Sidney Coleman's Dirac Lecture "Quantum Mechanics in Your Face"
aa r X i v : . [ phy s i c s . h i s t - ph ] A ug Sidney Coleman’s Dirac Lecture “Quantum Mechanics in Your Face”
Sidney ColemanTranscript and slides edited by Martin Greiter
Institute for Theoretical Physics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany (Dated: November 26, 2020)This is a write-up of Sidney Coleman’s classic lecture first given as a Dirac Lecture at CambridgeUniversity and later recorded when repeated at the New England sectional meeting of the AmericanPhysical Society (April 9, 1994). My sources have been this recording and a copy of the slidesSidney send to me after he gave the lecture as a Physics Colloquium at Stanford University sometime between 1995 and 1998. To preserve both the scientific content and most of the charm, Ihave kept the editing to a minimum, but did add a bibliography containing the references Sidneymentioned.—MG This lecture has a history. It’s essentially a rerun of alecture I gave as the Dirac Lecture at Cambridge Univer-sity a little under a year ago.There’s a story there. I had been asked to give thislecture several years ago, when it was two years in thefuture. And of course when someone asks you to do some-thing two years in the future, that’s never. You’ll alwayssay yes.And when the time came I got a communication fromPeter Goddard at St. John’s College, who was runningthe operation. He said: “What do you want to talkabout?” I said: “Who’s the audience?” And he said: “Oh,it’s pretty mixed—you’ll get physics graduate students,physics undergraduates, people from chemistry and phi-losophy and mathematics.” And I thought: “Hmmm,these are not the people whom to address on the subjectof non-abelian quantum hair on black holes”, which waswhat I was working on at the moment. Quantum Mechanics in Your FaceIt’s Quantum Mechanics, StupidAnd Now for Something CompletelyDifferent: Quantum Reality
Quantum Mechanics with the Gloves Off
So I said: “Look, I’ve always been interested in givinga lecture on quantum mechanics—what a strange thingit is, and exactly what strange thing it is—do you thinksuch a lecture would be suitable?” And he said: “Yes, giveus a clever title!” So I emailed back:
Quantum Mechanicsin Your Face , because I wanted to really confront peoplewith quantum mechanics. [Coleman places Slide 1 onthe projector, with all the titles except for the first onecovered up. He then uncovers the titles as he presentsthem.] And Peter said: “No good”. He said a Britishaudience would not understand the locution and indeedmight think it was obscene.“All to the good!” I said. But he was adamant. So, since one of the themes of the proposed lecture wasthat a lot of confusion arises because people keep tryingto think of quantum mechanics as classical mechanics, Isuggested this alternative title
It’s Quantum Mechanics,Stupid . And he said—I have all of this on disks: this isa true story—“Nope; a British audience wouldn’t get it;too American”. So I said: “Well, all right, if you wantsomething British:
And Now for Something CompletelyDifferent: Quantum Reality .” He said: “too facetious”.So finally we settled on the title
Quantum Mechanicswith the Gloves Off , which, as you can see, is a littlewimpier than the others.But now I’m back in the land of free speech, so thetitle of the talk is
Quantum Mechanics in Your Face .The talk will fall into three parts. Outline (1) A quick review of (vernacular) quantummechanics(2) Better than Bell: the GHZM effect(3) The return of Schrödinger’s catThere is no representation, expressed or implied, thatany part of this lecture is original aa or that any account is taken of classical or quantum gravity. There will be a preliminary where I give a quick re-view of quantum mechanics—I would say the Copen-hagen interpretation, or the interpretation in somebody’stextbook, but it’s not really that—it’s looser and moresloppy.Architects and architectural historians, when they’rediscussing kinds of buildings that were being built in acertain place in a certain time, but aren’t in any partic-ular well defined style, but just what builders threw upin the United States in ca. 1948—they call it “vernaculararchitecture”. This will be a quick review of vernacularquantum mechanics. That’s more to establish notation,and make sure we’re all on the same wavelength.Then the two main parts of the lecture will be,rstly, a review of a pedagogical improvement on JohnBell’s famous analysis of hidden variables in quantummechanics . It is easier to explain than Bell’s originalargument, and deserves to be widely publicized. It wasbuilt by David Mermin . He’s the “M” out of someearlier work by Greenberger, Horn, and Zeilinger .In the second [main] part of the lecture I will turn tothe much vexed question sometimes called “the interpre-tation of quantum mechanics”, although, as I will argue,that’s really a bad name for it. I want to stress that I havemade no original contributions to this subject. There isnothing I will say in this lecture, with the exception ofthe carefully prepared spontaneous jokes—that was oneof them—that cannot be found in the literature.Of course, such is the nature of the subject that there isnothing I will say where the contradiction cannot also befound in the literature. So I claim a measure of respon-sibility, if no credit—the reverse of the usual scholarlyprocedure.I will stick strictly to quantum mechanics in flat spaceand not worry about either classical or quantum grav-ity. We will have problems enough keeping these thingsstraight there without worrying about what happenswhen the geometry of space-time is itself a quantum vari-able. (1) Some Things Everyone Knows (Even if not everyone believes them)(i) The state of a physical system at a fixed time isa vector in a Hilbert space, | ψ i , normalized suchthat h ψ | ψ i = 1 .(ii) It evolves in time according toi ∂∂t | ψ i = H | ψ i where H is “the Hamiltonian”, some self-adjointlinear operator. Now to begin with, the very quick review. These trans-parencies are going to go by extremely fast. A pointermight be handy.... I’m always worried about these things:I’ll point them the wrong way and zap a member of theaudience.The state of a physical system at a fixed time is avector in Hilbert space. Following Dirac we call it ψ . Wenormalize it to unit norm. It evolves in time accordingto the Schrödinger equation, where the Hamiltonian issome self-adjoint linear operator—a simple one if we’retalking about a single atom, and a complicated one ifwe’re talking about a quantum field theory.Now if there is anyone who has any questions aboutthe material on the screen at this moment, please leavethe auditorium, because you won’t be able to understandanything else in the lecture. | ψ i is an eigenstate of theobservable A with eigenvalue a , A | ψ i = a | ψ i then we say “the value of A is certain to beobserved to be a ”.[Strictly speaking, just a definition, but there isan implicit promise (c.f. F = ma ).] Now some, maybe all, self-adjoint operators are “ob-servables”. If the state is an eigenstate of an observable A , with eigenvalue a , then we say the value of A is a ,is certain to be observed to be a . Now, strictly speak-ing, this is just a definition of what I mean by “observ-able” and “observed”, but of course that’s because thosewords have not occurred on any previous transparencyso I can call them what I want. Of course, that’s likesaying Newton’s second law F = ma , as it appears intextbooks on mechanics, is just a definition of what youmean by “force”. That’s true, strictly speaking, but welive in a landscape where there is an implicit promise thatwhen someone writes that down, when they begin talk-ing about particular dynamical systems that they willgive laws for the force, and not, say, for some quantityinvolving the 17th time derivative of the position.Likewise, the words “observable” and “observed” havea history before quantum mechanics. People like to sayall these things have a meaning in classical mechanics,but really it goes way earlier than classical mechanics.I’m sure the pre-Columbian inhabitants of Massachusettswere capable of saying, in their language, “I observe adeer”, despite their scanty knowledge of Newtonian me-chanics. Indeed, I even suspect that the deer was capa-ble of observing the Native Americans despite its evenweaker grasp on action and angle variables.So there’s an implicit promise in here that, whenyou put the whole theory together and start calculatingthings, that the words “observes” and “observable” willcorrespond to entities that act in the same way as thoseentities do in the language of everyday speech under thecircumstances in which the language of everyday speechis applicable. Now to show that is a long story. It’s notsomething I’m going to focus on here, involving thingslike the WKB approximation and von Neumann’s anal-ysis of an ideal measuring device , but I just wanted topoint out that that’s there.Now we come to the fourth thing: every measurementthat happens when the state ψ is not an eigenstate of theobservable yields one of the eigenvalues, with the proba-bility of finding a particular eigenvalue a proportional tothe magnitude of the part of the wave function that lieson the subspace of states with eigenvalue a . (I’m assum-ing here just for notational simplicity that the eigenvaluespectrum is discrete.) If a has been measured, then the2 (iv) Every measurement of A yields one of theeigenvalues of A . The probability of finding aparticular eigenvalue, a , is k P ( A ; a ) | ψ ik where P ( A ; a ) is the projection operator on thesubspace of states with eigenvalue a . (I assume,for notational simplicity, that A has a discretespectrum.) If a has been measured, then thestate of the system after the measurement is P ( A ; a ) | ψ ik P ( A ; a ) | ψ ik (Much) more about this later. state of the system after the measurement is just thatpart of the wave function—all the rest of it has been an-nihilated. And, of course, it has to be rescaled, or beinga quantum field theorist, I suppose I should say renor-malized, so it has unit norm again. This is the famousprojection postulate. It is sometimes called “the reduc-tion of the wave packet”.It’s very different from the previous three statementsI’ve put on the board because it contradicts one of them:causal time evolution according to Schrödinger’s equa-tion. Schrödinger’s equation, d | ψ i dt = − i H | ψ i , (1)is totally causal: given the initial wave function—giventhe initial state of the system—the final state is com-pletely determined. Furthermore, this causality is timereversal invariant: given the final state the initial state iscompletely determined.This operation is something other than Schrödinger’sequation. It is not deterministic. It is probabilistic. Itisn’t just that you cannot predict the future from thepast. Even when you know the future, you don’t knowwhat the past was. If I measure an electron and discoverit is an eigenstate of σ z with σ z = +1 , I have no way ofknowing what its initial state was. Maybe it was σ z =+1 , maybe it was σ x = +1 , and it turned out that I wasin the 50% probability branch that got the measurement σ z = +1 .That ends the preliminary.Before I go into the first of the two main parts of thelecture, the GHZM analysis, are there any questionsabout this?I will in the second part return to a critical analysis ofthe “reduction of the wave packet”, but for the first part ofthis lecture I’d like to take it as given. Now there’s refer-ences, but actually I call them credits, because I noticednobody ever writes down the references. It is just hereto avoid the speaker being sued. This whole analysis, as (2) Credits for the next part [i] A. Einstein, B. Podolsky, and N. Rosen,Phys. Rev. 47, (1935) 777.[ii] J. S. Bell, Rev. Mod. Phys. 38, 447 (1966);—, Physics 1 (1964) 195.[iii] N.D. Mermin, Physics Today 38, April 1985, p.38.[iv] D.M. Greenberger, M.A. Horne, A. Shimony, andA. Zeilinger, Am. J. Phys. 58 (1990) 1131.[v] N.D. Mermin, Am. J. Phys. 58 (1990) 731;—, Physics Today 43, June 1990, p. 9. everyone knows, starts with the work of Einstein, Rosen,and Podolsky , which sat around as an irritant for someyears, until John Bell , picking up an idea from DavidBohm , was able to turn it into a conclusive argumentagainst hidden variables.A pedagogical improvement was made by DavidMermin who, at least to my mind, really clarifiedwhat was going on in Bell’s analysis. And then a com-pletely different experiment was suggested by Green-berger, Horn, and Zeilinger. I’ve got a reference hereto a paper they wrote with Abner Shimoni , not becausethat was the original paper, but the original paper isa brief report in conference proceedings. That one pol-ishes it up. This is my version of Mermin’s version of Greenberger, Horn, and Zeilinger’s Gedanken exper-iment inspired by John Bell based on Bohm and Ein-stein, Rosen, and Podolsky . And I’ve left out ninetypercent of the references.The way I like to think of this analysis is by imagininga physicist, whom I call “Dr. Diehard”, who was aroundat the time of the discovery of quantum mechanics inthe late 20s, and didn’t believe it. Although some timehas passed since then, he’s still around—quite old butintellectually vigorous, and he still doesn’t believe in it.Our task is to convince him that quantum mechanicsis right and classical ideas are wrong, or as I say evenprimitive pre-classical ideas.There’s no point in trying to wow him with the anoma-lous magnetic moment of the electron or the behavior ofartificial atoms that we just heard about or anything likethat, because he is so deeply opposed to quantum me-chanics and so old and stubborn that as soon as youstart putting a particular quantum mechanical equationon the board his brain turns off, rather like my brain ina seminar on string theory. So the only way to convincehim is on very general grounds—not by doing particularcalculations.At first thought you say: “It’s easy—quantum mechan-ics is probabilistic, classical mechanics is deterministic. IfI have that electron in an eigenstate of σ x and choose tomeasure σ z , I can’t tell whether I’m going to get +1 or3 . There’s no way anyone can tell. That’s very differ-ent from classical mechanics, and it seems to describe thereal world”. A = A ( α ) where α = “subquantum” or “hidden” variables; may bevery many; may involve “apparatus” as well as “system”.Prob { A ≤ a } = Z θ ( a − A ( α )) dµ ( α ) where µ ( α ) = probability distribution for the hiddenvariables—“a result of our ignorance not some quantumnonsense!”Noncommuting Observables? “Just interfering measure-ments!” But Dr. Diehard is not convinced for a second by that.“Probability has nothing to do with this fancy quantummechanics. Jérôme Cardan was writing down the rulesof probability when he analyzed games of chance in thelate Renaissance. When I flip a coin or go to Las Vegasand have a spin on the roulette wheel, the results seemto be perfectly probabilistic. But I don’t see Planck’sconstant playing any significant role there”, he says. “Thereason the roulette wheel gives me a probabilistic resultis that there are all sorts of sensitive initial conditionswhich I can’t measure well enough—initial conditions towhich the final state of the ball is sensitive—there are allsorts of degrees of freedom of the system which I cannotcontrol, and because of my ignorance, not because of anyfundamental physics, I get a probabilistic result.”This is sometimes called the hidden variable position.“Really, you don’t know everything about the state of theelectron when you measure its momentum and its spinalong the x -axis. There are zillions of unknown hiddenvariables which you can’t control; maybe they are alsoin the system that is measuring the electron. (There’sno separation in this viewpoint between the observer ob-serving the system and the quantity being observed.) Ifyou knew those quantities exactly, then you know exactlywhat the electron was going to do in any future exper-iment. But since you only know them probabilistically,you only have a probabilistic distribution.”Here I’ve written it down in somewhat fancy-shmancymathematical notation. [Coleman points at Slide 7.] Infact, this is right—you can get probability from classicalmechanics. John von Neumann way back was aware ofthis. He said: “No, that’s not the real difference betweenclassical mechanics and quantum mechanics. The realdifference is that in quantum mechanics you have non-commuting observables: If you measure σ x repeatedly for an electron and take care to keep it isolated from theexternal world, you always get the same result. But if youthen measure σ z and get a probabilistic result, when youmeasure σ x again you will again get a probabilistic resultthe first time—the first measurement of σ z has interferedwith the measurement of σ x . That’s because you havenon-commuting observables, and those are characteristicof quantum mechanics.Dr. Diehard says: “Absolute nonsense! We’re big,clumsy guys. When we think we’re doing a nice cleanmeasurement of σ x we might be messing up all of thosehidden observables. When we measure your σ z we thenget a different result because we’ve messed things up.My friends the anthropologists talk about this a lot whenthey discuss how an anthropologist can affect an isolatedsociety he or she believes they’re observing. And, forsome reason I don’t understand, they call it the uncer-tainty principle”.And Dr. Diehard continues: “My friends the social psy-chologists tell me that if you do an opinion survey, unlessyou construct it very carefully, the answers you will getto the questions will depend upon the order in which theyare asked”. (This is true, by the way.) He doesn’t see anydifference between that and measurements of σ x and σ z .That’s Dr. Diehard’s position.As John Bell pointed out in the first written of thosetwo articles I cited —which is not the one with the fa-mous inequality—this is in fact an irrefutable position,despite all the stuff to the contrary that has been saidin the literature. On this level there is no way of refut-ing it. He gave a specific example of a classical theorythat on this level reproduced all the results of quantummechanics—the de Broglie pilot wave theory .However, if Dr. Diehard admits one more thing, we cantrap him. I will now explain what that one thing is. x (lt.yrs.) t (yrs.)A BB’But spacelike-separated measurements can not interferewith each other (unless we have propagation ofinfluence backward in time).We have now a contradiction with the predictions ofquantum mechanics for simple systems. t is measured in years and x inlight-years, therefore the paths of light rays are 45-degreelines.Now let’s consider two measurements on possibly twodifferent systems done in two regions A and B—forget B’for the moment, its role will emerge later. Thus theseblack dots represent actually substantial regions in spacetime, during which an experiment has been conducted.Now one thing Dr. Diehard will have to admit is thatalthough the results of an experiment in A may interferewith an experiment in B, the results of an experiment inB can hardly interfere with the results of an experimentin A unless information can travel backwards in time,which we will assume he does not accept. That’s becauseA is over and done with and its results recorded in thelog book before B occurs.On the other hand, if we imagine another Lorentz ob-server with another coordinate system, B will appear asB’ here. B and B’, as you can see by eyeball, are onthe same space-like hyperbola—there is a Lorentz trans-formation that leaves A at the origin of coordinates un-changed and turns B into B’. B and B’ are space-likeseparated from A. A light signal cannot get from A to B,and nothing traveling slower than the speed of light canget from A to B.Now that second Lorentz observer would give the sameargument I gave, except he would interchange the rolesof A and B’. He would say the results of an experimentat A cannot interfere with the act of doing an experimentat B’ because B’ is earlier than A. But B’ is B, just Bseen by a different observer.Therefore, if you believe in the principal of Lorentzinvariance, and if you believe you cannot send informa-tion backwards in time, you have to conclude that ex-periments done at space-like separated locations suffi-ciently far apart from each other cannot interfere witheach other. It can’t matter what order you ask the ques-tions if this question is being asked of an earthman andthis one of an inhabitant of the Andromeda Nebula, andthey’re both being asked today.Are there any questions about this? This is thegroundwork from which the rest will proceed.On everything else we accept the Diehard position.Now here is the experimental proposal—this is a drawingfrom an imaginary proposal to the Department of En-ergy for the Diehard experiment. Three of Dr. Diehard’sgraduate students are assigned to experimental stations,as you see from the scale they are several light-minutesfrom each other. The graduate students, with lack ofimagination, are called numbers 1, 2, and 3. They’re al-most as old as Dr. Diehard—it’s difficult to get a thesisunder him.They are informed that once a minute something willbe sent from a mysterious central station to each ofthe three Diehard teams—what something is, they don’t ? − + Fig. 2: The Acme “Little Wonder” Dual Cryptometer know. However, they’re armed with measuring deviceswhose structure they again do not know. They are calleddual cryptometers because they can measure each of twothings, but what those two things are nobody knows—atleast the Diehards don’t know. They can turn a switch toeither measure A or measure B . They make this decisiononce a minute shortly before the announced time of thesignal, and sure enough, a light bulb lights up that sayseither A is +1 or A is − if they are measuring A , or thesame thing for B . They have no idea what A or B is.It’s possible the central station is sending them elemen-tary particles. It’s possible the central station is sendingthem blood samples, which they have the choice of an-alyzing for either high blood cholesterol or high bloodglucose. It is possible the whole thing is a hoax, thereis no central station, and a small digital computer insidethe cryptometer is making the lights go on and off. Theydo not know.In this way, however, they obtain a sequence of mea-surements, which they record as this. [Coleman pointsto Slide 11.] The first line means observer 1 has decidedto measure A and obtained the result +1 ; observer 2 has5 A = 1 B = − B = − A = 1 A = − B = − B = 1 B = 1 A = 1 . . . They find whenever they measure A B B it is +1 .Likewise for B A B and B B A .They deduce that A A A = 1 decided to measure B and obtained the result − ; andobserver 3 has decided to measure B and obtained theresult − . They have obtained in this way zillions ofmeasurements on a long tape. They record them in thisway because they really believe that whatever this thingis doing, A = 1 , that is to say, the value of quantity A that would be measured at station 1 is +1 independent ofwhat is going on on stations 2 and 3, because these threemeasurements are space-like separated. That’s what theyhave to believe if they’re Diehards. They have to believethere’s really some predictable value of this thing whichthey would know if they knew all the hidden variables.In this particular case, they don’t know what B is butthey know what A is.Now as they go through their measurements, they findin that roughly 3/8 of the measurements—they’re mak-ing random decisions about which things they measure—whenever they measure one A and two B ’s the result ofthe product of the measurements is +1 . Now they’remaking their choices at random and since they believethat these things have well-defined meanings independentof their measurements, they have to believe, if they be-lieve in normal empirical principles, that all the time thevalue of one A and two B ’s—the value that would be ob-tained if they had done the measurement—the product is +1 . Sometimes all three of these numbers are +1 . Some-times one of them is +1 and two are − . But the productis always +1 . It’s as if I gave you a zillion boxes and youturned up 3/8 of them and discovered each of them hada penny in it, you would assume within 1 over the squareroot of N —negligible error—that if you opened up allthe other boxes, they would also have pennies in them.By the miracle of modern arithmetic—that is to say bymultiplying these three numbers together and using thefact that each B squared is 1—they deduce that if theylook on their tape for those experiments in which they’vechosen to measure the product of three A ’s, they wouldobtain the answer +1 .Now let’s look behind the scenes and see what’s ac-tually going on. Maybe a little suspense would help...[Coleman covers most of Slide 12]. Behind the Scenes | ψ i = 1 √ (cid:2) |↑↑↑i − |↓↓↓i (cid:3) A = σ (1) x B = σ (1) y etc. A B B | ψ i = σ (1) x σ (2) y σ (3) y | ψ i = | ψ i etc. for B A B and B B A .But . . . A A A | ψ i = σ (1) x σ (2) x σ (3) x | ψ i = − | ψ i What spooky action-at-a-distance?
It’s not blood samples we’re sending to them after all,it’s three spin one-half particles arranged in the followingpeculiar initial state: one over the square root of two allspins up minus all spins down: √ (cid:2) |↑↑↑i − |↓↓↓i (cid:3) (2) A is simply σ x for the particle that arrives at the appro-priate station, and B is σ y .Let’s first check that A B B acting on this state is +1 . By the third statement about quantum mechanics Iput on the board in my preliminary section [see Slide 4],this quantity is definitely always going to be measured tobe +1 . Well, we have σ x (1) σ y (2) σ y (3) by my transcrip-tion table. σ x turns up into down. σ y turns up into downwith a factor of i or maybe − i, I can never remember, butthat’s no problem here because you have two of them sothe square is always − . Acting on the first component ofthis state this operator produces the second componentincluding the minus sign while acting on the second com-ponent this operator produces the first. So this state isindeed an eigenstate of this operator with eigenvalue +1 .And, of course, since everything is permutation invariant,the same is true for the other two operators.But A A A is σ x (1) σ x (2) σ x (3) , and σ x ’s turns and upinto a down without a minus sign. Therefore this stateis also an eigenstate of A A A , but with eigenvalue − .The Diehards using only these proto-classical ideas—they aren’t even so well developed to be called classi-cal physics, they’re sort of the underpinnings of classicalreasoning—deduce that they will always get A A A =+1 , sometimes a +1 and two − ’s, but always +1 . Infact, if quantum mechanics is right, they will always get − .This is pedagogically superior to the original Bell ar-gument for two reasons: Firstly, it doesn’t involve corre-lation coefficients—it’s not that classical mechanics saysthis will happen 47% of the time and quantum mechan-ics says it happens 33% of the time. Secondly, it is easyto remember—whenever I lecture on the Bell inequality6 have to look it up again because I can never rememberthe derivation. This thing—the ingredients in it are sosimple that if someone awakens you in the middle of thenight four years from now, and puts a gun to your head,and says: “show me the GHZM argument”, you shouldbe able to do it.We have shown that there are quantum mechanicalexperiments where the conclusions cannot be explainedby classical mechanics—even the most general sense ofclassical mechanics—unless, of course, the classical me-chanical person is willing to assume transmission of in-formation faster than the speed of light, which, with therelativity principle, is tantamount to transmission of in-formation backwards in time.This is, of course, also John Bell’s conclusion. This is,I must say, much misrepresented in the popular litera-ture and even in some of the not so popular literature.That’s not coming out right. I mean: some technicalliterature, where people talk about quantum mechanicsnecessarily implying connections between space-like sep-arated regions of space and time. That’s getting it ab-solutely backwards. There are no connections betweenspace-like separated regions of space and time in this ex-periment. In fact, there’s no interaction Hamiltonian,let alone one that transmits information faster than thespeed of light, except maybe an interaction Hamiltonianbetween the individual cryptometers and the particles.But, otherwise, it’s either quantum mechanics or super-luminal transmission of information, not both.Why on earth do people—I’m trying to see insideother people’s heads, which is always a dangerous op-eration, but let me do it—why, why on earth do peopleget so confused, so wrong about such a simple point?Why do they write long books about quantum mechanicsand non-locality full of funny arrows pointing in differ-ent directions? Okay, that’s the technical philosophers.They really—well, I’ll avoid the laws of libel—so, any-way, why do they do this? It’s because, I think, secretlyin their heart of hearts they believe it’s really classicalmechanics—that we’re really putting something over onthem—deep, deep down it’s really classical mechanics. People get things backwards and they shouldn’t—ithas been said, and wisely said, that every successful phys-ical theory swallows its predecessors alive. By that wemean that in the appropriate domain—for example theway statistical mechanics swallowed thermodynamics— in the appropriate domain of experience, the fundamen-tal concepts of thermodynamics—entropy for example,or heat—were explained in terms of molecular motions,and then we showed that if you defined heat in terms ofmolecular motion it acted under appropriate conditionspretty much the way it acted in thermodynamics. It’snot the other way around. The thing you want to do isnot to interpret the new theory in terms of the old, butthe old theory in terms of the new.The other day I was looking at a British videotape ofFeynman explaining elementary concepts in science to aninterrogator, whom I think was the producer ChristopherSykes. He asked Feynman to explain the force betweenmagnets. Feynman hemmed and hawed for a while, andthen he got on the right track, and he said somethingthat’s dead on the nail. He said: “You’ve got it all back-wards, because you’re not asking me to explain the forcebetween your pants and the seat of your chair. Youwant me, when you say the force between magnets, toexplain the force between magnets in terms of the kindsof forces you think of as being fundamental—those be-tween bodies in contact”. Obviously, I’m not phrasing itas wonderfully as Feynman. But, well, as Picasso saidin other circumstances, it doesn’t have to be a master-piece for you to get the idea. We physicists all know it’sthe other way around: the fundamental force betweenatoms is the electromagnetic force which does fall off asone over R squared. Christopher Sykes was confused be-cause he was asking something impossible. He shouldhave asked to explain the pants-chair force in terms ofthe force between magnets. Instead he asked to derivethe fundamental quantity in terms of the derived one.Likewise, a similar error is being made here. Theproblem is not the interpretation of quantum mechan-ics. That’s getting things just backwards. The problemis the interpretation of classical mechanics.Now, I’m going to address this, and in particular thefamous, or infamous, projection postulate. (3) Credits for the next part [vi] J. von Neumann, Mathematische Grundlagen derQuantenmechanik (1932).[vii] H. Everett (1957), Rev. Mod. Phys. 29, 454(1957).[viii] J. Hartle, Am. J. Phys. 36 (1968) 704.[ix] E. Farhi, J. Goldstone, and S. Gutmann, Ann.Phys. (NY) 192, (1989) 368.
The fundamental analysis is von Neumann’s. I don’tread two words of German, but I wanted to put downthis early publication . I read it in English translation .The position I am going to advocate is associated withHugh Everett in a classic paper . Some of the thingsI’ll say about probability later come from a paper by7im Hartle , and one by Cambridge’s own Eddie Farhi,Jeffrey Goldstone, and Sam Gutmann .I’d like to begin by recapitulating von Neumann’s anal-ysis of the measurement chain. The Measurement Chain (after von Neuman)(1) Electron prepared in σ x eigenstate: | ψ i = 1 √ (cid:2) |↑i + |↓i (cid:3) I measure σ z : | ψ i = ( |↑i|↓i equal probabilitiesNon-deterministic “reduction of the wavefunction”(2) Electron as before, measuring device in groundstate: | ψ i = 1 √ (cid:2) |↑ , M i + |↓ , M i (cid:3) Electron interacts with the device: | ψ i → √ (cid:2) |↑ , M + i + |↓ , M − i (cid:3) (normal deterministic time evolution)I observe device: | ψ i = ( |↑ , M + i|↓ , M − i equal probabilities I prepare an electron in a σ x eigenstate and I mea-sure σ z —the famous non-deterministic “reduction of thewave packet” takes place, and with equal probabilities, Icannot tell which, the spin either goes up or down.But this is rather unrealistic even for a highly ideal-ized measurement. An electron is a little tiny thing, andI have bad eyes. I probably won’t be able to see directlywhat it’s spin is. There has to be an intervening measur-ing device. So we complicate the system.The initial state is the same as before, as far as theelectron goes, but the measuring device is in some neutralstate [M on Slide 14]. The electron interacts with themeasuring device. Von Neumann showed us how to setthings up with the interaction Hamiltonian so if the elec-tron is spinning up the measuring device goes—maybeit’s one of those dual cryptometers—the light bulb say-ing +1 flashes, if the electron is spinning down the lightbulb saying − flashes. This is normal deterministic timeevolution according to Schrödinger’s equation.Now I come by. I can’t see the electron, but I observethe device. By the usual projection postulate, I eithersee it in state +1 or state − . I make the observation, ifI see the state +1 , and the rest of the wave function is annihilated. I get with either probability these two things[Coleman points at the state vectors |↑ , M + i and |↓ , M − i on the bottom of Slide 14]. The result is the same asbefore because the electron is entangled with the device.I measure the device. The electron comes along for theride.Now let’s complicate things a bit more. Let’s sup-pose I cannot do the measurement because I’m givingthis lecture. However, I have a colleague, a very cleverexperimentalist—for purposes of definiteness, let’s say itsPaul Horowitz—who has constructed an ingenious robot.I’ll call him Gort. It’s a good name for a robot. I say“Gort, I want you during the lecture to go and see whatthe measuring device says about the electron”. And soGort goes and does this. Although he’s an extremely in-genious and complicated robot, he’s still just a big quan-tum mechanical system, like anything else. So it’s thesame story. Things starts out with electron in a super-position of up and down, measuring device neutral, acertain register and a RAM chip inside Gort’s belly alsohas nothing written on it. Then everything interacts andthe state of this world is: electron “up”, measuring de-vices says “up”, Gort’s RAM chip’s register says “up”,plus the same thing with “up” replaced by “down”, all di-vided by the square root of two. And Gort comes rollingin the door there with his rollers, and I say: “Hey, Gort,which way is the electron spinning?” And he tells me.And wham-o, it either goes into one or the other of thesestates fifty percent probability.But Gort is very polite. He observes that I am lectur-ing. So rather than coming to me directly, he rolls up tomy colleague Professor Nelson sitting there in the cornerand hands him a clip of printout that says either up ordown, and says: “Pass this on to Sidney when the lectureis over”. And he rolls away.Well, of course, vitalism was an intellectually live po-sition early in the 19th century. Dr. Lydgate in Middle-march, which will be appearing on TV tomorrow, heldthat living creatures are not simply complicated mechan-ical systems. But it hasn’t had many advocates this cen-tury. I think most of us would admit that David [Nelson]is just another quantum mechanical system, althoughperhaps more complicated than the electron and Gort,and certainly more likeable. Anyway, there he is.So it’s the same story as before: the state of the worldafter all this has happened is: electron “up”, measuringdevice says “up”, Gort’s RAM chip says “up”, David’sslip of paper says “up”, plus the same thing with down,divided by the square root of two. After the lecture I goup to them and say: “What’s up, David?” Wham-o! Hetells me. And the whole wave function collapses.Now this is getting a little silly, especially if you con-sider the possibility that—after all, I’m getting on inyears, I’m not in perfect health, here I am running arounda lot—maybe I have a heart attack before the lecture isover and die. What happens then? Who reduces thewave packet?Yakir Aharonov, who has of course since acquired great8ame for himself, was a young postdoc at Brandeis whenI was a young postdoc at Harvard. I had been readingvon Neumann and thinking about this, and come to aconclusion which I did not like, which was solipsism: Iwas the only creature in the world which could reducewave packets. Otherwise it didn’t make any sense. I wasnot totally happy with this position, even though I wasas egotistical as any young man—indeed probably moreegotistical than most—I was still unhappy with the po-sition. I was discussing this with Aharonov. Even in hisyouth he would smoke these enormous cigars, which hewould use to punctuate the conversation; he would takehuge drafts on them; he was and is sort of the quantumGeorge Burns.Anyway, I explained this position to him, and he said:“I see. [Coleman imitates Aharonov inhaling and blowingout smoke of his cigar] Tell me: before you were born,could your father reduce wave packets?” NO special measurement process NO reduction of the wave function NO indeterminancy NOTHING probabilisticin quantum mechanics.
ONLY deterministic evolutionaccording to Schrödinger’s Equation“Ridiculous”—E. Schrödinger (1935)“Absurd”—E.P. Wigner (1961)“Why do I, the observer, perceive only one of theoutcomes?”—W.H. Zurek (1991)
Now I will argue that in fact there is no special mea-surement process, there is no reduction of the wave func-tion in quantum mechanics, there is no indeterminacy,and nothing probabilistic—only deterministic evolutionaccording to Schrödinger’s equation.This is not a novel position. In the famous paper on thecat, Schrödinger raised this position, the position thatthe cat is in fact in the coherent superposition of beingdead and being alive, and instantly said it’s ridiculous:“We reject the ridiculous possibility ...”Some years later in the paper on Wigner’s friend, whereWigner attempted to resolve the ancient mind-bodyproblem through the quantum theory of measurement,he also raised this position, and said it was “absurd”.There is a recent paper by Zurek in Physics To-day —Zurek has made major contributions to the theoryof decoherence—where instead of just saying it’s ridicu- lous or absurd, he actually raised a question one can talkabout. He said: “If this is so, why do I the observer per-ceive only one of the outcomes?” This is now the questionI will attempt to address: Zurek’s question. If there is noreduction of the wave packet, why do I feel at the end ofthe day that I have observed a definite outcome, that theelectron is spinning up or the electron is spinning down? | C i be the state of the cloud chamber.Define a “linearity operator” L , such that L | C i = | C i if track is straight (t.w.s.s.a.) ,L | C i = 0 on states orthogonal to these. | ψ i i = | φ k , C i → | ψ f , k i where φ k = state where the particle is concentratednear the center in position and near k in momentum. L | ψ f , k i = | ψ f , k i Now consider: | ψ i i = Z d Ω k | φ k , C i → | ψ f i = Z d Ω k | ψ f , k i L | ψ f i = | ψ f i In order to ease into this, I’d like to begin with ananalysis of Neville Mott . Neville Mott worried wayback in 1929 about cloud chambers. He said: “Look, anatom releases an ionizing particle at the center of a cloudchamber in an s-wave. And it makes a straight line track.Why should it make a straight line track? If I think aboutan s-wave, it is spherically symmetric. Why do they notget some spherically symmetric random distribution ofsprinkles? Why should the track be a straight line?”Now we’re going to answer that question, and in afaster and slicker way then Neville Mott did. Of course,we have the advantage of 65 years of hindsight.We must assume that the scattering between the par-ticle and an atom when it ionizes it is unchanged orchanged only within some small angle to begin with;otherwise, of course, even classically the particle wouldbounce around like a pinball on a pinball table.Let | C i be the state of the cloud chamber. We define alinearity operator L —a projection operator so that L on | C i equals | C i if there is a track and it forms a straightline to within some small angle, and L on | C i equals zeroif the track is not a straight line, or there is no track forthat matter. Now let’s imagine we start out the problem9n some initial state where the particle is concentratednear the center of the chamber and near some momen-tum k , and the cloud chamber in a neutral condition, allunionized ready to make tracks. This evolves into somefinal state.Now we all believe that if you started out with the par-ticle in a narrow beam it would of course make a straightline track along that beam. The final state would bean eigenstate of this linearity operator and would haveeigenvalue +1 .Now here comes the tricky part: not tricky to followbut tricky-clever. I consider an initial state that’s an in-tegral over the angles of k of this state. This is a statewhere the particle is initially in an s-wave, and the cloudchamber is still in a neutral state—that’s independent of k . That state evolves by the linearity—the causal lin-earity of Schrödinger’s equation—into the correspondingsuperposition of these final states here [Coleman pointsto the state | ψ f i on the bottom of Slide 15a]. But if Ihave a linear superposition of eigenstates of the particlewith respect to the operator L , each of which is an eigen-state with eigenvalue +1 , then the combination is also aneigenstate with eigenvalue +1 . So this also has straightline tracks in it.That’s the short version of Mott’s argument. Mottsaid the problem is that people think of the Schrödingerequation as a wave in a three-dimensional space ratherthan a wave in a multi-dimensional space. I would phrasethat, making a gloss on this—he’s dead, so I can’t checkwhether it’s an accurate phrasing—I would make a glosson this and say: the problem is that people think of theparticle as a quantum mechanical system but of the cloudchamber as a classical mechanical system. If you’re will-ing to realize that both the particle and the cloud cham-ber are two interacting parts of one quantum mechanicalsystem, then there’s no problem. It’s an s-wave not be-cause the tracks are not straight lines but because there isa rotationally invariant correlation between the momen-tum of the particle and where the straight line points.But it’s always an eigenstate of this linearity operator.Any questions about this?Nobody doubts it—the tracks in cloud chambers, orbubble chambers if you’re young enough, are straightlines, even if the initial state is an s-wave.Now I will give an argument due to David Albert with respect to Zurek’s question. Zurek asked: “Why doI always have the perception that I have observed a def-inite outcome?” To answer this question, no cheating:we can’t assume Zurek is some vitalistic spirit loadedwith élan vital unobeying the laws of quantum mechan-ics. We have to say the observer—well I don’t want tomake it Zurek, that would be using him without his per-mission, I’ll make it me, Sidney—has some Hilbert spaceof states, and some condition in Sidney’s consciousnesscorresponds to the perception that he has observed a def-inite outcome, so there is some projection operator on it,the definiteness operator. If you want, we could give itan operational definition: the state where the definite- | S i ∈ H S Hilbert space of statesof observer (Sidney)Introduce D , “the definiteness operator”: D | S i = | S i if observer feels he has perceivedonly one of the outcomes D | S i = 0 on states orthogonal to these. (1) | ψ i i = |↑ , M , S i → | ψ f i = |↑ , M + , S + i D | ψ f i = | ψ f i (2) | ψ i i = |↓ , M , S i → | ψ f i = |↓ , M − , S − i D | ψ f i = | ψ f i (3) | ψ i i = 1 √ (cid:2) |↑ , M , S i + |↓ , M , S i (cid:3) → | ψ f i = 1 √ (cid:2) |↑ , M + , S + i + |↓ , M − , S − i (cid:3) D | ψ f i = | ψ f i ness operator is +1 is one where a hypothetical politeinterrogator asks Sidney: “Have you observed a definiteoutcome?”, and he says: “Yes”. In the orthogonal stateshe would say: “No, gee, I was looking someplace else whenthat sign flashed” or “I forgot” or “Don’t bother me, man,I’m stoned out of my mind” or, you know, any of thosethings.Now let’s begin. Our same old system as before: elec-tron, measuring apparatus, and Sidney. If the electron isspinning up, the measuring apparatus measures spin inthe up direction, and we get a definite state—no prob-lem of superposition—and Sidney thinks: “I’ve observeda definite outcome”. Same if everything is down. What ifwe start out with a superposition? Same story as NevilleMott’s cloud chamber. The same reason the cloud cham-ber always shows the track to be a straight line is thereason Sidney always has the feeling he has observed adefinite outcome.[Coleman answers an inaudible question:] That’s notwhat Zurek said. Zurek didn’t say: “It’s a matter of com-mon experience that in this experiment we always ob-serve the electron spinning up”, and Neville Mott didn’tsay: “It’s a matter of common experience that in thecloud chamber the straight line is always pointing alongthe z-axis”. The matter of common experience is thatSidney always has the perception that he has observeda definite outcome if you set up the initial conditionscorrectly. The matter of common experience is that thecloud chamber is always in a straight line. If you don’tlike this argument [the argument why Sidney perceivesdefinite outcomes], you can’t like that one [the argument10hy the cloud chamber detects straight lines]. If you likethat one, you have to like this one.The problem there—the confusion Nevil Mottremoved—was refusing to think of the cloud chamber as aquantum mechanical system. The problem here is refus-ing to think of Sidney as a quantum mechanical system.Because of the pressure of time I will remove thesetransparencies now and go on to discuss the question ofprobability. What about probability?
Classical probability theory reviewed:Suppose we have an infinite sequence of coin flips, or,equivalently, a sequence σ r ( r = 1 , , . . . ) of plus andminus ones. We have a sequence of independentrandom flips of a fair coin if ¯ σ = lim N →∞ ¯ σ N = lim N →∞ N N X r =1 σ r = 0 and ¯ σ a = lim N →∞ ¯ σ N,a = lim N →∞ N N X r =1 σ r σ r + a = 0 for all a . Also lim N →∞ N N X r =1 σ r σ r + a σ r + b = 0 for all a, b . Etc.(some mathematical niceties ignored.) Probability is a difficult question to discuss because itrequires us to look at something counterfactual. If I askwhether a given sequence is or is not random, I can’tdo that even in classical probability theory for a finitesequence. For example, if I consider a binary sequencewhere the entries are either +1 or − , and ask whetherthe sequence +1 is a random sequence, obviously thereis no way of answering that question. But if I have aninfinite sequence I can ask whether it’s random. So letme talk about that.Let me suppose I have an infinite sequence of +1 and − ’s, which might represent heads and tails. I want tosee if these sequences can be interpreted as a fair coinflip. Firstly, I want the average value of this quantity σ r , which is of course the limit of the average of thefirst N terms as N goes to infinity, to converge to zero.Also, if I were an experimenter, I would probably lookat correlations. I would take the r -th value σ r times the ( r + a ) − th value σ r + a for some value of a , and look atthe limit of this correlation, and ask that this quantitybe also 0 for any value of a. That way I could rejectsequences like +1 , +1 , − , − , +1 , +1 , − , − . . . , which noone would call random. I could also look for triple andhigher correlations. And if all those things were zero thenI say there is a pretty good chance of a random sequence. I would actually have to provide even further tests if Iwanted the real definition of randomness, the Martin-Löfdefinition of randomness , but this will be good enoughfor a lecture where I only have five minutes left. |→i ≡ √ (cid:0) |↑i + |↓i (cid:1) Consider | ψ i = |→i ⊗ |→i ⊗ |→i · · · This is an infinite sequence of electrons, each with σ x = 1 . Let these interact with a σ z -measuring deviceand an observer, as before. Does the observer perceivea sequence of independent random flips?Equivalently, is | ψ i an eigenstate of ¯ σ z = lim N →∞ ¯ σ N z = lim N →∞ N N X r =1 σ ( r ) z = 0 with eigenvalue zero? (And likewise for σ a z etc.) Now we want to ask the parallel question in quantummechanics. We start out with an electron in the state I’llcall sidewise—just our good old σ x -eigenstate, the samestate I’ve used before. I consider an infinite sequence ofelectrons heading towards my σ z -measuring apparatus,and I do the usual routine with the measuring system inSidney’s head and turn it into a sequence of memoriesin Sidney’s head or maybe Sidney has a notebook andhe writes down +1 , − , +1 , +1 , − . I obtain a sequence ofrecords correlated with the z-component of spin. I ask:“Does this observer observe this as a random sequence?That is to say, is this state here an eigenstate of the cor-responding quantum observables with eigenvalue zero?”Well, we know it’s all correlated with σ z . In order tokeep the transparency from overflowing its boundaries,I just looked at σ z , rather than the operator, for therecords. I define the average value of σ z exactly the sameway as it is done up here. [Coleman points to the lowerequation of Slide 18.] Then I ask: Is this an eigenstateof this operator with eigenvalue zero? If it is, we cansay—despite the fact that there is nothing probabilisticin here—that the average value of σ z is guaranteed to beobserved to be zero.Well, the calculation is sort of trivial. Let’s computethe norm of the state obtained by applying this operatorto this state. [Coleman points to the lower equation ofSlide 19.] It’s two sums, and here I’ve written them out.Each of them is an individual thing, there’s a 1 over N -squared, there’s a sum on r , and a sum on s . Now inthis particular state of course if r is not equal to s this“expectation value” is equal to zero, because you get justthe product of the independent expectation values whichare individually zero. On the other hand, if r is equalto s , then this is σ z squared, which we all know is +1 .Therefore, the limit of this thing up here is the limit of11 (cid:13)(cid:13) ¯ σ N z | ψ i (cid:13)(cid:13) = 1 N h ψ | N X r =1 N X s =1 σ ( r ) z σ ( s ) z | ψ ih ψ | σ ( r ) z σ ( s ) z | ψ i = δ rs ⇒ lim N →∞ (cid:13)(cid:13) ¯ σ N z | ψ i (cid:13)(cid:13) = lim N →∞ N N = 0 Likewise for σ a z etc.A definite deterministic state, definitely a randomsequence. (An impossibility in classical physics—butthis is not classical physics.)Stoppard’s Wittgenstein. N squared—the double sum collapses to a singlesum, only the terms with r equals s contribute, and eachentry with r equals s contributes 1, so you get N . Thusthe result is N over N squared, which is of course 0.And the same thing happens for all those correlators,because each one is a sum of terms with a 1 over N squared in front and only the entries that match per-fectly will give you a nonzero contribution. So this defi-nitely quantum mechanical state completely determinedby the initial conditions nevertheless matches this ex-perimenter’s definition of randomness—something thatwould be impossible in classical mechanics, but it’s quan-tum mechanics, stupid.Now one final remark: In Tom Stoppard’s play Jumpers , there’s an anecdote about the philosopher Lud-wig Wittgenstein. I have no idea whether it’s a real story or a Cambridge folk story . Anyway, it goes like this. Afriend is walking down the street in Cambridge and seesWittgenstein standing on a street corner lost in thought,and said: “What’s bothering you, Ludwig?” Wittgensteinsays: “I was just wondering why people said it was natu-ral to believe the sun went around the earth rather thanthe other way around”. The friend says: “Well, that’sbecause it looks like the Sun goes around the earth”.Wittgenstein thinks for a moment and says: “Tell me:What would it have looked like if it had looked like if itwas the other way around?”Now people say the reduction of the wave packet occursbecause it looks like the reduction of the wave packet oc-curs, and that is indeed true. What I’m asking you in thesecond main part of this lecture is to consider seriouslywhat it would look like if it were the other way around—if all that ever happened was causal evolution accordingto quantum mechanics. What I have tried to convinceyou is that what it looks like is ordinary everyday life.Welcome home. Thank you for your patience. Acknowledgments
MG wishes to thank Diana Coleman granting per-mission to publish this writeup, and Tobias Helbigfor his critical reading of the manuscript. The edito-rial effort was supported by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation)—Project-ID 258499086—SFB 1170 and through theWürzburg-Dresden Cluster of Excellence on Complexityand Topology in Quantum Matter— ct.qmat
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