need to use the Born prob­a­bil­ities.

You could de­scribe a brain by say­ing “Use the Born prob­a­bil­ities to find hu­man so­ciety, and then use this other sam­pling method to find a brain” or maybe “Use the Born prob­a­bil­ities ev­ery­where ex­cept for this ex­per­i­men­tal out­come.” But this is only true in the same sense that you could spec­ify a con­figu­ra­tion for the uni­verse by say­ing “Use these laws of physics for a while, and then switch to these other laws.” We don’t ex­pect it be­cause non-unifor­mity sig­nifi­cantly in­creases com­plex­ity.

As far as I can tell, the re­main­ing mys­tery is the same as “why these laws of physics?” An ob­ser­va­tion like “If you use the prob­a­bil­ities cubed, you get one messed up uni­verse.” would be helpful to this ques­tion, as would an ob­ser­va­tion like “it turns out that there is a sim­ple way to sam­ple con­figu­ra­tions with prob­a­bil­ity pro­por­tional to am­pli­tude squared, but not am­pli­tude,” but nei­ther ob­ser­va­tion is any more use­ful or nec­es­sary than “If you used clas­si­cal prob­a­bil­ities in­stead of quan­tum prob­a­bil­ities, you wouldn’t have life” or “it turns out that there is a very sim­ple way to de­scribe quan­tum me­chan­ics, but not clas­si­cal prob­a­bil­ities.”

This ques­tion no longer seems mys­te­ri­ous to me; some­one would have to give a con­vinc­ing ar­gu­ment for me to keep think­ing about it.

• Does your ar­gu­ment work as a post hoc ex­pla­na­tion of any reg­u­lar sys­tem of physics and sam­pling laws, pro­vided you’re an ob­server that finds it­self within it?

• First of all—great se­quence! I had a lot of ‘I see!’-mo­ments read­ing it. I study physics, but of­ten the clear pic­ture gets lost in the stan­dard ap­proach and one is left with a lot of calcu­lat­ing tech­niques with­out any in­tu­itive grasp of the sub­ject. After read­ing this I be­came very fond of tu­tor­ing the course on quan­tum me­chan­ics and always tried to give some deeper in­sight (many of which was taken from here) in ad­di­tion to just ex­plain­ing the ex­er­cises. If I am cor­rect, the world man­gling the­ory just tries to ex­plain some anoma­lies, but the rule of squared mod­uli is well es­tab­lished and can be de­rived. Let me try an easy ex­pla­na­tion:

The ba­sic prin­ci­ple is that if one defines how the mea­sure­ment equip­ment re­acts to all pure states (am­pli­tude 1 for one con­figu­ra­tion, 0 for all else), one has no free­dom left to define how it re­acts to mixed states. I think the only pre­req­ui­site is that time evolu­tion is lin­ear. From here one can de­rive the No-Clon­ing the­o­rem: Sup­pose you have two sys­tems, one be­ing in the ‘ready to store a copy’ state |0> and one hav­ing the two pos­si­bil­ities |1> and |2> (and of course ev­ery lin­ear com­bi­na­tion of those, so a com­bi­na­tion of a|1>+b|2> will have an am­pli­tude of a for the con­figu­ra­tion |1> and b for |2>). Now you set up some in­ter­ac­tion which tries to copy the state of the sec­ond sys­tem onto the first. So:

• |0>|1> evolves into |1>|1>.

• |0>|2> evolves into |2>|2>.

But if we have a combination

• |0>(a|1>+b|2>)=a|0>|1>+b|0>|2>,

this will be mapped onto

• a|1>|1>+b|2>|2>

and not just clone the state, which would give

• (a|1>+b|2>)(a|1>+b|2>)=a²|1>|1>+ab|1>|2>+ab|2>|1>+b²|2>|2>.

So it is not pos­si­ble to copy the whole state of a sys­tem, but it is pos­si­ble to choose a ba­sis and then copy the state if it is one of the ba­sis vec­tors. So the ba­sic mea­sure­ment pro­cess would just copy the state of the sys­tem onto an­other sys­tem as good as pos­si­ble (hence the so-called Heisen­berg Uncer­tainty Prin­ci­ple—one has to choose ac­cord­ing to which ba­sis the mea­sure­ment is cou­pled to the sys­tem). From the ba­sis states of the com­pos­ite sys­tem (|0>|x>, |x>|x>, x=1,2) one can con­struct a scalar product such that ev­ery vec­tor has length 1 and they are or­thog­o­nal to each other:

• |x>=1, |x>=0 etc.

So the time evolu­tion ob­vi­ously con­serves the length of the ba­sis vec­tors—but since we could also have cho­sen an­other ba­sis, it has to con­serve also the length of mixed states (this step may be not so rigor­ous but at least makes the square rule much more plau­si­ble that any other). So the state (a|1>|1>+b|2>|2>) has to have length 1 and if we com­pute it we get

• 1=(|1>+b|2>|2>)=|a|²|1>+|b|²|2>+0=|a|²+|b|².

So the squared mod­uli add to 1 (Pythago­ras sends his re­gards). Fur­ther­more, if the ‘origi­nal’ sys­tem had three pos­si­bil­ities, but the copy pro­cess mapped

• |0>|1> onto |1>|1>

• |0>|2> onto |2>|2>

• |0>|3> onto |1>|3> (!),

• |0>(a|1>+b|2>+c|3>) --> |1>(a|1>+c|3>)+b|2>|2>.

Math­e­mat­i­cally, one can ‘trace out’ the in­fluence of the origi­nal sys­tem—graph­i­cly one just sees that the length of the part with the copied sys­tem in |1> is the length of the vec­tor a|1>+c|3>, namely |a|²+|c|², while the other part has the length |b|². Thus the Born prob­a­bil­ities are added when group­ing states to­gether in the pro­cess of copy­ing them—which could be re­spon­si­ble for the con­nec­tion of the Born rule to the pro­cess of cre­at­ing an­ti­ci­pa­tions and so forth. Of course a mea­sure­ment and the cou­pling of our brains to a sys­tem is not just copy­ing the states—but the same ar­gu­men­ta­tion holds since ev­ery sen­si­tive cou­pling of an­other sys­tem to the origi­nal sys­tem can only be defined on some ba­sis—the way the mea­sure­ment re­acts to com­bi­na­tions of states is de­ter­mined from there and is not open to ma­nipu­la­tion. So the Born rule is not a great mys­tery—al­though some of the steps may lack some rigor, it is far more plau­si­ble than for ex­am­ple just the mod­u­lus or some other power of it.

I hope this clears up some con­fu­sion,

Viktor

• Hm, just read the ar­ti­cle again and saw that many of this was already ex­plained there. But the es­sen­tial point is that al­though the full in­for­ma­tion of a sys­tem is given by the am­pli­tude dis­tri­bu­tion over all pos­si­ble con­figu­ra­tions, this in­for­ma­tion is not ac­cessible to an­other sys­tem. When we try to cou­ple the sys­tem to an­other (for ex­am­ple, by copy­ing the state), this only re­spects the pure ‘clas­si­cal’ states as de­scribed above. Thus it is pos­si­ble to ask the ques­tion ‘how much have these two states in com­mon’, where one clas­si­cal state com­pared with it­self gives one and with an­other one 0. If we want to also be able to com­pare mixed states, the no­tion of a scalar product comes in. The squared mod­u­lus is just the com­par­i­son of a state with it­self, which is con­stantly 1 - ob­vi­ously, the state has a hell lot in com­mon with it­self.

• Weren’t the Born prob­a­bil­ities suc­cess­fully de­rived from de­ci­sion the­ory for the MWI in 2007 by Deutsch: “Prob­a­bil­ities used to be re­garded as the biggest prob­lem for Everett, but iron­i­cally, they are now its most pow­er­ful suc­cess”—http://​​fo­rum.as­tro­ver­sum.nl/​​view­topic.php?p=1649

• There are a cou­ple of re­cent pa­pers on this topic:

I per­son­ally find Finkel­stein’s re­sponse/​coun­ter­ar­gu­ment con­vinc­ing.

• Hm, Wei_Dai(2009) seems to have a no­tion of ra­tio­nal­ity that is quite per­mis­sive if he’s con­vinced by Finkel­stein. If ra­tio­nal­ity isn’t in fact per­mis­sive and in­stead stringently re­quires di­achronic con­sis­tency (ex­cep­tion­less­ness, up­date­less­ness, pre-ra­tio­nal pri­ors) then I don’t think Finkel­stein’s ar­gu­ments are con­vinc­ing. And there are pos­i­tive ar­gu­ments, e.g. by Derek Parfit, that ra­tio­nal­ity is nor­ma­tively “thick”.

• are all the norms in­var­i­ant un­der per­mu­ta­tion of the in­dices p-norms?

Well, you an­swered that ex­act ques­tion, but here’s a de­scrip­tion of all norms (on a finite di­men­sional real vec­tor space): a norm de­ter­mines the set of all vec­tors of norm less than or equal to 1. This is con­vex and sym­met­ric un­der in­vert­ing sign (if you wanted com­plex, you’d have to al­low mul­ti­pli­ca­tion by com­plex units). It de­ter­mines the norm: the norm of a vec­tor is the amount you have to scale the set to en­velope the vec­tor. Any set satis­fy­ing those con­di­tions de­ter­mines a norm.

So there are a lot of norms out there. eg, you can take a cylin­der in 3-space (one of your ex­am­ples). You could take a hexagon in the plane. This norm al­lows the in­ter­change of co­or­di­nates, but it has a big­ger sym­me­try group, though still finite. (I guess one could write this as max(|x|,|y|,|x-y|))

• “Given the Born rule, it seems rather ob­vi­ous, but the Born rule it­self is what is cur­rently ap­pears to be sus­pi­ciously out of place. So, if that arises out of some­thing more ba­sic, then why the uni­tary rule in the first place?”

While not an an­swer, I know of a rele­vant com­ment. Sup­pose you as­sume that a the­ory is lin­ear and pre­serves some norm. What norm might it be? Be­fore ad­dress­ing this, let’s say what a norm is. In math­e­mat­ics a norm is defined to be some func­tion on vec­tors that is only zero for the all ze­ros vec­tor, and obeys the tri­an­gle in­equal­ity: the norm of a+b is no more than the norm of a plus the norm of b. The func­tions satis­fy­ing these ax­ioms seem to cap­ture ev­ery­thing that we would in­tu­itively re­gard as some sort of length or mag­ni­tude.

The Eu­cli­dian norm is ob­tained by sum­ming the squares of the ab­solute val­ues of the vec­tor com­po­nents, and then tak­ing the square root of the re­sult. The other norms that arise in math­e­mat­ics are usu­ally of the type where you raise the each of the ab­solute val­ues of the vec­tor com­po­nents to some power p, then sum them up, and then take the pth root. The cor­re­spond­ing norm is called the p-norm. (Does some­body know: are all the norms in­var­i­ant un­der per­mu­ta­tion of the in­dices p-norms?) Scott Aaron­son proved that for any p other than 1 or 2, the only norm-pre­serv­ing lin­ear trans­for­ma­tions are the per­mu­ta­tions of the com­po­nents. If you choose the 1-norm, then the sum of the ab­solute val­ues of the com­po­nents are pre­served, and the norm pre­serv­ing trans­for­ma­tions cor­re­spond to the stochas­tic ma­tri­ces. This is es­sen­tially prob­a­bil­ity the­ory. If you choose the 2-norm then the Eu­clidean length of the vec­tors is pre­served, and the al­lowed lin­ear trans­for­ma­tions cor­re­spond to the uni­tary ma­tri­ces. This is es­sen­tially quan­tum me­chan­ics. (Scott always has­tens to add that his the­o­rem about p-norms and per­mu­ta­tions was prob­a­bly known by math­e­mat­i­ci­ans for a long time. The new part is the ap­pli­ca­tion to foun­da­tions of QM.)

• Scott Aaron­son proved that for any p other than 1 or 2, the only norm-pre­serv­ing lin­ear trans­for­ma­tions are the per­mu­ta­tions of the com­po­nents.

This seems to be true, but with the small note that you should add mul­ti­pi­ca­tion of the co­or­di­nates by −1 [by any num­ber from unit cir­cle if the space is taken over com­plex num­bers] and their com­po­si­tions with per­mu­ta­tions to the al­lowed iso­mor­phisms. Never heard about this though, in­ter­est­ing.

How­ever this does not gen­er­al­ize to all the norms. As Dou­glas noted be­low one can imag­ine norm sim­ply as a cen­tral-sym­met­ric con­vex body. And there are plenty of those. Now if we can fix a finite sub­group of space ro­ta­tions and sym­me­tries that strictly con­tains all the co­or­di­nate per­mu­ta­tions and cen­tral-sym­me­try then we are done, since one can sim­ply take con­vex hull of the or­bit of some point as your de­sired norm. Sym­me­tries and ro­ta­tions of reg­u­lar 100-gon on the plane would work for ex­am­ple.

If you choose the 1-norm, then the sum of the ab­solute val­ues of the com­po­nents are pre­served, and the norm pre­serv­ing trans­for­ma­tions cor­re­spond to the stochas­tic ma­tri­ces.

Hmm, some­thing fishy is go­ing with signs in the whole ar­gu­ment and here I am com­pletely lost. What if I take 2x2 ma­trix with all en­tries equal to 12 and a vec­tor (1/​2, −1/​2)? Prob­a­bly the full for­mu­la­tion by Scott would help. Does any­body have a link?

• Prob­a­bly the full for­mu­la­tion by Scott would help. Does any­body have a link?

This.

• Thank you.

Nice pa­per. Signs are treated ac­cu­rately there of course. How­ever call to “for­mal func­tions” in the end of the proof seems wacky at best. For­mal­iz­ing it looks harder to me than the ini­tial state­ment. At this point it should be eas­ier to just look at the smooth­ness de­grees of the norm on x_i = 0 hy­per­planes.

If any­body knows what was meant, how­ever, please clar­ify.

• As I un­der­stand it (some­one cor­rect me if I’m wrong), there are two prob­lems with the Born rule: 1) It is non-lin­ear, which sug­gests that it’s not fun­da­men­tal, since other fun­da­men­tal laws seem to be linear

2) From my read­ing of Robin’s ar­ti­cle, I gather that the prob­lem with the many-wor­lds in­ter­pre­ta­tion is: let’s say a world is cre­ated for each pos­si­ble out­come (countable or un­countable). In that case, the vast ma­jor­ity of wor­lds should end up away from the peaks of the dis­tri­bu­tion, just be­cause the peaks only oc­cupy a small part of any dis­tri­bu­tion.

Robin’s solu­tion seems to me equiv­a­lent to the Quan­tum Spaghetti Mon­ster eat­ing the un­likely wor­lds that we find our­selves not to end up in. The key line is “sud­den and ther­mo­dy­nam­i­cally ir­re­versible.” Ac­tu­ally, that should be enough to bury the the­ory since aren’t fun­da­men­tal phys­i­cal laws ther­mo­dy­nam­i­cally neu­tral?

We could prob­a­bly elimi­nate this dis­trac­tion of con­scious­ness, couldn’t we? I mean, let’s say that Math­e­mat­ica ver­sion 5000 comes out in a few cen­turies and in ad­di­tion to its other sym­bolic alge­bra ca­pa­bil­ities, it comes with a phys­i­cal-law-prover: you ask it ques­tions and it sets up ex­per­i­ments to an­swer those ques­tions. So you ask it about quan­tum me­chan­ics, it does a bunch of dou­ble-slit-ex­per­i­ments in a robotic lab, and gives you the an­swer, which in­cludes the Born rule. Con­scious­ness was never in­volved.

Ac­tu­ally it seems to me like this whole busi­ness of quan­tum prob­a­bil­ities is way over­rated (for the non-physi­cist), be­cause it only re­ally man­i­fests it­self in clev­erly con­structed ex­per­i­ments . . . right? I mean, set­ting aside ex­actly how Born’s rule de­rives from the un­der­ly­ing physics, is there any rea­son to be­lieve that we would learn any­thing new by find­ing out?

• The ob­server’s con­scious­ness is still in­volved. Imag­ine that the Born rule isn’t a law of the uni­verse it­self, but of con­scious­ness. The uni­verse eval­u­ates all branches. Con­scious­ness fol­lows the branches in weights fol­low­ing the Born rule. The con­scious ob­server always finds them­selves down a se­ries of branches that were se­lected by the Born rule, and it’s easy for them to take mea­sure­ments to con­firm this. The Math­e­mat­ica 5000 ma­chine that’s come down this se­ries of branches has made mea­sure­ments from ex­per­i­ments and has found that the Born rule has held. It only comes up with this re­sult be­cause this is the ver­sion of the ma­chine that has fol­lowed the ob­server’s con­scious­ness through the branches. In the raw uni­verse, most wor­lds have the Math­e­mat­ica 5000 ma­chine find­ing that Born’s rule does not hold; these aren’t the wor­lds that con­scious ob­servers usu­ally find them­selves in though.

• could the flow of am­pli­tude be­tween blobs we nor­mally think of as sep­a­rated fol­low­ing a mea­sure­ment pos­si­bly ex­plain the quan­tum field the­ory pre­dic­tion/​phe­nomenon of vac­uum fluc­tu­a­tions?

• Nope. Vacuum fluc­tu­a­tions hap­pen be­cause the field that tells you whether there’s a par­ti­cle there or not be­haves like a quan­tum thing and not a clas­si­cal thing, and you end up with a non-bor­ing vac­uum state for the same rea­son atoms have non-bor­ing ground states rather than col­laps­ing in on them­selves. Weird as all get out, but not quan­tum-me­chan­ics-break­ing, and mea­sured rea­son­ably well by the Casimir effect (though also hor­ribly wrong be­cause of the cos­molog­i­cal con­stant prob­lem, but that’s a prob­lem for quan­tum grav­ity to sort out, not one that can be solved by big changes to already-tested parts of quan­tum me­chan­ics).

• First of all—great se­quence! I had a lot of ‘I see!‘-mo­ments read­ing it. I study physics, but of­ten the clear pic­ture gets lost in the stan­dard ap­proach and one is left with a lot of calcu­lat­ing tech­niques with­out any in­tu­itive grasp of the sub­ject. After read­ing this I be­came very fond of tu­tor­ing the course on quan­tum me­chan­ics and always tried to give some deeper in­sight (many of which was taken from here) in ad­di­tion to just ex­plain­ing the ex­er­cises. If I am cor­rect, the world man­gling the­ory just tries to ex­plain some anoma­lies, but the rule of squared mod­uli is well es­tab­lished and can be de­rived. Let me try an easy ex­pla­na­tion: The ba­sic prin­ci­ple is that if one defines how the mea­sure­ment equip­ment re­acts to all pure states (am­pli­tude 1 for one con­figu­ra­tion, 0 for all else), one has no free­dom left to define how it re­acts to mixed states. I think the only pre­req­ui­site is that time evolu­tion is lin­ear. From here one can de­rive the No-Clon­ing the­o­rem: Sup­pose you have two sys­tems, one be­ing in the ‘ready to store a copy’ state |0> and one hav­ing the two pos­si­bil­ities |1> and |2> (and of course ev­ery lin­ear com­bi­na­tion of those, so a com­bi­na­tion of a|1>+b|2> will have an am­pli­tude of a for the con­figu­ra­tion |1> and b for |2>). Now you set up some in­ter­ac­tion which tries to copy the state of the sec­ond sys­tem onto the first. So:

• |0>|1> evolves into |1>|1>.

• |0>|2> evolves into |2>|2>. But if we have a combination

• |0>(a|1>+b|2>)=a|0>|1>+b|0>|2>, this will be mapped onto

• a|1>|1>+b|2>|2> and not just clone the state, which would give

• (a|1>+b|2>)(a|1>+b|2>)=a²|1>|1>+ab|1>|2>+ab|2>|1>+b²|2>|2>. So it is not pos­si­ble to copy the whole state of a sys­tem, but it is pos­si­ble to choose a ba­sis and then copy the state if it is one of the ba­sis vec­tors. So the ba­sic mea­sure­ment pro­cess would just copy the state of the sys­tem onto an­other sys­tem as good as pos­si­ble (hence the so-called Heisen­berg Uncer­tainty Prin­ci­ple—one has to choose ac­cord­ing to which ba­sis the mea­sure­ment is cou­pled to the sys­tem). From the ba­sis states of the com­pos­ite sys­tem (|0>|x>, |x>|x>, x=1,2) one can con­struct a scalar product such that ev­ery vec­tor has length 1 and they are or­thog­o­nal to each other:

• |x>=1, |x>=0 etc. So the time evolu­tion ob­vi­ously con­serves the length of the ba­sis vec­tors—but since we could also have cho­sen an­other ba­sis, it has to con­serve also the length of mixed states (this step may be not so rigor­ous but at least makes the square rule much more plau­si­ble that any other). So the state (a|1>|1>+b|2>|2>) has to have length 1 and if we com­pute it we get

• 1=(|1>+b|2>|2>)=|a|²|1>+|b|²|2>+0=|a|²+|b|². So the squared mod­uli add to 1 (Pythago­ras sends his re­gards). Fur­ther­more, if the ‘origi­nal’ sys­tem had three pos­si­bil­ities, but the copy pro­cess mapped

• |0>|1> onto |1>|1>

• |0>|2> onto |2>|2>

• |0>|3> onto |1>|3> (!), we had

• |0>(a|1>+b|2>+c|3>) maps onto |1>(a|1>+c|3>)+b|2>|2>. Math­e­mat­i­cally, one can ‘trace out’ the in­fluence of the origi­nal sys­tem—graph­i­cly one just sees that the length of the part with the copied sys­tem in |1> is the length of the vec­tor a|1>+c|3>, namely |a|²+|c|², while the other part has the length |b|². Thus the Born prob­a­bil­ities are added when group­ing states to­gether in the pro­cess of copy­ing them—which could be re­spon­si­ble for the con­nec­tion of the Born rule to the pro­cess of cre­at­ing an­ti­ci­pa­tions and so forth. Of course a mea­sure­ment and the cou­pling of our brains to a sys­tem is not just copy­ing the states—but the same ar­gu­men­ta­tion holds since ev­ery sen­si­tive cou­pling of an­other sys­tem to the origi­nal sys­tem can only be defined on some ba­sis—the way the mea­sure­ment re­acts to com­bi­na­tions of states is de­ter­mined from there and is not open to ma­nipu­la­tion. So the Born rule is not a great mys­tery—al­though some of the steps may lack some rigor, it is far more plau­si­ble than for ex­am­ple just the mod­u­lus or some other power of it. I hope this clears up some con­fu­sion, Viktor

• The Trans­ac­tional In­ter­pre­ta­tion of QM re­solves the mys­tery of where this non­lin­ear squared mod­u­lus comes from quite neatly. On that ba­sis alone, I’m sur­prised that Eliezer doesn’t even men­tion it as a se­ri­ous ri­val to MWI.

• Don’t the trans­ac­tional in­ter­pre­ta­tion’s fol­low­ers claim that stan­dard QM gives the wrong re­sult on the Afshar ex­per­i­ment? Or is that not all of them?

• Cramer ar­gues that both Copen­hagen and MWI are in­con­sis­tent with the re­sults of the Afshar ex­per­i­ment.

• Yeah, but he’s wrong. Al­most no physi­cists ac­cept his ar­gu­ment as math­e­mat­i­cally valid. If the trans­ac­tional in­ter­pre­ta­tion does give differ­ent re­sults, then it is in­com­pat­i­ble with ex­per­i­ment.

• Al­most no physi­cists ac­cept his ar­gu­ment as math­e­mat­i­cally valid.

If you’re talk­ing about the Afshar ex­per­i­ment, Un­ruh de­mol­ished that con­vinc­ingly. We don’t need to take it on trust that Afshar is wrong.

How­ever, Afshar and Cramer were only ever ar­gu­ing about the in­ter­pre­ta­tion of the re­sults of Afshar’s ex­per­i­ment, not what those re­sults would be. It would be most un­wise to rule out the trans­ac­tional in­ter­pre­ta­tion just be­cause its in­ven­tor sub­se­quently said some­thing fool­ish.

• See the grand­par­ent; Cramer jus­tified the trans­ac­tional in­ter­pre­ta­tion by say­ing that it was the only in­ter­pre­ta­tion able to give the cor­rect re­sult for the Afshar ex­per­i­ment. This be­ing wrong re­moves much of the claimed ev­i­dence.

• Sure. I think the bot­tom line is that the Afshar ex­per­i­ment doesn’t give em­piri­cal sup­port, or even ‘philo­soph­i­cal sup­port’, to any in­ter­pre­ta­tion. It’s a wild goose chase.

• Con­sider for ex­am­ple what “scat­ter­ing ex­per­i­ments” show, in a con­text of imag­in­ing that the uni­verse is made of fields and that only “ob­ser­va­tion” makes a man­i­fes­ta­tion in a small re­gion of space? I mean, sup­pose we think of the “ob­ser­va­tions” as be­ing our de­tect­ing the im­pacts of the “scat­tered” elec­trons rather than the scat­ter­ings them­selves. (IOW, we don’t con­sider “mere” in­ter­ac­tions to be ob­ser­va­tions—what­ever that means.) But then why and how did the waves rep­re­sent­ing the elec­trons scat­ter as if off lit­tle con­cen­tra­tions when they were in­ter­pen­e­trat­ing? And, what of the find­ing that elec­trons are “points” as far as we can tell, from scat­ter­ing ex­per­i­ments? Note that the scat­ter­ing is based on imag­in­ing one charge “source” be­ing af­fected by an­other source’s cen­tral in­verse-square field, noth­ing that makes a lot of sense in terms of spread-out waves. Note also that the scat­ter­ing is not a spe­cific “im­pact” like that of billiard balls, since it is a mat­ter of de­gree (how close one elec­tron ap­proaches an­other, still not touch­ing since they don’t have ex­ten­sions with a dis­con­ti­nu­ity like a hard ball—and the very term “how close” be­trays an ex­ist­ing point­ness.) And so on … IOW, it’s worse than you think.

On a differ­ent note, it is sup­posed to be im­pos­si­ble to find out cer­tain things about the wave func­tion, like its par­tic­u­lar shape. We are sup­posed to only be able to find out, whether it passed or failed to pass the test for chance of a par­tic­u­lar eigen­state (like, a lin­ear po­larized pho­ton hav­ing a greater chance of pass­ing a lin­ear filter of similar ori­en­ta­tion, but we wouldn’t be able to find out di­rectly it had been pro­duced with a 20 de­gree ori­en­ta­tion of po­lariza­tion.) How­ever, I thought of a way to per­haps do such a thing. It in­volves pass­ing a po­larized pho­ton through two half-wave plates over and over, say with re­flec­tions. The first plate col­lects a lit­tle bit of av­er­age spin from each pass of the pho­ton, due to the in­vert­ing of pho­ton spin by such a HWP. The sec­ond HWP re­verts the pho­ton’s spin (su­per­posed value, the “cir­cu­lar­ity”) back to it’s origi­nal value so it will reen­ter the first HWP with the same value of cir­cu­lar­ity each time.

After many passes, an­gu­lar mo­men­tum trans­fer S should ac­cu­mu­late in the first plate along a range of val­ues. S = 2nC hbar, where n is num­ber of passes, and C is the “cir­cu­lar­ity” based on how much RH and LH is su­per­posed in that pho­ton. So for ex­am­ple, a pho­ton that came out of a lin­ear pol. filter would show zero net spin in such a de­vice, el­lip­ti­cal pho­tons would show in­ter­me­di­ate spin, and CP pho­tons would show full spin of S = 2n hbar. It isn’t at all like hav­ing eigen­state filters. Hav­ing an in­di­ca­tion along a range is not sup­posed to be pos­si­ble (pro­jec­tion pos­tu­late), and is rem­i­nis­cent of Y. Aharonov’s “weak mea­sure­ment” ideas.

• If any­one can pro­duce a cel­lu­lar au­tomata model that can cre­ate cir­cles like those which re­late to the in­verse square of dis­tance or the stuff of early wave me­chan­ics, I think I can bridge the MWI view and the one uni­verse of many fid­get­ings view that I cling to. I know of one other per­son who has a similar idea, un­for­tu­nately his idea has a bizarre quan­tity which is the square root of a me­ter.

• Stephen: Aaah, okay. And yeah, that’s why I said no rescal­ing.

I mean, if one didn’t already have the “prob­a­bil­ity of ex­pe­rienc­ing some­thing is lin­ear in p-norm...” thing, would one still be able to ar­gue su­per­pow­ers?

From your de­scrip­tion, it looks like he still has to use the princ­ple of “prob­a­bil­ity of ex­pe­rienc­ing some­thing pro­por­tional to p-norm” to jus­tify the su­per­pow­ers thing.

Browsed through the pa­per, and, if I in­ter­preted it right, that is kinda what it was do­ing… As­sume there’s some p-norm cor­re­spond­ing to prob­a­bil­ity. But maybe I mi­s­un­der­stood.

Eliezer: oh, mind elab­o­rat­ing on ‘His­tor­i­cal note: If “ob­serv­ing a par­ti­cle’s po­si­tion” in­voked a mys­te­ri­ous event that squeezed the am­pli­tude dis­tri­bu­tion down to a delta point, or flat­tened it in one sub­space, this would give us a differ­ent fu­ture am­pli­tude dis­tri­bu­tion from what de­co­her­ence would pre­dict. All in­ter­pre­ta­tions of QM that in­volve quan­tum sys­tems jump­ing into a point/​flat state, which are both testable and have been tested, have been falsified.’? Thanks.

• “If you didn’t know squared am­pli­tudes cor­re­sponded to prob­a­bil­ity of ex­pe­rienc­ing a state, would you still be able to de­rive “nonuni­tary op­er­a­tor → su­per­pow­ers?”″

Scott looks at a spe­cific class of mod­els where you as­sume that your state is a vec­tor of am­pli­tudes, and then you use a p-norm to get the cor­re­spond­ing prob­a­bil­ities. If you de­mand that the time evolu­tions be norm-pre­serv­ing then you’re stuck with per­mu­ta­tions. If you al­low non-norm-pre­serv­ing time evolu­tion, then you have to read­just the nor­mal­iza­tion be­fore calcu­lat­ing the prob­a­bil­ities in or­der to make them add up to 1. This read­just­ment of the norm is non­lin­ear. It re­sults in su­per­pow­ers. The pa­per in pdf and other for­mats is here.

• Stephen: I don’t have a postscript viewer.

Wait, I thought the su­per­power stuff only hap­pens if you al­low non­lin­ear trans­forms, not just nonuni­tary. Let’s add an ad­di­tional re­stric­tion: let’s ac­tu­ally throw in some no­tion of lo­cal­ity, but even with the lo­cal­ity, aban­don uni­tary­ness. So our rules are “lin­ear, lo­cal, in­vertable” (no rescal­ing af­tar­wards… not defin­ing a norm to pre­serve in the first place)… or does lo­cal­ity ne­ces­si­tate uni­tar­ity? (is uni­tar­ity a word? Well, you know what I mean. Maybe I should say or­thog­nal­ity in­stead?)

Well, ac­tu­ally, also same ques­tion here I asked Eliezer. If you didn’t know squared am­pli­tudes cor­re­sponded to prob­a­bil­ity of ex­pe­rienc­ing a state, would you still be able to de­rive “nonuni­tary op­er­a­tor → su­per­pow­ers?”

Any­ways, let’s turn it around again. Let’s say we didn’t know the Born rule, but we did already know some other way that all state vec­tors must evolve via a uni­tary op­er­a­tor.

So from there we may no­tice sum/​in­te­gral of squared am­pli­tude is con­served, and that by ap­pro­pri­ate scal­ing, to­tal squared am­pli­tude = 1 always.

Looks like we may even no­tice that it hap­pens to obey the ax­ioms of prob­a­bil­ity. (it looks like the quanity in ques­tion does au­to­mat­i­cally do so, given only uni­tary trans­forms are al­lowed.)

Is the mere fact that the quan­tity does “just hap­pen” to obey the ax­ioms of prob­a­bil­ity, on its own, help us here? Would that at least help an­swer the “why” for the Born rule? I’d think it would be rele­vant, but, think­ing about it, I don’t see any ob­vi­ous way to go from there to “there­fore it’s the prob­a­bil­ity we’ll ex­pe­rience some­thing...”

Yep, my con­fu­sion is defi­nately shuffled.

hrgflargh… (That’s the noise of frus­trated cu­ri­ousity. :D)

• Psy-Kosh:

“Or did I com­pletely and ut­terly mi­s­un­der­stand what you were try­ing to say?”

No, you are cor­rectly in­ter­pret­ing me and notic­ing a gap in the rea­son­ing of my pre­ceed­ing post. Sorry about that. I re-looked-up Scott’s pa­per to see what he ac­tu­ally said. If, as you pro­pose, you al­low in­vert­ible but non-norm-pre­serv­ing time evolu­tions and just re-ad­just the norm af­ter­wards then you get FTL sig­nal­ling, as well as ob­scene com­pu­ta­tional power. The pa­per is here.

• @Roland: My physics and maths is patchy but I’m still just about fol­low­ing (the posts—some com­ments are way too ad­vanced) though it is hard work for some bits. Lots of slow re-read­ing, look­ing things up and re­vis­ing old posts, but it’s worth it.

If you’re de­ter­mined enough, try read­ing the posts a few at a time (in­stead of one a day) start­ing a few posts be­fore where you got stuck, and make sure you “get” each one be­fore you move on, even if it means an hour on an­other web source study­ing the thing you don’t un­der­stand in Eliezer’s ex­pla­na­tion.

• Stephen: Is the point you’re mak­ing ba­si­cally along the lines of “vec­tor as ge­o­met­ric ob­ject rather than list of num­bers”?

Sure, I buy that. Heck, I’m nat­u­rally in­clined to­ward that per­spec­tive at this time. (In part be­cause have been study­ing GR lately)

Aaany­ways, so I guess ba­si­cally what you’re say­ing is that all op­er­a­tors cor­re­spond­ing to time evolu­tion or what­ever are just ro­ta­tions or such in the space? And why the 2-norm in­stead of, say, the 1-norm? why would the uni­verse “pre­fer” to pre­serve the sum of the squared mag­ni­tudes rather than the sum of the mag­ni­tudes? ie, why is the rule “uni­tary” rather than “stochas­tic”, for in­stance? (Well, I have a par­tial an­swer for that my­self… re­versibil­ity. Stochas­tic isn’t nec­es­sar­ally re­versible, right? uni­tary is though, so there is that...)

If I’m un­der­stand­ing what you’re try­ing to say, ba­si­cally you’re say­ing “it’s as if you use any ole trans­form, then just di­vide by the fac­tor the norm’s been changed by, so you may as well have that ‘already in’ the trans­form”… But if the trans­form isn’t some mul­ti­ple of a uni­tary trans­form, then there won’t be any sin­gle scalar value that takes care of that, right? Why in­stead of “norm pre­serv­ing” isn’t the rule “any in­vertable lin­ear trans­form”?

Or did I com­pletely and ut­terly mi­s­un­der­stand what you were try­ing to say?

• I’m struck by guilt for hav­ing spo­ken of “ra­tios of am­pli­tudes”. It makes the pro­posal sound more spe­cific and fully worked-out than it is. Let me just re­place that phrase in my pre­vi­ous post with the va­guer no­tion of “rel­a­tive am­pli­tudes”.

• Psy-Kosh:

Good ex­am­ple with the Lorentz met­ric.

In­var­i­ance of norm un­der per­mu­ta­tions seems a rea­son­able as­sump­tion for state spaces. On the other hand, I now re­al­ize the an­swer to my ques­tion about whether per­mu­ta­tion in­var­i­ance nar­rows things down to p-norms is no. A sim­ple coun­terex­am­ple is a lin­ear com­bi­na­tion of two differ­ent p-norms.

I think there might be a good rea­son to think in terms of norm-pre­serv­ing maps. Namely, sup­pose the norms can be any­thing but the in­di­vi­d­ual am­pli­tudes don’t mat­ter, only their ra­tios do. That is, states are iden­ti­fied not with vec­tors in the Hilbert space, but rays in the Hilbert space. This is the way von Neu­mann for­mu­lated QM, and it is equiv­a­lent to the now more com­mon norm=1 for­mu­la­tion. This also seems to be the for­mu­la­tion Eli was im­plic­itly us­ing in some of his pre­vi­ous posts.

The usual way to for­mu­late QM these days is, rather than ig­nor­ing the nor­mal­iza­tions of the state vec­tors, one can in­stead just de­cree that the norms must always have a cer­tain value (speci­fi­cally, 1). Then we can as­sign mean­ing to the in­di­vi­d­ual am­pli­tudes rather than only their ra­tios. It seems likely to me that the­o­ries where only the ra­tios of the “am­pli­tudes” mat­ter, gener­i­cally can be equiv­a­lently for­mu­lated as a the­ory with fixed norm. Think­ing that only ra­tios mat­ter seems a more in­tu­itive start­ing point.

• Stephen: Thanks. First, not ev­ery­thing cor­re­spond­ing to a length or such obeys that par­tic­u­lar rule… con­sider the Lorenz met­ric… any “lightlike” vec­tor has a norm of zero, for in­stance, and yet that par­tic­u­lar ma­tric is rather use­ful phys­i­cally. :) (ad­mit­tedly, you get that via the minus sign, and if your norm is such that it treats all the com­po­nents in some sense equiv­a­lently, you don’t get that… well, what about norms in­volv­ing cross terms?)

More to the sub­ject… why is any norm pre­served? That is, why only al­low norm pre­serv­ing trans­forms?

Which brings be to Eliezer:

So? Why does the uni­verse “choose” rules that say “no out­come pump”? That’s way up the lad­der of stuff built out of other stuff. (as far as com­mu­ni­cat­ing faster than light, I’d think “out­come pump” type things are the main ‘crazy’ re­sult of FTL in the first place)

Ac­tu­ally, I think I didn’t com­mu­ni­cate my ques­tion ac­cu­rately. You de­rived it would be an out­come pump by not­ing it would change the Born de­rived prob­a­bil­ities (At least, that’s my un­der­stand­ing of the sig­nifi­cance of you not­ing that the ra­tios of the squared mag­ni­tudes chang­ing.) But the Born prob­a­bil­ities are already the “odd rule out”… so I wanted to know if there was any other rea­son/​ar­gu­ment you could think of as to why we have norm preser­va­tion with­out ap­peal­ing to the Born rule. (Does that clar­ify my ques­tion?)

I mean, if I was let­ting my­self use the Born rule, I could just say that the prob­a­bil­ities have to sum to 1, and that hands me the uni­tary­ness. But my whole point was “the re­stric­tion to uni­tary trans­forms it­self seems to be re­lated to squared mag­ni­tude stuff. So by un­der­stand­ing why that re­stric­tion ex­ists in re­al­ity, maybe I’d have a bet­ter idea where the Born rule is com­ing from”

• Stephen, thanks for your thoughts on Eli’s thoughts. I’m go­ing to have to think on them fur­ther—af­ter all these helpful posts I can pre­tend I un­der­stand quan­tum me­chan­ics, but pre­tend­ing to un­der­stand how con­scious minds per­ceive a sin­gle point in con­figu­ra­tion space in­stead of blobs of am­pli­tude is go­ing to take more work.

I will point out, though, that the ques­tion of how con­scious­ness is bound to a par­tic­u­lar branch (and thus why the Born rule works like it does) doesn’t seem that much differ­ent from how con­scious­ness is tied to a par­tic­u­lar point in time or to a par­tic­u­lar brain when the Spaghetti Mon­ster can see all brains in all times and would have to be given ex­tra in­for­ma­tion to know that my con­scious­ness seems to be liv­ing in this par­tic­u­lar brain at this par­tic­u­lar time.

Fi­nally: “it is a com­mon mis­con­cep­tion that should be ad­dressed at some point any­way”—it ap­pears to me that Robin’s pa­per is based on this same mis­con­cep­tion, or some­thing like it: the Born rule (and ex­per­i­ment!) give one re­sult while count­ing wor­lds gives an­other, there­fore we have to add a new rule (“wor­lds that are too small get man­gled”) in or­der to make count­ing wor­lds match ex­per­i­ment. Whereas with­out the mis­con­cep­tion we wouldn’t be count­ing wor­lds in the first place. Do you think I’m un­der­stand­ing Robin’s po­si­tion and/​or QM cor­rectly?

• Here’s a differ­ent ques­tion which may be rele­vant: why uni­tary trans­forms?

That is, if you didn’t in the first place know about the Born rule, what would be a (even semi) in­tu­itive jus­tifi­ca­tion for the re­stric­tion that all “rea­son­able” trans­forms/​time evolu­tion op­er­a­tors have to con­serve the squared mag­ni­tude?

Given the Born rule, it seems rather ob­vi­ous, but the Born rule it­self is what is cur­rently ap­pears to be sus­pi­ciously out of place. So, if that arises out of some­thing more ba­sic, then why the uni­tary rule in the first place?

• In this case epiphe­nom­e­nal­ism would be true (since qualia have no effect on the phys­i­cal world), but the cor­re­la­tion would not be a co­in­ci­dence (since the phys­i­cal world di­rectly causes qualia).

But the na­ture of the ex­pe­riences we claimed to have would not de­pend in any way on the prop­er­ties of these hy­po­thet­i­cal ‘qualia’. There would be no event in the phys­i­cal world that would be af­fected by them—they would not, in fact, ex­ist.

Epiphe­nom­e­nal­ism is never true, be­cause it con­tains a con­tra­dic­tion in terms.

• Nick: I don’t un­der­stand the con­nec­tion to quan­tum me­chan­ics.

The ar­gu­ment that I com­monly see re­lat­ing quan­tum me­chan­ics to an­thropic rea­son­ing is deeply flawed. Some peo­ple seem to think that many wor­lds means there are many “branches” of the wave­func­tion and we find our­selves in them with equal prob­a­bil­ity. In this case, they ar­gue, we should ex­pect to find our­selves in a di­s­or­derly uni­verse. How­ever, this is ex­actly what the Born rule (and ex­per­i­ment!) does not say. Rather, the Born rule says that we are only likely to find our­selves in states with large am­pli­tude. Also, stan­dard quan­tum me­chan­ics al­lows the prob­a­bil­ities to fall on a con­tinuum. They aren’t ar­rived at by count­ing, so the whole con­cept of count­ing branches is not stan­dard QM any­way.

(I don’t know whether you hold this view, but it is a com­mon mis­con­cep­tion that should be ad­dressed at some point any­way.)

• Un­less there is a sur­pris­ing amount of co­her­ence be­tween wor­lds with differ­ent lot­tery out­comes, this man­gled wor­lds model should still be vuln­er­a­ble to my lot­tery win­ning tech­nique (split the world a bunch of times if you win).

• Per­haps I’m be­ing too sim­plis­tic, but I see a de­cent ex­pla­na­tion that doesn’t get as far into the weeds as some of the oth­ers. It’s pro­por­tional to the square be­cause both the event be­ing ob­served and the ob­server need to be in the same uni­verse. If the par­ti­cle can be in A or B, the odds are:

P(A)&O(A) = A^2

P(B)&O(B) = B^2

P(A)&O(B) = Would be AB, but this is phys­i­cally im­pos­si­ble.

P(B)&O(A) = Would be AB, but this is phys­i­cally im­pos­si­ble.

Squares fall out nat­u­rally.

• There are a num­ber of rea­sons this solu­tion does not work. Here is one prob­lem with the solu­tion that does not re­quire any dis­cus­sion of the for­mal­ism or in­ter­pre­ta­tion of quan­tum the­ory:

Ac­cord­ing to you, the lo­ca­tion of the par­ti­cle and the lo­ca­tion of the ob­server are cor­re­lated (this fol­lows from the fact that some com­bi­na­tions are phys­i­cally im­pos­si­ble). If that’s the case, you can’t calcu­late the prob­a­bil­ity of the con­junc­tion by mul­ti­ply­ing the prob­a­bil­ities of the con­juncts. That only works if the con­juncts are un­cor­re­lated.

More broadly, based on what you pro­pose here I don’t think you have suffi­cient un­der­stand­ing of quan­tum me­chan­ics to fully ap­pre­ci­ate the na­ture of the prob­lem or the kind of solu­tion that would be re­quired. Your com­ment sug­gests sev­eral fairly fun­da­men­tal mi­s­un­der­stand­ings about the the­ory. I hope this doesn’t come off as im­po­lite or con­de­scend­ing. It’s the kind of thing I’d want some­one to say to me if they gen­uinely be­lieved it (al­though that in it­self doesn’t en­tail that it isn’t im­po­lite or con­de­scend­ing).

• I didn’t ex­pect some­thing that sim­ple had es­caped ev­ery­one’s no­tice(though I sup­pose I should have said that more ex­plic­itly in my post) - I threw it out there be­cause it made sense at first glance and had no im­me­di­ately ob­vi­ous prob­lems, not be­cause I figured I had definitely cracked the prob­lem. Easier to see if there’s a known re­sponse than to try to figure it out my­self. So no, I’m not an­noyed by your re­sponse.

And I do think I see what you’re get­ting at. Oh well, it was worth a shot.