Configurations and Amplitude

So the uni­verse isn’t made of lit­tle billiard balls, and it isn’t made of crests and troughs in a pool of aether… Then what is the stuff that stuff is made of?

In Figure 1, we see, at A, a half-silvered mir­ror, and two pho­ton de­tec­tors, De­tec­tor 1 and De­tec­tor 2.

Early sci­en­tists, when they ran ex­per­i­ments like this, be­came con­fused about what the re­sults meant. They would send a pho­ton to­ward the half-silvered mir­ror, and half the time they would see De­tec­tor 1 click, and the other half of the time they would see De­tec­tor 2 click.

The early sci­en­tists—you’re go­ing to laugh at this—thought that the silver mir­ror deflected the pho­ton half the time, and let it through half the time.

Ha, ha! As if the half-silvered mir­ror did differ­ent things on differ­ent oc­ca­sions! I want you to let go of this idea, be­cause if you cling to what early sci­en­tists thought, you will be­come ex­tremely con­fused. The half-silvered mir­ror obeys the same rule ev­ery time.

If you were go­ing to write a com­puter pro­gram that was this ex­per­i­ment— not a com­puter pro­gram that pre­dicted the re­sult of the ex­per­i­ment, but a com­puter pro­gram that re­sem­bled the un­der­ly­ing re­al­ity—it might look sort of like this:

At the start of the pro­gram (the start of the ex­per­i­ment, the start of time) there’s a cer­tain math­e­mat­i­cal en­tity, called a con­figu­ra­tion. You can think of this con­figu­ra­tion as cor­re­spond­ing to “there is one pho­ton head­ing from the pho­ton source to­ward the half-silvered mir­ror,” or just “a pho­ton head­ing to­ward A.”

A con­figu­ra­tion can store a sin­gle com­plex value—“com­plex” as in the com­plex num­bers , with i defined as . At the start of the pro­gram, there’s already a com­plex num­ber stored in the con­figu­ra­tion “a pho­ton head­ing to­ward A.” The ex­act value doesn’t mat­ter so long as it’s not zero. We’ll let the con­figu­ra­tion “a pho­ton head­ing to­ward A” have a value of .

All this is a fact within the ter­ri­tory, not a de­scrip­tion of any­one’s knowl­edge. A con­figu­ra­tion isn’t a propo­si­tion or a pos­si­ble way the world could be. A con­figu­ra­tion is a vari­able in the pro­gram—you can think of it as a kind of mem­ory lo­ca­tion whose in­dex is “a pho­ton head­ing to­ward A”—and it’s out there in the ter­ri­tory.

As the com­plex num­bers that get as­signed to con­figu­ra­tions are not pos­i­tive real num­bers be­tween 0 and 1, there is no dan­ger of con­fus­ing them with prob­a­bil­ities. “A pho­ton head­ing to­ward A” has com­plex value −1, which is hard to see as a de­gree of be­lief. The com­plex num­bers are val­ues within the pro­gram, again out there in the ter­ri­tory. We’ll call the com­plex num­bers am­pli­tudes.

There are two other con­figu­ra­tions, which we’ll call “a pho­ton go­ing from A to De­tec­tor 1” and “a pho­ton go­ing from A to De­tec­tor 2.” Th­ese con­figu­ra­tions don’t have a com­plex value yet; it gets as­signed as the pro­gram runs.

We are go­ing to calcu­late the am­pli­tudes of “a pho­ton go­ing from A to­ward 1” and “a pho­ton go­ing from A to­ward 2” us­ing the value of “a pho­ton go­ing to­ward A,” and the rule that de­scribes the half-silvered mir­ror at A.

Roughly speak­ing, the half-silvered mir­ror rule is “mul­ti­ply by 1 when the pho­ton goes straight, and mul­ti­ply by i when the pho­ton turns at a right an­gle.” This is the uni­ver­sal rule that re­lates the am­pli­tude of the con­figu­ra­tion of “a pho­ton go­ing in,” to the am­pli­tude that goes to the con­figu­ra­tions of “a pho­ton com­ing out straight” or “a pho­ton be­ing deflected.”[1]

So we pipe the am­pli­tude of the con­figu­ra­tion “a pho­ton go­ing to­ward A,” which is , into the half-silvered mir­ror at A, and this trans­mits an am­pli­tude of to “a pho­ton go­ing from A to­ward 1,” and also trans­mits an am­pli­tude of to “a pho­ton go­ing from A to­ward 2.”

In the Figure 1 ex­per­i­ment, these are all the con­figu­ra­tions and all the trans­mit­ted am­pli­tude we need to worry about, so we’re done. Or, if you want to think of “De­tec­tor 1 gets a pho­ton” and “De­tec­tor 2 gets a pho­ton” as sep­a­rate con­figu­ra­tions, they’d just in­herit their val­ues from “A to 1” and “A to 2” re­spec­tively. (Ac­tu­ally, the val­ues in­her­ited should be mul­ti­plied by an­other com­plex fac­tor, cor­re­spond­ing to the dis­tance from A to the de­tec­tor; but we will ig­nore that for now, and sup­pose that all dis­tances trav­eled in our ex­per­i­ments hap­pen to cor­re­spond to a com­plex fac­tor of 1.)

So the fi­nal pro­gram state is:

Con­figu­ra­tion “a pho­ton go­ing to­ward A”: (−1+0i)
Con­figu­ra­tion “a pho­ton go­ing from A to­ward 1”: (0−i)
Con­figu­ra­tion “a pho­ton go­ing from A to­ward 2”: (−1+0i)

and optionally

Con­figu­ra­tion “De­tec­tor 1 gets a pho­ton”: (0−i)
Con­figu­ra­tion “De­tec­tor 2 gets a pho­ton”: (−1+0i).

This same re­sult oc­curs—the same am­pli­tudes stored in the same con­figu­ra­tions—ev­ery time you run the pro­gram (ev­ery time you do the ex­per­i­ment).

Now, for com­pli­cated rea­sons that we aren’t go­ing to go into here— con­sid­er­a­tions that be­long on a higher level of or­ga­ni­za­tion than fun­da­men­tal quan­tum me­chan­ics, the same way that atoms are more com­pli­cated than quarks—there’s no sim­ple­mea­sur­ing in­stru­ment that can di­rectly tell us the ex­act am­pli­tudes of each con­figu­ra­tion. We can’t di­rectly see the pro­gram state.

So how do physi­cists know what the am­pli­tudes are?

We do have a mag­i­cal mea­sur­ing tool that can tell us the squared mod­u­lus of a con­figu­ra­tion’s am­pli­tude. If the origi­nal com­plex am­pli­tude is , we can get the pos­i­tive real num­ber . Think of the Pythagorean the­o­rem: if you imag­ine the com­plex num­ber as a lit­tle ar­row stretch­ing out from the ori­gin on a two-di­men­sional plane, then the magic tool tells us the squared length of the lit­tle ar­row, but it doesn’t tell us the di­rec­tion the ar­row is point­ing.

To be more pre­cise, the magic tool ac­tu­ally just tells us the ra­tios of the squared lengths of the am­pli­tudes in some con­figu­ra­tions. We don’t know how long the ar­rows are in an ab­solute sense, just how long they are rel­a­tive to each other. But this turns out to be enough in­for­ma­tion to let us re­con­struct the laws of physics—the rules of the pro­gram. And so I can talk about am­pli­tudes, not just ra­tios of squared mod­uli.

When we wave the magic tool over “De­tec­tor 1 gets a pho­ton” and “De­tec­tor 2 gets a pho­ton,” we dis­cover that these con­figu­ra­tions have the same squared mod­u­lus—the lengths of the ar­rows are the same. Thus speaks the magic tool. By do­ing more com­pli­cated ex­per­i­ments (to be seen shortly), we can tell that the origi­nal com­plex num­bers had a ra­tio of i to 1.

And what is this mag­i­cal mea­sur­ing tool?

Well, from the per­spec­tive of ev­ery­day life—way, way, way above the quan­tum level and a lot more com­pli­cated—the mag­i­cal mea­sur­ing tool is that we send some pho­tons to­ward the half-silvered mir­ror, one at a time, and count up how many pho­tons ar­rive at De­tec­tor 1 ver­sus De­tec­tor 2 over a few thou­sand tri­als. The ra­tio of these val­ues is the ra­tio of the squared mod­uli of the am­pli­tudes. But the rea­son for this is not some­thing we are go­ing to con­sider yet. Walk be­fore you run. It is not pos­si­ble to un­der­stand what hap­pens all the way up at the level of ev­ery­day life, be­fore you un­der­stand what goes on in much sim­pler cases.

For to­day’s pur­poses, we have a mag­i­cal squared-mod­u­lus-ra­tio reader. And the magic tool tells us that the lit­tle two-di­men­sional ar­row for the con­figu­ra­tion “De­tec­tor 1 gets a pho­ton” has the same squared length as for “De­tec­tor 2 gets a pho­ton.” That’s all.

You may won­der, “Given that the magic tool works this way, what mo­ti­vates us to use quan­tum the­ory, in­stead of think­ing that the half-silvered mir­ror re­flects the pho­ton around half the time?”

Well, that’s just beg­ging to be con­fused—putting your­self into a his­tor­i­cally re­al­is­tic frame of mind like that and us­ing ev­ery­day in­tu­itions. Did I say any­thing about a lit­tle billiard ball go­ing one way or the other and pos­si­bly bounc­ing off a mir­ror? That’s not how re­al­ity works. Real­ity is about com­plex am­pli­tudes flow­ing be­tween con­figu­ra­tions, and the laws of the flow are sta­ble.

But if you in­sist on see­ing a more com­pli­cated situ­a­tion that billiard-ball ways of think­ing can’t han­dle, here’s a more com­pli­cated ex­per­i­ment.

In Figure 2, B and C are full mir­rors, and A and D are half-mir­rors. The line from D to E is dashed for rea­sons that will be­come ap­par­ent, but am­pli­tude is flow­ing from D to E un­der ex­actly the same laws.

Now let’s ap­ply the rules we learned be­fore:

At the be­gin­ning of time “a pho­ton head­ing to­ward A” has am­pli­tude .

We pro­ceed to com­pute the am­pli­tude for the con­figu­ra­tions “a pho­ton go­ing from A to B” and “a pho­ton go­ing from A to C”:

“a pho­ton go­ing from A to B” = a pho­ton head­ing to­ward A” =

Similarly,

“a pho­ton go­ing from A to C” = 1 a pho­ton head­ing to­ward A” =

The full mir­rors be­have (as one would ex­pect) like half of a half-silvered mir­ror—a full mir­ror just bends things by right an­gles and mul­ti­plies them by i. (To state this slightly more pre­cisely: For a full mir­ror, the am­pli­tude that flows, from the con­figu­ra­tion of a pho­ton head­ing in, to the con­figu­ra­tion of a pho­ton head­ing out at a right an­gle, is mul­ti­plied by a fac­tor of i.)

So:

“a pho­ton go­ing from B to D = “a pho­ton go­ing from A to B = ,
“a pho­ton go­ing from C to D = “a pho­ton go­ing from A to C =

“B to D and “C to D are two differ­ent con­figu­ra­tions—we don’t sim­ply write “a pho­ton at D—be­cause the pho­tons are ar­riv­ing at two differ­ent an­gles in these two differ­ent con­figu­ra­tions. And what D does to a pho­ton de­pends on the an­gle at which the pho­ton ar­rives.

Again, the rule (speak­ing loosely) is that when a half-silvered mir­ror bends light at a right an­gle, the am­pli­tude that flows from the pho­ton-go­ing-in con­figu­ra­tion to the pho­ton-go­ing-out con­figu­ra­tion, is the am­pli­tude of the pho­ton-go­ing-in con­figu­ra­tion mul­ti­plied by i. And when two con­figu­ra­tions are re­lated by a half-silvered mir­ror let­ting light straight through, the am­pli­tude that flows from the pho­ton-go­ing-in con­figu­ra­tion is mul­ti­plied by 1.

So:

From the con­figu­ra­tion “a pho­ton go­ing from B to D,” with origi­nal am­pli­tude(1+0i)

Am­pli­tude of flows to “a pho­ton go­ing from D to E.
Am­pli­tude of flows to “a pho­ton go­ing from D to F. ”

From the con­figu­ra­tion “a pho­ton go­ing from C to D,” with origi­nal am­pli­tude(0−i)

Am­pli­tude of flows to “a pho­ton go­ing from D to F.
Am­pli­tude of flows to “a pho­ton go­ing from D to E.

There­fore:

  • The to­tal am­pli­tude flow­ing to con­figu­ra­tion “a pho­ton go­ing from D to E” is .

  • The to­tal am­pli­tude flow­ing to con­figu­ra­tion “a pho­ton go­ing from D to F” is .

(You may want to try work­ing this out your­self on pen and pa­per if you lost track at any point.)

But the up­shot, from that su­per-high-level “ex­per­i­men­tal” per­spec­tive that we think of as nor­mal life, is that we see no pho­tons de­tected at E. Every pho­ton seems to end up at F. The ra­tio of squared mod­uli be­tween “D to E” and “D to F” is 0 to 4. That’s why the line from D to E is dashed, in this figure.

This is not some­thing it is pos­si­ble to ex­plain by think­ing of half-silvered mir­rors deflect­ing lit­tle in­com­ing billiard balls half the time. You’ve got to think in terms of am­pli­tude flows.

If half-silvered mir­rors deflected a lit­tle billiard ball half the time, in this setup, the lit­tle ball would end up at De­tec­tor 1 around half the time and De­tec­tor 2 around half the time. Which it doesn’t. So don’t think that.

You may say, “But wait a minute! I can think of an­other hy­poth­e­sis that ac­counts for this re­sult. What if, when a half-silvered mir­ror re­flects a pho­ton, it does some­thing to the pho­ton that en­sures it doesn’t get re­flected next time? And when it lets a pho­ton go through straight, it does some­thing to the pho­ton so it gets re­flected next time.”

Now re­ally, there’s no need to go mak­ing the rules so com­pli­cated. Oc­cam’s Ra­zor, re­mem­ber. Just stick with sim­ple, nor­mal am­pli­tude flows be­tween con­figu­ra­tions.

But if you want an­other ex­per­i­ment that dis­proves your new al­ter­na­tive hy­poth­e­sis, it’s Figure 3.

Here, we’ve left the whole ex­per­i­men­tal setup the same, and just put a lit­tle block­ing ob­ject be­tween B and D. This en­sures that the am­pli­tude of “a pho­ton go­ing from B to D” is 0.

Once you elimi­nate the am­pli­tude con­tri­bu­tions from that con­figu­ra­tion, you end up with to­tals of in “a pho­ton go­ing from D to F, ” and in “a pho­ton go­ing from D to E.”

The squared mod­uli of and are both 1, so the magic mea­sur­ing tool should tell us that the ra­tio of squared mod­uli is 1. Way back up at the level where physi­cists ex­ist, we should find that De­tec­tor 1 goes off half the time, and De­tec­tor 2 half the time.

The same thing hap­pens if we put the block be­tween C and D. The am­pli­tudes are differ­ent, but the ra­tio of the squared mod­uli is still 1, so De­tec­tor 1 goes off half the time and De­tec­tor 2 goes off half the time.

This can­not pos­si­bly hap­pen with a lit­tle billiard ball that ei­ther does or doesn’t get re­flected by the half-silvered mir­rors.

Be­cause com­plex num­bers can have op­po­site di­rec­tions, like 1 and −1, or i and −i, am­pli­tude flows can can­cel each other out. Am­pli­tude flow­ing from con­figu­ra­tion X into con­figu­ra­tion Y can be can­celed out by an equal and op­po­site am­pli­tude flow­ing from con­figu­ra­tion Z into con­figu­ra­tion Y. In fact, that’s ex­actly what hap­pens in this ex­per­i­ment.

In prob­a­bil­ity the­ory, when some­thing can ei­ther hap­pen one way or an­other, X or ¬X, then . And all prob­a­bil­ities are pos­i­tive. So if you es­tab­lish that the prob­a­bil­ity of Z hap­pen­ing given X is , and the prob­a­bil­ity of X hap­pen­ing is , then the to­tal prob­a­bil­ity of Z hap­pen­ing is at least no mat­ter what goes on in the case of ¬X. There’s no such thing as nega­tive prob­a­bil­ity, less-than-im­pos­si­ble cre­dence, or cred­i­bil­ity, so de­grees of be­lief can’t can­cel each other out like am­pli­tudes do.

Not to men­tion that prob­a­bil­ity is in the mind to be­gin with; and we are talk­ing about the ter­ri­tory, the pro­gram-that-is-re­al­ity, not talk­ing about hu­man cog­ni­tion or states of par­tial knowl­edge.

By the same to­ken, con­figu­ra­tions are not propo­si­tions, not state­ments, not ways the world could con­ceiv­ably be. Con­figu­ra­tions are not se­man­tic con­structs. Ad­jec­tives like prob­a­ble do not ap­ply to them; they are not be­liefs or sen­tences or pos­si­ble wor­lds. They are not true or false but sim­ply real.

In the ex­per­i­ment of Figure 2, do not be tempted to think any­thing like: “The pho­ton goes to ei­ther B or C, but it could have gone the other way, and this pos­si­bil­ity in­terferes with its abil­ity to go to E…”

It makes no sense to think of some­thing that “could have hap­pened but didn’t” ex­ert­ing an effect on the world. We can imag­ine things that could have hap­pened but didn’t—like think­ing, “Gosh, that car al­most hit me”—and our imag­i­na­tion can have an effect on our fu­ture be­hav­ior. But the event of imag­i­na­tion is a real event, that ac­tu­ally hap­pens, and that is what has the effect. It’s your imag­i­na­tion of the un­real event—your very real imag­i­na­tion, im­ple­mented within a quite phys­i­cal brain—that af­fects your be­hav­ior.

To think that the ac­tual event of a car hit­ting you—this event which could have hap­pened to you, but in fact didn’t—is di­rectly ex­ert­ing a causal effect on your be­hav­ior, is mix­ing up the map with the ter­ri­tory.

What af­fects the world is real. (If things can af­fect the world with­out be­ing “real,” it’s hard to see what the word “real” means.) Con­figu­ra­tions and am­pli­tude flows are causes, and they have visi­ble effects; they are real. Con­figu­ra­tions are not pos­si­ble wor­lds and am­pli­tudes are not de­grees of be­lief, any more than your chair is a pos­si­ble world or the sky is a de­gree of be­lief.

So what is a con­figu­ra­tion, then?

Well, you’ll be get­ting a clearer idea of that in later es­says.

But to give you a quick idea of how the real pic­ture differs from the sim­plified ver­sion we saw in this es­say…

Our ex­per­i­men­tal setup only dealt with one mov­ing par­ti­cle, a sin­gle pho­ton. Real con­figu­ra­tions are about mul­ti­ple par­ti­cles. The next es­say will deal with the case of more than one par­ti­cle, and that should give you a much clearer idea of what a con­figu­ra­tion is.

Each con­figu­ra­tion we talked about should have de­scribed a joint po­si­tion of all the par­ti­cles in the mir­rors and de­tec­tors, not just the po­si­tion of one pho­ton bop­ping around.

In fact, the re­ally real con­figu­ra­tions are over joint po­si­tions of all the par­ti­cles in the uni­verse, in­clud­ing the par­ti­cles mak­ing up the ex­per­i­menters. You can see why I’m sav­ing the no­tion of ex­per­i­men­tal re­sults for later es­says.

In the real world, am­pli­tude is a con­tin­u­ous dis­tri­bu­tion over a con­tin­u­ous space of con­figu­ra­tions. This es­say’s “con­figu­ra­tions” were blocky and digi­tal, and so were our “am­pli­tude flows.” It was as if we were talk­ing about a pho­ton tele­port­ing from one place to an­other.

If none of that made sense, don’t worry. It will be cleared up in later es­says. Just wanted to give you some idea of where this was head­ing.


1. [Edi­tor’s Note: Strictly speak­ing, a stan­dard half-silvered mir­ror would yield a rule “mul­ti­ply by −1 when the pho­ton turns at a right an­gle,” not “mul­ti­ply by i.” The ba­sic sce­nario de­scribed by the au­thor is not phys­i­cally im­pos­si­ble, and its use does not af­fect the sub­stan­tive ar­gu­ment. How­ever, physics stu­dents may come away con­fused if they com­pare the dis­cus­sion here to text­book dis­cus­sions of Mach–Zehn­der in­terferom­e­ters. We’ve left this idiosyn­crasy in the text be­cause it elimi­nates any need to spec­ify which side of the mir­ror is half-silvered, sim­plify­ing the ex­per­i­ment.]