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.