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.

Part of The Quan­tum Physics Sequence

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