The Quantum Arena

Pre­vi­ously in se­ries: Clas­si­cal Con­figu­ra­tion Spaces

Yes­ter­day, we looked at con­figu­ra­tion spaces in clas­si­cal physics. In clas­si­cal physics, con­figu­ra­tion spaces are a use­ful, but op­tional, point of view.

To­day we look at quan­tum physics, which in­her­ently takes place in­side a con­figu­ra­tion space, and can­not be taken out.

For a start, as you might guess, in quan­tum physics we deal with dis­tri­bu­tions of com­plex am­pli­tudes, rather than prob­a­bil­ity dis­tri­bu­tions made up of pos­i­tive real num­bers. At left, I’ve used up 3 di­men­sions draw­ing a com­plex dis­tri­bu­tion over the po­si­tion of one par­ti­cle, A.

You may re­call that yes­ter­day, 3 di­men­sions let us dis­play the po­si­tion of two 1-di­men­sional par­ti­cles plus the sys­tem evolu­tion over time. To­day, it’s tak­ing us 3 di­men­sions just to vi­su­al­ize an am­pli­tude dis­tri­bu­tion over the po­si­tion of one 1-di­men­sional par­ti­cle at a sin­gle mo­ment in time. Which is why we did clas­si­cal con­figu­ra­tion spaces first.

To clar­ify the mean­ing of the above di­a­gram, the left-to-right di­rec­tion is the po­si­tion of A.

The up-and-down di­rec­tion, and the in­visi­ble third di­men­sion that leaps out of the pa­per, are de­voted to the com­plex am­pli­tudes. Since a com­plex am­pli­tude has a real and imag­i­nary part, they use up 2 of our 3 di­men­sions.

Richard Feyn­man said to just imag­ine the com­plex am­pli­tudes as lit­tle 2-di­men­sional ar­rows. This is as good a rep­re­sen­ta­tion as any; lit­tle 2D ar­rows be­have just the same way com­plex num­bers do. (You add lit­tle ar­rows by start­ing at the ori­gin, and mov­ing along each ar­row in se­quence. You mul­ti­ply lit­tle ar­rows by adding the an­gles and mul­ti­ply­ing the lengths. This is iso­mor­phic to the com­plex field.) So we can think of each po­si­tion of the A par­ti­cle as hav­ing a lit­tle ar­row as­so­ci­ated to it.

As you can see, the po­si­tion of A bul­ges in two places—a big bulge to the left, and a smaller bulge at right. Way up at the level of clas­si­cal ob­ser­va­tion, there would be a large prob­a­bil­ity (in­te­grat­ing over the squared mod­u­lus) of find­ing A some­where to the left, and a smaller prob­a­bil­ity of find­ing it at the small bulge to the right.

Draw­ing a neat lit­tle graph of the A+B sys­tem would in­volve hav­ing a com­plex am­pli­tude for each joint po­si­tion of the A and B par­ti­cles, which you could vi­su­al­ize as a hy­per­sur­face in 4 di­men­sions. I’d draw it for you, but I left my 4-di­men­sional pen­cil in the pocket of the 3rd leg of my other pants.

You may re­call from yes­ter­day that a plaid rec­t­an­gu­lar prob­a­bil­ity dis­tri­bu­tion fac­tor­izes into the product of two in­de­pen­dent prob­a­bil­ity dis­tri­bu­tions.

This kind of in­de­pen­dence-struc­ture is one of sev­eral keys to re­cov­er­ing the illu­sion of in­di­vi­d­ual par­ti­cles from quan­tum am­pli­tude dis­tri­bu­tions. If the am­pli­tude dis­tri­bu­tion roughly fac­tor­izes, has sub­sys­tems A and B with Am­pli­tude(X,Y) ~ Am­pli­tude(X) * Am­pli­tude(Y), then X and Y will seem to evolve roughly in­de­pen­dently of each other.

But main­tain­ing the illu­sion of in­di­vi­d­u­al­ity is harder in quan­tum con­figu­ra­tion spaces, be­cause of the iden­tity of par­ti­cles. This iden­tity cuts down the size of a 2-par­ti­cle con­figu­ra­tion space by 12, cuts down the size of a 3-par­ti­cle con­figu­ra­tion space by 16, and so on. Here, the diminished con­figu­ra­tion space is shown for the 2-par­ti­cle case:


The quan­tum con­figu­ra­tion space is over joint pos­si­bil­ities like “a par­ti­cle here, a par­ti­cle there”, not “this par­ti­cle here, that par­ti­cle there”. What would have been a neat lit­tle plaid pat­tern gets folded in on it­self.

You might think that you could re­cover the struc­ture by figur­ing out which par­ti­cle is “re­ally which”—i.e. if you see a “par­ti­cle far for­ward, par­ti­cle in mid­dle”, you can guess that the first par­ti­cle is A, and the sec­ond par­ti­cle is B, be­cause only A can be far for­ward; B just stays in the mid­dle. (This con­figu­ra­tion would lie in at the top of the origi­nal plaid pat­tern, the part that got folded over).

The prob­lem with this is the lit­tle tri­an­gu­lar re­gion, where the folded plaid in­ter­sects it­self. In this re­gion, the folded-over am­pli­tude dis­tri­bu­tion gets su­per­posed, added to­gether. Which makes an ex­per­i­men­tal differ­ence, be­cause the squared mod­u­lus of the sum is not the sum of the squared mod­uli.

In that lit­tle tri­an­gu­lar re­gion of quan­tum con­figu­ra­tion space, there is sim­ply no fact of the mat­ter as to “which par­ti­cle is which”. Ac­tu­ally, there never was any such fact; but there was an illu­sion of in­di­vi­d­u­al­ity, which in this case has bro­ken down.

But even that isn’t the ul­ti­mate rea­son why you can’t take quan­tum physics out of con­figu­ra­tion space.

In clas­si­cal con­figu­ra­tion spaces, you can take a sin­gle point in the con­figu­ra­tion space, and the sin­gle point de­scribes the en­tire state of a clas­si­cal sys­tem. So you can take a sin­gle point in clas­si­cal con­figu­ra­tion space, and ask how the cor­re­spond­ing sys­tem de­vel­ops over time. You can take a sin­gle point in clas­si­cal con­figu­ra­tion space, and ask, “Where does this one point go?”

The de­vel­op­ment over time of quan­tum sys­tems de­pends on things like the sec­ond deriva­tive of the am­pli­tude dis­tri­bu­tion. Our laws of physics de­scribe how am­pli­tude dis­tri­bu­tions de­velop into new am­pli­tude dis­tri­bu­tions. They do not de­scribe, even in prin­ci­ple, how one con­figu­ra­tion de­vel­ops into an­other con­figu­ra­tion.

(I pause to ob­serve that physics books make it way, way, way too hard to figure out this ex­tremely im­por­tant fact. You’d think they’d tell you up front, “Hey, the evolu­tion of a quan­tum sys­tem de­pends on stuff like the sec­ond deriva­tive of the am­pli­tude dis­tri­bu­tion, so you can’t pos­si­bly break it down into the evolu­tion of in­di­vi­d­ual con­figu­ra­tions.” When I first saw the Schröd­inger Equa­tion it con­fused the hell out of me, be­cause I thought the equa­tion was sup­posed to ap­ply to sin­gle con­figu­ra­tions.)

If I’ve un­der­stood the laws of physics cor­rectly, quan­tum me­chan­ics still has an ex­tremely im­por­tant prop­erty of lo­cal­ity: You can de­ter­mine the in­stan­ta­neous change in the am­pli­tude of a sin­gle con­figu­ra­tion us­ing only the in­finites­i­mal neigh­bor­hood. If you for­get that the space is con­tin­u­ous and think of it as a mesh of com­puter pro­ces­sors, each pro­ces­sor would only have to talk to its im­me­da­tien neigh­bors to figure out what to do next. You do have to talk to your neigh­bors—but only your next-door neigh­bors, no tele­phone calls across town. (Tech­ni­cal term: “Markov neigh­bor­hood.”)

Con­way’s Game of Life has the dis­crete ver­sion of this prop­erty; the fu­ture state of each cell de­pends only on its own state and the state of neigh­bor­ing cells.

The sec­ond deriva­tive—Lapla­cian, ac­tu­ally—is not a point prop­erty. But it is a lo­cal prop­erty, where know­ing the im­me­di­ate neigh­bor­hood tells you ev­ery­thing, re­gard­less of what the rest of the dis­tri­bu­tion looks like. Po­ten­tial en­ergy, which also plays a role in the evolu­tion of the am­pli­tude, can be com­puted at a sin­gle po­si­tional con­figu­ra­tion (if I’ve un­der­stood cor­rectly).

There are math­e­mat­i­cal trans­for­ma­tions physi­cists use for their con­ve­nience, like view­ing the sys­tem as an am­pli­tude dis­tri­bu­tion over mo­menta rather than po­si­tions, which throw away this neigh­bor­hood struc­ture (e.g. by mak­ing po­ten­tial en­ergy a non-lo­cally-com­putable prop­erty). Well, math­e­mat­i­cal con­ve­nience is a fine thing. But I strongly sus­pect that the phys­i­cally real wave­func­tion has lo­cal dy­nam­ics. This kind of lo­cal­ity seems like an ex­tremely im­por­tant prop­erty, a can­di­date for some­thing hard­wired into the na­ture of re­al­ity and the struc­ture of cau­sa­tion. Im­pos­ing lo­cal­ity is part of the jump from New­to­nian me­chan­ics to Spe­cial Rel­a­tivity.

The tem­po­ral be­hav­ior of each am­pli­tude in con­figu­ra­tion space de­pends only on the am­pli­tude at neigh­bor­ing points. But you can­not figure out what hap­pens to the am­pli­tude of a point in quan­tum con­figu­ra­tion space, by look­ing only at that one point. The fu­ture am­pli­tude de­pends on the pre­sent sec­ond deriva­tive of the am­pli­tude dis­tri­bu­tion.

So you can’t say, as you can in clas­si­cal physics, “If I had in­finite knowl­edge about the sys­tem, all the par­ti­cles would be in one definite po­si­tion, and then I could figure out the ex­act fu­ture state of the sys­tem.”

If you had a point mass of am­pli­tude, an in­finitely sharp spike in the quan­tum arena, the am­pli­tude dis­tri­bu­tion would not be twice differ­en­tiable and the fu­ture evolu­tion of the sys­tem would be un­defined. The known laws of physics would crum­ple up like tin­foil. In­di­vi­d­ual con­figu­ra­tions don’t have quan­tum dy­nam­ics; am­pli­tude dis­tri­bu­tions do.

A point mass of am­pli­tude, con­cen­trated into a sin­gle ex­act po­si­tion in con­figu­ra­tion space, does not cor­re­spond to a pre­cisely known state of the uni­verse. It is phys­i­cal non­sense.

It’s like ask­ing, in Con­way’s Game of Life: “What is the fu­ture state of this one cell, re­gard­less of the cells around it?” The im­me­di­ate fu­ture of the cell de­pends on its im­me­di­ate neigh­bors; its dis­tant fu­ture may de­pend on dis­tant neigh­bors.

Imag­ine try­ing to say, in a clas­si­cal uni­verse, “Well, we’ve got this prob­a­bil­ity dis­tri­bu­tion over this clas­si­cal con­figu­ra­tion space… but to find out where the sys­tem evolves, where the prob­a­bil­ity flows from each point, we’ve got to twice differ­en­ti­ate the prob­a­bil­ity dis­tri­bu­tion to figure out the dy­nam­ics.”

In clas­si­cal physics, the po­si­tion of a par­ti­cle is a sep­a­rate fact from its mo­men­tum. You can know ex­actly where a par­ti­cle is, but not know ex­actly how fast it is mov­ing.

In Con­way’s Game of Life, the ve­loc­ity of a glider is not a sep­a­rate, ad­di­tional fact about the board. Cells are only “al­ive” or “dead”, and the ap­par­ent mo­tion of a glider arises from a con­figu­ra­tion that re­peats it­self as the cell rules are ap­plied. If you know the life/​death state of all the cells in a glider, you know the glider’s ve­loc­ity; they are not sep­a­rate facts.

In quan­tum physics, there’s an am­pli­tude dis­tri­bu­tion over a con­figu­ra­tion space of par­ti­cle po­si­tions. Quan­tum dy­nam­ics spec­ify how that am­pli­tude dis­tri­bu­tion evolves over time. Maybe you start with a blob of am­pli­tude cen­tered over po­si­tion X, and then a time T later, the am­pli­tude dis­tri­bu­tion has evolved to have a similarly-shaped blob of am­pli­tude at po­si­tion X+D. Way up at the level of hu­man re­searchers, this looks like a par­ti­cle with ve­loc­ity D/​T. But at the quan­tum level this be­hav­ior arises purely out of the am­pli­tude dis­tri­bu­tion over po­si­tions, and the laws for how am­pli­tude dis­tri­bu­tions evolve over time.

In quan­tum physics, if you know the ex­act cur­rent am­pli­tude dis­tri­bu­tion over par­ti­cle po­si­tions, you know the ex­act fu­ture be­hav­ior of the am­pli­tude dis­tri­bu­tion. Ergo, you know how blobs of am­pli­tude ap­pear to prop­a­gate through the con­figu­ra­tion space. Ergo, you know how fast the “par­ti­cles” are “mov­ing”. Full knowl­edge of the am­pli­tude dis­tri­bu­tion over po­si­tions im­plies full knowl­edge of mo­menta.

Imag­ine try­ing to say, in a clas­si­cal uni­verse, “I twice differ­en­ti­ate the prob­a­bil­ity dis­tri­bu­tion over these par­ti­cles’ po­si­tions, to phys­i­cally de­ter­mine how fast they’re go­ing. So if I learned new in­for­ma­tion about where the par­ti­cles were, they might end up mov­ing at differ­ent speeds. If I got very pre­cise in­for­ma­tion about where the par­ti­cles were, this would phys­i­cally cause the par­ti­cles to start mov­ing very fast, be­cause the sec­ond deriva­tive of prob­a­bil­ity would be very large.” Doesn’t sound all that sen­si­ble, does it? Don’t try to in­ter­pret this non­sense—it’s not even analo­gously cor­rect. We’ll look at the hor­ribly mis­named “Heisen­berg Uncer­tainty Prin­ci­ple” later.

But that’s why you can’t take quan­tum physics out of con­figu­ra­tion space. In­di­vi­d­ual con­figu­ra­tions don’t have physics. Am­pli­tude dis­tri­bu­tions have physics.

(Though you can re­gard the en­tire state of a quan­tum sys­tem—the whole am­pli­tude dis­tri­bu­tion—as a sin­gle point in a space of in­finite di­men­sion­al­ity: “Hilbert space.” But this is just a con­ve­nience of vi­su­al­iza­tion. You imag­ine it in N di­men­sions, then let N go to in­finity.)

Part of The Quan­tum Physics Sequence

Next post: “Feyn­man Paths

Pre­vi­ous post: “Clas­si­cal Con­figu­ra­tion Spaces