Joint Configurations

The key to un­der­stand­ing con­figu­ra­tions, and hence the key to un­der­stand­ing quan­tum me­chan­ics, is re­al­iz­ing on a truly gut level that con­figu­ra­tions are about more than one par­ti­cle.

Con­tin­u­ing from the pre­vi­ous es­say, Figure 1 shows an al­tered ver­sion of the ex­per­i­ment where we send in two pho­tons to­ward D at the same time, from the sources B and C.

The start­ing con­figu­ra­tion then is:

“a pho­ton go­ing from B to D,
and a pho­ton go­ing from C to D.”

Again, let’s say the start­ing con­figu­ra­tion has am­pli­tude .

And re­mem­ber, the rule of the half-silvered mir­ror (at D) is that a right-an­gle deflec­tion mul­ti­plies by i, and a straight line mul­ti­plies by 1.

So the am­pli­tude flows from the start­ing con­figu­ra­tion, sep­a­rately con­sid­er­ing the four cases of deflec­tion/​non-deflec­tion of each pho­ton, are:

1. The “B to D” pho­ton is deflected and the “C to D” pho­ton is deflected. This am­pli­tude flows to the con­figu­ra­tion “a pho­ton go­ing from D to E, and a pho­ton go­ing from D to F.” The am­pli­tude flow­ing is .

2. The “B to D” pho­ton is deflected and the “C to D” pho­ton goes straight. This am­pli­tude flows to the con­figu­ra­tion “two pho­tons go­ing from D to E.” The am­pli­tude flow­ing is .

3. The “B to D” pho­ton goes straight and the “C to D” pho­ton is deflected. This am­pli­tude flows to the con­figu­ra­tion “two pho­tons go­ing from D to F.” The am­pli­tude flow­ing is .

4. The “B to D” pho­ton goes straight and the “C to D” pho­ton goes straight. This am­pli­tude flows to the con­figu­ra­tion “a pho­ton go­ing from D to F, and a pho­ton go­ing from D to E.” The am­pli­tude flow­ing is .

Now—and this is a very im­por­tant and fun­da­men­tal idea in quan­tum me­chan­ics—the am­pli­tudes in cases 1 and 4 are flow­ing to the same con­figu­ra­tion. Whether the B pho­ton and C pho­ton both go straight, or both are deflected, the re­sult­ing con­figu­ra­tion is one pho­ton go­ing to­ward E and an­other pho­ton go­ing to­ward F.

So we add up the two in­com­ing am­pli­tude flows from case 1 and case 4, and get a to­tal am­pli­tude of .

When we wave our magic squared-mod­u­lus-ra­tio reader over the three fi­nal con­figu­ra­tions, we’ll find that “two pho­tons at De­tec­tor 1” and “two pho­tons at De­tec­tor 2” have the same squared mod­u­lus, but “a pho­ton at De­tec­tor 1 and a pho­ton at De­tec­tor 2” has squared mod­u­lus zero.

Way up at the level of ex­per­i­ment, we never find De­tec­tor 1 and De­tec­tor 2 both go­ing off. We’ll find De­tec­tor 1 go­ing off twice, or De­tec­tor 2 go­ing off twice, with equal fre­quency. (As­sum­ing I’ve got­ten the math and physics right. I didn’t ac­tu­ally perform the ex­per­i­ment.)

The con­figu­ra­tion’s iden­tity is not, “the B pho­ton go­ing to­ward E and the C pho­ton go­ing to­ward F. ” Then the re­sul­tant con­figu­ra­tions in case 1 and case 4 would not be equal. Case 1 would be, “B pho­ton to E, C pho­ton to F” and case 4 would be “Bpho­ton to F, C pho­ton to E.” Th­ese would be two dis­t­in­guish­able con­figu­ra­tions, if con­figu­ra­tions had pho­ton-track­ing struc­ture.

So we would not add up the two am­pli­tudes and can­cel them out. We would keep the am­pli­tudes in two sep­a­rate con­figu­ra­tions. The to­tal am­pli­tudes would have non-zero squared mod­uli. And when we ran the ex­per­i­ment, we would find (around half the time) that De­tec­tor 1 and De­tec­tor 2 each reg­istered one pho­ton. Which doesn’t hap­pen, if my calcu­la­tions are cor­rect.

Con­figu­ra­tions don’t keep track of where par­ti­cles come from. A con­figu­ra­tion’s iden­tity is just, “a pho­ton here, a pho­ton there; an elec­tron here, an elec­tron there.” No mat­ter how you get into that situ­a­tion, so long as there are the same species of par­ti­cles in the same places, it counts as the same con­figu­ra­tion.

I say again that the ques­tion “What kind of in­for­ma­tion does the con­figu­ra­tion’s struc­ture in­cor­po­rate?” has ex­per­i­men­tal con­se­quences. You can de­duce, from ex­per­i­ment, the way that re­al­ity it­self must be treat­ing con­figu­ra­tions.

In a clas­si­cal uni­verse, there would be no ex­per­i­men­tal con­se­quences. If the pho­ton were like a lit­tle billiard ball that ei­ther went one way or the other, and the con­figu­ra­tions were our be­liefs about pos­si­ble states the sys­tem could be in, and in­stead of am­pli­tudes we had prob­a­bil­ities, it would not make a differ­ence whether we tracked the ori­gin of pho­tons or threw the in­for­ma­tion away.

In a clas­si­cal uni­verse, I could as­sign a 25% prob­a­bil­ity to both pho­tons go­ing to E, a 25% prob­a­bil­ity of both pho­tons go­ing to F, a 25% prob­a­bil­ity of the B pho­ton go­ing to E and the C pho­ton go­ing to F, and 25% prob­a­bil­ity of the B pho­ton go­ing to Fand the C pho­ton go­ing to E. Or, since I per­son­ally don’t care which of the two lat­ter cases oc­curred, I could de­cide to col­lapse the two pos­si­bil­ities into one pos­si­bil­ity and add up their prob­a­bil­ities, and just say, “a 50% prob­a­bil­ity that each de­tec­tor gets one pho­ton.”

With prob­a­bil­ities, we can ag­gre­gate events as we like—draw our bound­aries around sets of pos­si­ble wor­lds as we please—and the num­bers will still work out the same. The prob­a­bil­ity of two mu­tu­ally ex­clu­sive events always equals the prob­a­bil­ity of the first event plus the prob­a­bil­ity of the sec­ond event.

But you can’t ar­bi­trar­ily col­lapse con­figu­ra­tions to­gether, or split them apart, in your model, and get the same ex­per­i­men­tal pre­dic­tions. Our mag­i­cal tool tells us the ra­tios of squared mod­uli. When you add two com­plex num­bers, the squared mod­u­lus of the sum is not the sum of the squared mod­uli of the parts:

E.g.

Or in the cur­rent ex­per­i­ment of dis­course, we had flows of and can­cel out, adding up to 0, whose squared mod­u­lus is 0, where the squared mod­u­lus of the parts would have been 1 and 1.

If in place of Squared_Mo­du­lus, our mag­i­cal tool was some lin­ear func­tion— any func­tion where —then all the quan­tum­ness would in­stantly van­ish and be re­placed by a clas­si­cal physics. (A differ­ent clas­si­cal physics, not the same illu­sion of clas­si­cal­ity we hal­lu­ci­nate from in­side the higher lev­els of or­ga­ni­za­tion in our own quan­tum world.)

If am­pli­tudes were just prob­a­bil­ities, they couldn’t can­cel out when flows col­lided. If con­figu­ra­tions were just states of knowl­edge, you could re­or­ga­nize them how­ever you liked.

But the con­figu­ra­tions are nailed in place, in­di­visi­ble and un­merge­able with­out chang­ing the laws of physics.

And part of what is nailed is the way that con­figu­ra­tions treat mul­ti­ple par­ti­cles. A con­figu­ra­tion says, “a pho­ton here, a pho­ton there,” not “this pho­ton here, that­pho­ton there.” “This pho­ton here, that pho­ton there” does not have a differ­ent iden­tity from “that pho­ton here, this pho­ton there.”

The re­sult, visi­ble in to­day’s ex­per­i­ment, is that you can’t fac­tor­ize the physics of our uni­verse to be about par­ti­cles with in­di­vi­d­ual iden­tities.

Part of the rea­son why hu­mans have trou­ble com­ing to grips with perfectly nor­malquan­tum physics, is that hu­mans bizarrely keep try­ing to fac­tor re­al­ity into a sum of in­di­vi­d­u­ally real billiard balls.

Ha ha! Silly hu­mans.

• Je­sus Christ, the com­plex plane. I half-re­mem­ber that.

Eliezer, this may come as a shock, but I sus­pect there ex­ists at least some minor­ity of in­di­vi­d­u­als be­yond just me who will find the con­sis­tent use of com­plex num­bers at all to be the the most mi­graine-in­duc­ing part of this. You might also find that even those of us who sup­pos­edly know how to do some com­pu­ta­tion on the com­plex plane are likely to have lit­tle to no in­tu­itive grasp of com­plex num­bers. Em­pha­sis on com­pu­ta­tion over un­der­stand­ing in math­e­mat­ics teach­ing, while per­va­sive, does not tend to serve stu­dents well. I could be the only one for whom this ap­plies, but I wouldn’t bet heav­ily on it.

Thank­fully, there’s already a cou­ple of “in­tu­itive ex­pla­na­tions” of com­plex num­bers on the ’net. And I dug up the links. And the same site has a few ar­ti­cles about gen­er­al­iza­tions of the Pythagorean The­o­rem, on a re­lated note. Ba­si­cally, any­one who is hav­ing any trou­ble with the math­e­mat­i­cal side of this is likely to find it a bit of a help. It’s also a lot like over­com­ing bias in its ex­plana­tory ap­proach, so there’s that.

Imag­i­nary/​com­plex num­bers:

http://​​bet­ter­ex­plained.com/​​ar­ti­cles/​​a-vi­sual-in­tu­itive-guide-to-imag­i­nary-num­bers/​​

http://​​bet­ter­ex­plained.com/​​ar­ti­cles/​​in­tu­itive-ar­ith­metic-with-com­plex-num­bers/​​

Gen­eral math:

http://​​bet­ter­ex­plained.com/​​ar­ti­cles/​​cat­e­gory/​​math/​​

I should prob­a­bly also take this op­por­tu­nity to swear up and down that I’m not try­ing to gen­er­ate ad rev­enue for that site, but you’d have to take my word on it. I might also add that I’m quite cer­tain I still don’t un­der­stand com­plex num­bers mean­ingfully, but that’s a sep­a­rate thing.

• Jor­dan, the three an­swers to your ques­tion are:

1) Each pho­ton is ac­tu­ally spread out in con­figu­ra­tion space—I’ll talk about this later—so an in­finites­i­mal er­ror in timing only cre­ates an in­finites­i­mal prob­a­bil­ity of both de­tec­tors go­ing off, rather than a dis­con­tin­u­ous jump.

2) Physi­cists have got­ten good at do­ing things with bloody pre­cise timing, so they can run ex­per­i­ments like this.

3) I didn’t ac­tu­ally perform the ex­per­i­ment.

• What con­fuses me about the ac­tual ver­ifi­ca­tion of these ex­per­i­ments is that they re­quire perfect timing and dis­tanc­ing. How ex­actly do you make two pho­tons hit a half-silvered mir­ror at ex­actly the same time? It seems that if you were off only slightly then the uni­verse would nec­es­sar­ily have to keep track of both as in­di­vi­d­u­als. In prac­tice you’re always go­ing to be off slightly so you would think by your ex­pla­na­tion above that this would in fact change the re­sult of the ex­per­i­ment, plac­ing a 25% prob­a­bil­ity on all four cases. Why doesn’t it?

• I thought I un­der­stood quan­tum me­chan­ics. I have stud­ied it for years. Passed ex­ams. Got a de­gree.

This is the first time I have ever heard of the HOM ex­per­i­ment, and it is caus­ing a crisis in my mind, since it does not match what I know about quan­tum me­chan­ics.

They re­ally should at least tell us of this ex­per­i­ment and its re­sults when they start teach­ing us. Even af­ter these years I was still un­der the im­pres­sion that quan­tum me­chan­ics was about un­break­able limi­ta­tions on the mea­sure­ment of data.

This ex­per­i­ment proves that the uni­verse it­self does not work in an in­tu­itive way. All the peo­ple who are try­ing to out­wit quan­tum me­chan­ics by mea­sur­ing po­si­tion and en­ergy at the same time are not just at­tempt­ing some­thing im­pos­si­ble, what they are try­ing doesn’t even make sense.

AAAAAAAAAAAGH!

• Re­gard­ing Larry’s ques­tion about how close the pho­tons have to be be­fore they merge --

The solu­tion to that prob­lem comes from the fact that Eliezer’s ex­per­i­ment is (nec­es­sar­ily) sim­plify­ing things. I’m sure he’ll get to this in a later post so you might be bet­ter off wait­ing for a bet­ter ex­pla­na­tion (or read­ing Feyn­man’s QED: The Strange The­ory of Light and Mat­ter, which I think is a fan­tas­ti­cally clear ex­pla­na­tion of this stuff.) But if you’re will­ing to put up with a poor ex­pla­na­tion just to get it quicker...

In re­al­ity, you don’t have just one ini­tial am­pli­tude of a pho­ton at ex­actly time T. To get the full solu­tion, you have to add in the am­pli­tude of the pho­ton ar­riv­ing a lit­tle ear­lier, or a lit­tle later, and with a lit­tle smaller or a lit­tle larger wave­length, and even trav­el­ling faster or slower or not in a straight line, and pos­si­bly in­ter­act­ing with some stray elec­tron along the way, and so on for a ridicu­lously in­tractable set of com­pli­ca­tions. Each vari­a­tion or in­ter­ac­tion shows up as a small mul­ti­plier to your ini­tial am­pli­tude.

But for­tu­nately, most of these in­ter­ac­tions can­cel out over long dis­tances or long times, just like the case of the two pho­tons hit­ting op­po­site de­tec­tors, so in this ex­per­i­ment you can treat the pho­ton as just hav­ing ar­rived at a cer­tain time and you’ll get very close to the right an­swer.

Or in other words—eas­ier to vi­su­al­ize but per­haps mis­lead­ing—the am­pli­tude of the pho­tons is “smeared out” a lit­tle in space and time, so it’s not too hard to get them to over­lap enough for the ex­per­i­ment to work.

--Jeff

• Added refer­ence to the writeup of this ex­per­i­ment in Wikipe­dia: http://​​en.wikipe­dia.org/​​wiki/​​Hong%E2%80%93Ou%E2%80%93Man­del_effect

HT Man­fred.

• (The ver­sion in this com­ment doesn’t work, though the main one in the ar­ti­cle does. There’s a pe­riod in­side the URL. -- You should have http://​​en.wikipe­dia.org/​​wiki/​​Hong%E2%80%93Ou%E2%80%93Man­del_effect

• Thx fixed.

• Thank you for that. This is one of most in­ter­est­ing ex­per­i­ments I’ve seen, be­cause in my in­ter­pre­ta­tion, it’s re­fut­ing a quan­tum on­tolog­i­cal ran­dom­ness more than con­firm­ing it.

Con­sider the case of 1 pho­ton. It hits the split­ter, the split­ter es­tab­lishes bound­ary con­di­tions on the pho­ton wave packet such that there is only pos­si­ble mode com­pat­i­ble with the split­ter at any given time, and only 2 modes gen­er­ally.

Now, two pho­tons. The ar­ti­cle says they have to match in phase, time, and po­lariza­tion. Since they match, they will be deflected in the same way all the time, be­cause the beam split­ter is only com­pat­i­ble with one mode at a par­tic­u­lar in­stance of time (for a par­tic­u­lar phase and po­lariz­tion?).

Yes, I know, Bell’s The­o­rem, no hid­den vari­ables, yadda yadda yadda. I’m not con­vinced. Nei­ther was Jaynes, and I find him clearer and clev­erer than those who think the quan­tum world is mag­i­cal and mys­te­ri­ous, and the world runs on telekine­sis. He wasn’t con­vinced by Bell, and in par­tic­u­lar charged that Bell’s anal­y­sis didn’t in­clude time vary­ing hid­den vari­ables, which is of course the nat­u­ral way to get the ap­pear­ance of on­tolog­i­cal ran­dom­ness—have the hid­den vari­able vary at smaller time scales than you are able to mea­sure.

Although ap­par­ently not. Looks like the HOM effect has mea­sured the time in­ter­val down to the rele­vant time scales. Hur­rah! On­tolog­i­cal ran­dom­ness is dead! Long live the Bayesian Con­spir­acy!

But I’d like to see the ex­per­i­ment done with­out the split­ter. Do the pho­tons ever go the same way with­out the split­ter there to es­tab­lish a bound­ary con­di­tion? If it’s all just about pho­ton en­tan­gle­ment and on­tolog­i­cal ran­dom­ness, shouldn’t they? My pre­dic­tion is that they wouldn’t.

And yes, I re­al­ize that it’s un­likely that I have re­solved all the mys­ter­ies of quan­tum physics be­fore break­fast. Still, that’s the way it looks to me.

Won­der­ing if Jaynes had ever com­mented on the HOM effect, I found no di­rect com­ment, but in­stead a wikipe­dia ar­ti­cle: “The Jaynes–Cum­mings model (JCM) is a the­o­ret­i­cal model in quan­tum op­tics. It de­scribes the sys­tem of a two-level atom in­ter­act­ing with a quan­tized mode of an op­ti­cal cav­ity...” Is he get­ting at the same thing here—of bound­ary con­di­tions ap­plied to wave pack­ets? I don’t know. Looks like in his last pa­per on quan­tum the­ory, Scat­ter­ing of Light by Free Elec­trons, he’s get­ting at the wave func­tion as be­ing phys­i­cally real, and not the prob­a­bil­ity dis­tri­bu­tion of teeny tiny billiard balls.

And while I was at it, I found that Hess and Philipp have been push­ing against Bell for time vari­a­tion. Some­thing to check out some­time.

• Op­po­site prob­lem—I know pretty much what an imag­i­nary num­ber is, and even some ap­pli­ca­tions of i. Num­bers can have real and imag­i­nary el­e­ments, fine. But I have no idea why they have an ap­pli­ca­tion here.

That said, this post makes a lot of in­tu­itive sense to me, a hu­man­i­ties grad­u­ate, so this se­ries is off to a pretty good start. If Eliezer’s good at one thing, it’s ex­plain­ing com­plex-seem­ing things I know a lit­tle about in a very sen­si­ble and use­ful way.

• Ac­tu­ally, I was in ex­actly the same po­si­tion.

Then I ac­tu­ally read the ar­ti­cle from the com­ment above (to re­fa­mil­iarise my­self with the maths), and was quite sur­prised to find that the ar­ti­cle makes this re­la­tion very clear.

Worth a look. :)

• Oh my god...imag­i­nary num­bers...they make sense now! Se­ri­ously, thank you for that link. I’ve got­ten all the way through high school calcu­lus with­out ever hav­ing imag­i­nary num­bers=ro­ta­tion ex­plained. Look­ing at the graph for 10 sec­onds com­pletely ex­plained that con­cept and why Eliezer was us­ing imag­i­nary num­bers to rep­re­sent when the pho­tons were deflected.

• I found the calcu­la­tion of the am­pli­tude flows for cases 1 to 4 con­fus­ing at first. I think the part I missed is that the re­flec­tion of a pho­ton at a half silvered mir­ror mul­ti­plies the con­figu­ra­tion (ie the state of both pho­tons) by i. So in case 1 we get each pho­ton mul­ti­plied by i twice, so the am­pli­tude at both de­tec­tors is 1.

• I am hav­ing a bit of trou­ble with this se­ries. I can see that you are ex­plain­ing that re­al­ity con­sists of states with “am­pli­tude” num­bers as­signed to each state.

1. You seem to as­sign ar­bi­trary num­bers to the ini­tial states and an ar­bi­trary am­pli­tude change rules to mir­rors. Why is this in any way ap­pli­ca­ble to ob­jec­tive re­al­ity? Or are these num­bers non-ar­bi­trary? Or am I just miss­ing some­thing el­e­men­tary?

2. Why states of pho­tons or de­tec­tors are com­plex num­bers and mir­ror is a func­tion?

3. How does time fac­tor into all of this?

• Well, an al­ter­na­tive way to sug­gest prob­a­bil­ity is the Right Way is stuff like dutch book ar­gu­ments, or more gen­er­ally, build­ing up de­ci­sion the­ory, and epistemic prob­a­bil­ities get gen­er­ated “along the way”

The ar­gu­ments I like are of the form that each step is ba­si­cally along the lines of “if you don’t fol­low this rule, you’ll be vuln­er­a­ble to that kind of stupid be­havrior, where stupid be­hav­ior ba­si­cally means ‘wast­ing re­sources with­out mak­ing progress to­ward fulfilling your goals, what­ever they are’”

Frankly, I also like those ar­gu­ments be­cause math­e­mat­i­caly, they’re cleaner. Each step gets you some­thing of the fi­nal re­sult, and gen­er­ally doesn’t re­quire any­thing more de­mand­ing than ba­sic lin­ear algeabra.

It’s nice to know Cox’s The­o­rem is there, but it’s not what, to me at least, would be a sim­ple clean deriva­tion.

• Richard: Cox’s the­o­rem is an ex­am­ple of a par­tic­u­lar kind of re­sult in math, where you have some par­tic­u­lar ob­ject in mind to rep­re­sent some­thing, and you come up with very plau­si­ble, very gen­eral ax­ioms that you want this rep­re­sen­ta­tion to satisfy, and then prove this ob­ject is unique in satis­fy­ing these. There are equiv­a­lent re­sults for en­tropy in in­for­ma­tion the­ory. The prob­lem with these re­sults, they are al­most always based on hind­sight, so a lot of the times you sneak in an ax­iom that only SEEMS plau­si­ble in hind­sight. For in­stance, Cox’s the­o­rem states that plau­si­bil­ity is a real num­ber. Why should it be a real num­ber?

• Another great ex­pla­na­tion. What you de­scribe sug­gests that the fabric of the uni­verse is not made of par­tic­u­late stuff, but rather in­for­ma­tional and/​or com­pu­ta­tional. Or am I read­ing to much into this?

• A lot of the dumbed down sci­ence I read/​watch (doc­u­men­taries, pop­u­lar sci­ency mag­a­z­ines, sci­ency web­sites, etc) sug­gests that this is ex­actly how a num­ber of physi­cists view the world these days.

For ex­am­ple, of­ten when there is de­bate about whether such and such the­o­ret­i­cal effect breaks the con­ser­va­tion laws they speak in terms of in­for­ma­tion be­ing con­served or de­stroyed, even though they are refer­ring to things like pho­tons and what­not (e.g. the Thorne-Hawk­ing-Preskill bet con­cern­ing Hawk­ing Ra­di­a­tion).

• Thanks for helping me re­move the clas­sic hal­lu­ci­na­tion to now un­der­stand the dou­ble slit phe­nomenon.

• Gray Area,

In re­ply to “why a real num­ber ques­tion”, we might want to weaken the the­ory to the point were only equal­ities and in­equal­ities can be stated. There are two weaker desider­ata one might hold. Let (A|X) be the plau­si­bil­ity of A given X.

Tran­si­tivity: if (A|X) > (B|X) and (B|X) > (C|X), then (A|X) > (C|X)

Univer­sal Com­pa­ra­bil­ity: one of the fol­low­ing must hold (A|X) > (B|X) (A|X) = (B|X) (A|X) < (B|X)

If you keep both, you might as well use a real num­ber—do­ing so will cap­ture all of the de­sired be­hav­ior. If you throw out Tran­si­tivity, I have a se­ries of wa­gers I’d like to make with you. If you throw out Univer­sal Com­pa­ra­bil­ity, then you get lat­tice the­o­ries in which propo­si­tions are ver­texes and per­mit­ted com­par­i­sons are edges.

On the other hand, you might find just a sin­gle real num­ber too re­stric­tive, so you use more than one. Then you get some­thing like Demp­ster-Shafer the­ory.

In short, there are al­ter­na­tives.

As for why should the plau­si­bil­ity of the nega­tion of a state­ment de­pend only on the plau­si­bil­ity of the state­ment, the an­swer (I be­lieve) is that we are con­sid­er­ing only the sorts of propo­si­tions in which the Law of the Ex­cluded Mid­dle holds. So if we are us­ing only a sin­gle real num­ber to cap­ture plau­si­bil­ity, we need f{(A|X),(!A|X)} = (truth|X) = con­stant, and we have no free­dom to let (!A|X) de­pend on the de­tails of A.

• Dan, Em­mett,

I hear you. Un­for­tu­nately, I can’t put as much work into this as I did for the in­tu­itive ex­pla­na­tion of Bayes’s The­o­rem, plus the sub­ject mat­ter is in­her­ently more com­pli­cated.

Feyn­man’s QED uses lit­tle ar­rows in 2D space in­stead of com­plex num­bers (the two are equiv­a­lent). And if I had the time and space, I’d draw differ­ent vi­sual di­a­grams for each con­figu­ra­tion, and show the am­pli­tude flow­ing from one to an­other...

But QM is also in­her­ently more com­pli­cated than Bayes’s The­o­rem, and takes more effort; plus I’m try­ing to ex­plain it in less time… I’m not figur­ing that all read­ers will be able to fol­low, I’m afraid, just hop­ing that some of them will be.

If the prob­lem is not that QM is con­fus­ing but that you can’t fol­low what is be­ing said at all, you prob­a­bly want to be read­ing Richard Feyn­man’s QED in­stead.

• I found your ex­pla­na­tions of Bayesian prob­a­bil­ity en­light­en­ing, and I’ve tried to read sev­eral ex­pla­na­tions be­fore. Your re­cent posts on quan­tum me­chan­ics, much less so. Un­like the prob­a­bil­ity posts, I find these ones very hard to fol­low. Every time I hit a block of ‘The “B to D” pho­ton is deflected and the “C to D” pho­ton is deflected.’ state­ments, my eyes glaze over and I lose you.

• Along the lines of what Larry asks: Ob­vi­ously the an­gles are not go­ing to be perfect. The two pho­tons will come in with slightly differ­ent an­gles. So you would think the pho­tons will not be perfectly in­dis­t­in­guish­able. Now I won­der if it is the case that if you put your de­tec­tors in close, so that they “see” a rel­a­tively wide range of an­gles, then you get the in­terfer­ence and the pho­tons are treated the same; whereas if you put your de­tec­tors far away, they might be­come sen­si­tive to a smaller range of an­gles, so that they could dis­t­in­guish the two pho­tons, then the in­terfer­ence might go away?

But in that case, you could make an FTL sig­nal­ing de­vice by putting one de­tec­tor at a dis­tance, and mov­ing the other de­tec­tor from close to far. When close, you get in­terfer­ence and get two pho­tons or none, while when far, you get sin­gle pho­tons, a differ­ence de­tectable by the re­mote de­tec­tor. Clearly this can’t hap­pen.

I’m sure Jeff is right and that a ful­ler in­ves­ti­ga­tion of the wave equa­tions would ex­plain ex­actly what hap­pens here. But it does point out one big prob­lem with this level of de­scrip­tion of the quan­tum world: the ab­sence of a pri­mary role for space and time. We just have events and con­figu­ra­tions. How does lo­cal­ity and causal­ity en­ter into this? Where do speed of light limi­ta­tions get en­forced? The world is fun­da­men­tally lo­cal, but some ways of ex­press­ing QM seem to ig­nore that seem­ingly im­por­tant foun­da­tion.

• With the setup you de­scribe, the re­mote de­tec­tor still has to wait un­til the pho­tons would be ex­pected to ar­rive to know whether the other de­tec­tor had been moved, right? This seems like it would be ex­actly at-light-speed sig­nal­ing.

• Here’s what bugs me: Those two pho­tons aren’t go­ing to be ex­actly the same, in terms of say fre­quency or maybe the an­gle they make against the table. So how close do they have to be for the con­figu­ra­tions to merge? Or is that a Wrong Ques­tion? Per­haps if we left the pho­ton em­miter the same but changed the de­tec­tor to one that could tell the differ­ence in an­gle, then the ex­per­i­men­tal re­sults would change? What if we use a an­gle-de­tect­ing pho­ton de­tec­tor but pro­gram it to dump the an­gle into /​dev/​null?

• Given that we can con­struct mul­ti­ple kinds of prob­a­bil­ity the­ory, on what grounds should we pre­fer one over the other to rep­re­sent what ‘be­lief’ ought to be?

Make sure you un­der­stand Cox’s the­o­rem, then ex­hibit for me two kinds of prob­a­bil­ity the­ory, then I will re­ply.

• Gray, fixed.

• If in place of Squared_Mo­du­lus, our mag­i­cal tool was some lin­ear func­tion—any func­tion where F(X + Y) = F(X) + F(Y)—then all the quan­tum­ness would in­stantly van­ish and be re­placed by a clas­si­cal physics.

I am hav­ing trou­ble work­ing out lin­ear­ity of func­tions. Let’s say we take a lin­ear func­tion F(x) = x + 5. Then we use the above lin­ear­ity you men­tion F(5 + 6) = F(5) + F(6).

We get F(11) = F(5) + F(6).

If we work that out we get ⇒ 11 + 5 = 5+5 + 6+5.

The re­sult is 16 =/​= 21.

So, the lin­ear func­tion doesn’t have lin­ear­ity as its prop­erty?

I am con­fused.

• The func­tion x → x+5 is not a lin­ear func­tion in the sense that’s rele­vant here. (The word “lin­ear” has mul­ti­ple uses, re­lated to one an­other but not iden­ti­cal.)

• It is not at all clear to me why a sin­gle half-mir­ror should re­sult in two mul­ti­pli­ca­tions and not an ad­di­tion in the case of there be­ing more than one pho­ton. After all they are added up when they strike the de­tec­tor.

It is com­pletely un­men­tioned why this would be the case, and would seem to bear ex­pla­na­tion.

• I’m not quite un­der­stand­ing it ei­ther, but if I’m slightly un­der­stand­ing cor­rectly: use sound wave as an ANALOGY. The half-silver mir­rors al­low it to “res­onate” (sound terms) and ri­co­chet off at the same time, while full silver only al­lows it to ri­co­chet. (this is most likely VERY WRONG. I just now got the rea­son­ing be­hind com­plex num­bers, ro­ta­tion of planes, 3d waves, etc)

• Cyan: I cer­tainly ad­mit that the ease of the math may be part of my re­ac­tion. Maybe if I was far more fa­mil­iar with the the­ory of func­tional equa­tions I’d find Cox’s the­o­rem more el­e­gant than I do.

(I’ve read that if one makes a minor tweak to Cox’s the­o­rem, just let­ting go of the real num­ber crite­ria and let­ting con­fi­dences be com­plex num­bers, the same line of deriva­tion more or less hands you quan­tum am­pli­tudes. I haven’t seen that deriva­tion though, but if that’s cor­rect, it makes Cox’s the­o­rem even more ap­peal­ing. QM for “free”! :))

The vuln­er­a­biliy ones though ac­tively mo­ti­vate the crite­ria rather than a list of rea­son­able sound­ing prop­er­ties an ex­ten­tion to boolean logic ought to have. The ba­sic crite­ria, I guess, could be de­scribes as “will you end up in a situ­a­tion in which you’d know­ingly will­ingly waste re­sources with­out in any way benefit­ing your goals?”

So each step is ba­si­cally a “Math­e­mat­i­cal Karma is go­ing to get you and take away your pen­nies if you don’t fol­low this rule.” :)

But yeah, the bit about only rea­son­able ex­ten­tion of vanilla logic does make the Cox thing a bit more ap­peal­ing. On the other hand, that may ac­tively dis­suade some from the Bayesian per­spec­tive. Speci­fi­cally, con­struc­tivists, in­tu­ition­ists in par­tic­u­lar, for in­stance, may be hes­i­tant of any­thing too de­pen­dant on law of ex­cluded mid­dle in the ab­stract. (This isn’t an ab­stract hy­po­thet­i­cal. I’ve ba­si­cally ended up in a friendly ar­gu­ment a while back with some­one that more or less had them re­ject­ing the no­tion of us­ing prob­a­bil­ity as a mea­sure of be­lief/​sub­jec­tive un­cer­tainty be­cause it was an ex­ten­tion of boolean logic, and the guy didn’t like law of ex­cluded mid­dle and ba­si­cally was, near as I can make out, an in­tu­ition­ist. (Some of the math­e­mat­i­cal con­cepts he was bring­ing up seem to im­ply that))

Per­son­ally, I just re­ally like the whole “math karma” fla­vor of each step of a vuln­er­a­bil­ity ar­gu­ment. Just a differ­ent fla­vor than most math­e­mat­i­cal deriva­tions for, well, any­thing, that I’ve seen. Not to men­tion, in some for­mu­la­tions, get­ting de­ci­sion the­ory all at once with it.

• Psy-Kosh,

Cu­ri­ously, I have just the op­po­site ori­en­ta­tion—I like the fact that prob­a­bil­ity the­ory can be de­rived as an ex­ten­sion of logic to­tally in­de­pen­dently of de­ci­sion the­ory. Cox’s The­o­rem also does a good job on the “pun­ish­ing stupid be­hav­ior” front. If some­one demon­strate a sys­tem that dis­agrees with Bayesian prob­a­bil­ity the­ory, when you find a Bayesian solu­tion you can go back to the desider­ata and say which one is be­ing vi­o­lated. But on the math front, I got noth­ing—there’s no get­ting around the fact that func­tional equa­tions are tougher than lin­ear alge­bra.

• Grey Area asked, “For in­stance, Cox’s the­o­rem states that plau­si­bil­ity is a real num­ber. Why should it be a real num­ber?” For that mat­ter, why should the plau­si­bil­ity of the nega­tion of a state­ment de­pend only on the plau­si­bil­ity of the state­ment? Mightn’t the state­ment it­self be rele­vant?

• So you calcu­late events for pho­tons that hap­pen in par­allel (e.g. one pho­ton be­ing deflected while an­other is deflected as well) the same way you would for when they oc­cur in se­ries (e.g. a pho­ton be­ing deflected, and then be­ing deflected again)? It seems that in both cases you are mul­ti­ply­ing the origi­nal con­figu­ra­tion by −1 (i.e. i * i).

FWIW, my first in­stinct when I saw the di­a­gram was to as­sume 2 start­ing con­figu­ra­tions, one for each pho­ton, though I guess the point of this post was that I can’t do stuff like that. In fact, when I did the math that way, I came up with the pho­tons hit­ting differ­ent de­tec­tors twice as much as when they both hit the same one.

I think I’ll be pick­ing up the Feyn­man book.....

• Feyn­man’s QED uses lit­tle ar­rows in 2D space in­stead of com­plex num­bers (the two are equiv­a­lent).

this is quite off­topic, but there are im­por­tant analo­gies [1] [2] be­tween Feyn­man di­a­grams, topol­ogy and com­puter sci­ence. be­ing aware of this may amke the topic eas­ier to un­der­stand.

• Jeff, I would not call your last para­graph mis­lead­ing. It is quan­tum me­chan­ics—as op­posed to quan­tum field the­ory—and I think QED is a gra­tu­itously com­pli­cated way to an­swer the ques­tion. Per­haps it’s a way to get from clas­si­cal no­tions of par­ti­cles to quan­tum me­chan­ics, but it seems to me to go against the spirit of try­ing to un­der­stand QM on its own terms, rather than as a mod­ifi­ca­tion of clas­si­cal par­ti­cles.

• If the pho­ton sources aren’t in a su­per po­si­tion I think you have to rep­re­sent the sys­tem with a vec­tor of com­plex num­bers. IANAP ei­ther.

• “The prob­a­bil­ity of two events equals the prob­a­bil­ity of the first event plus the prob­a­bil­ity of the sec­ond event.”

Mu­tu­ally ex­clu­sive events.

It is in­ter­est­ing that you in­sist that be­liefs ought to be rep­re­sented by clas­si­cal prob­a­bil­ity. Given that we can con­struct mul­ti­ple kinds of prob­a­bil­ity the­ory, on what grounds should we pre­fer one over the other to rep­re­sent what ‘be­lief’ ought to be?