ABC$in$Quantum$ Mechanics$ Jürg%Fröhlich% ETH%Zurich%

Size: px

Start display at page:

Download "ABC$in$Quantum$ Mechanics$ Jürg%Fröhlich% ETH%Zurich%"

Colin Brown
5 years ago
Views:

1 ABC$in$Quantum$ Mechanics$ JürgFröhlich ETHZurich

2 -(1!"#$&'( (56*+:;2##15))<=!$5*6:7)5><)26*56*2630:*26* (56*+:

3 Credits(and(Contents: Stimulation* and* encouragement* from* Rutgers( gang(((((((((((((((((( &*colleagues*at*eth*(graf*and*hepp);*some*motivation*from* recent* experiments,* in* particular* the* ones* of* the* Haroche7 Raimond( group;* papers* by* Bauer( &( Bernard,* Maassen( &( Kümmerer,* and* others;* joint* work* with* my* former* PhD* student* Baptiste( Schubnel;( (some* joint* efforts* with* Ballesteros,(Faupin,(Fraas,(Pickl,(Schilling,(Schubnel) ((((( 1. Introduction 2. QM(systems(and(projective(measurements 3. A(simple(model(of(a(quantum(system 4. Indirect(measurements(&( pointer(observables 5. Analogy(to(classical(statistical(mechanics( 6. Effective(dynamics(of(pointer(observables 7. Conclusions

4 1.Introduction QuestionstobeAddressed In* our* courses,* we* tend* to* describe* quantumh mechanical*systems*as*pairs*of*a*hilbert*space,******,*and* a* propagator,* U(t,s),( describing* timehevolution.* Unfortunately,* these* data* encode* almost* no( invariant( structure* (beyond* spectral* properties* of* U(t,s))* and* give* the* erroneous* impression* that* quantum* theory* might* be* deterministic.* Thus,* among* fundamental( problems(of(quantum(theory(are: What*do*we*have*to*add*to*the*usual*formalism*of* quantum*mechanics*in*order*to*arrive*at*a*matheh matical*structure*that*(through* interpretation )** can*be*given*physical*meaning,*independently(of( observers?

5 Questions,ctd. Where does intrinsic0 randomness0 in quantum mechanics come from, given the deterministic character of the Schrödinger equation? In which waydoesitdifferfromclassicalrandomness? Do we understand probabilistic phenomena in quantummechanics,suchas quantum0jumps0and0 tracks or Quantum0Brownian0Motion,etc.? What do we mean by a closed0 system 0 in quantummechanics,andwhyisthisanimportant notion? How does one prepare a system in a prescribedstate? 0Etc.0

QM0is0QM@as@QM0and0everything0else0is00 everything0else*0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*! The!one!thing!to!say!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!about!

6 everything0else*0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*! The!one!thing!to!say!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!about!art!is!that!it!is!one!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!thing.!Art!is!art4as4art!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!and!everything!else!is!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!everything!else.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Ad#Reinhardt!

7 2.QuantumSystems&projective measurements ((A(simple7minded(deZinition(of(quantum7mechanical(systems: A*closed(quantum*system,*S,*is*characterized*by*following* choices: *******(i)*************************************************************(u(=*unitary*propagator) ******(ii)**a*list,************************************,**of*bounded,*selfadjoint* *************operators*on*******representing*physical(quantities/ (((((((((((((potential(properties(of**s((that*can*be*measured*in* ************* projective(measurements ;*(S**must*be*a*closed **************system**chosen(large(enough(for*the*quantities **************represented*by**********************,*to*be*measureable).

8 QM0systems,0ctd.0 Choose*Tiducial*time,*t 0(,*and*deTine* to*be*the*operator*representing*the*pot.*prop.*corresp.* to*****************at*time*t(;((!(list*of*ops.(((((( Pot.*properties,**********,*measureable/observable*at*times** s( *t((generate*a**alg.(((((((((.(we*set*(see*also*appendix): ********************************************************************************************************* *************************************************************************************************************(1)*********************** ***************************************************************************************************************** *******************************************************"************* (((((((((((((((((((((((((((((((((((((((((((((((((((((((((!Information(Loss!****************************************** ******************************************************************

9 Projective0measurements0 LetbeapotentialpropertyofSthatis measured/observedaroundtimet;(i.e.,becomes objective /0 empirical aroundtimet).letbe theeigenvaluesof,andconsiderthespect.decomposition,(2) with{} j=1,,k00 thespectralprojections.letbethestate ofs0rightbeforeismeasured/observed.then(asone wouldexpect!) (3)

10 Projective0measurements,0ctd.0 i.e.,isanincoherent0superposition0(mixture)0of0 eigenstates0of00.thisiswhatisusuallycalleda projective0(orvon0neumann)0measurement Information0Loss canbea0mixed0state0(evenif,as astateon,maybeapure0state). Itisimportanttonotethatthechoiceoftimeevolution,andofastate,,determines00which potentialpropertyofs0willzirstbeobserved(become empirical)afterpreparationof! Detailsofcharacterizationof projectivemeasurements are somewhatsubtle;(seeappendix).

11 Projective0measurements,0ctd. Axiom0A0 Ifismeasuredaroundtimet00thenhasavalue0 aroundtimet.00 Thevalueofisobservedwithprobability ( Born0Rule ) (4) Ifisobservedaroundtimet0thenthestate # #####################################################################################################################00(5) shouldbeusedforimprovedpredictionsoffutureaftertimet.0 Origin0of0indeterminism0(randomness)0in0QM!

12 XI*P*2#$;&,*$(+,&*(0*'*G')"$*2?2",$*Y* "=,(.?*(0*#)+#.,-"*6')+*ZD7*$,'2.,$,)"2[

13 Main0features0of0model0 Atalltimes,onlyone00e`in T`channel Theonlypossibleprojectivemeasnt.inS00istoobserve whetherd L0 ord R0 hasclicked.thismeasurementis representedbyanoperator,x,0givenby (6) Thus,,inthisexample. Question:WhatcanonelearnaboutS,inparticularaboutP, byperforminglongsequencesofsuccessivemeasnts.ofx?0

14 Main0features0of0model,0ctd.00 UptoN00e sboundbyp00create Coulombblockade inarm of T 00reachingtoD R0,thus 00 moving00inside T tobescatteredontod R0.Simpleqmcalculationsyieldthe probabilities,p L (n)00andp R tobescatteredonto D L,D R0,respectively.Herenisthenumberofelectronsbound byp0.clearly p L (n)0+0p R (n)0= (7) Notethatn00isaneigenvalueoftheelectronnumber operatorwhoseeigenvaluescorrespondtothenumber ofelectronsboundbyp0.0 isinjected into T 0andgetsscatteredontooneofthetwodetectors, thusresultinginameasurement0of00x.letdenote theeigenvaluesofx(corresp.toclickofd L 0,0D R 0,0resp.)

15 Main0features0of0model,0ctd.0 Ameasurement0protocol0oflengthk00consistsofoutcomes (8) ofzirstk00measurementsofx.choosingastateofs00 enablesonetoassociateaprobability00(or frequency ) (9) witheachmeasurementprotocol,,oflengthk0=1,2,3,...0 (Theuniqueqmformulafortheprobabilities(9)hasZirst beenfoundbyschwingerandrediscoveredbywigner, ) Onehasthat Letdenotethespaceofalllongmeasurement protocols.thendezinesaprobabilitymeasureon.

16 4.Indirectmeasurements&pointer observables WedeZinethefrequency0of0clicks0ofD L0 inzirstkmeasnts. by (10) andthelimitof,as.weexpect: (11) ofthenumberoperatorintroducedafter(7). Important0observation:0 ThereisacloseanalogybetweenS,with(7)through(11), andtheclassical0stat.0mech.0ofspin0chains0(sc): timeofs d0space0ofsc;gibbsstateofsc0

17 Indirect0measurements,0ctd.0 Limit,as0(discrete)time TD0limit0ofSC Appealingto equivalence0of0ensembles 0inTDlimit,we expectthatthezluctuationsofaroundoneofthe possiblelimitingvaluesp L (n)00tendto0,as Precise0statement0 WedeZine (12) where

18 Indirect0measurements,0ctd. Theorem0 Hyp.(ND0measnt.):Operator(t)=const.intimet; &supposethat. Then: (A) Ifk0issolargethatthenthesets arealldisjointfromoneanother;and (B) Thesequencescan usually bechosen tobeindependentofthestate. General0insight0 Thinkofamoregeneralquantumsystem,S,with={X}.

19 5.Analogytoclassicalstatisticalmechanics Imaginethatmanysuccessiveprojectivemeasurementsof X0attimesaremade,withoutcomes whereisthevalueofan anobs.0x0measuredattimet j,0j0=1,,k,k0=1,2,3,. GivenanystateofS,onemaypredictthefrequency,,ofthemeasnt.protocol. Letbethesmallestsetofarbitrarilylong measnt.prot.s.t.forallstates,,ofs. WedeZinetobethealgebraoffunctions0on00 measureable0at,whicharealsocalled pointer0obs.. (IfS0isautonomousandthereiscompletedecoherence

20 Analogy0to0class.0Stat.0mech.,0ctd. Anexampleofapointerobservableis;(seeEq.(10)). Theprojections(characteristicfunctions)inthealgebra describeeventsinthesystems0detectedwiththe helpofverymanysuccessivemeasurementsofx.00 All0states00000/0probability0measures can0be0decomposed00 0into0a0convex0combination0of0mutually0singular0states/0 0prob.0measures00indexed0by0points0in0the0spectrum00 0of ().0 Inoursimplemodel,consistsofallfunctionson {1,,N},whichisthespectrumofthenumberoperator. ManysuccessivemeasurementsofX0(detectorclicks) boundbydotp.

21 6.Effectivedynamicsofpointerobservables Mostindirectmeasurementsarenot00non`demolition measurements.usuallyisempty.however,thereare functions,,ondependingonmeasurementprotocols oflengthk,whichapproximatepointerobservables example:`whoselevelsetsprovideadecom` positionofintodisjointsubsets, (13) withthepropertythatthe`measureofthecomplement oftheirunion,,istiny.

22 Effective0dynamics,0ctd.00 AmeasurementprotocoloflengthMk,0M0<nottoolarge, thendeterminesatrajectory(or history ) (14) ofevents,,m0=1,2,,m.0 Importantproblem Wewouldliketodeterminetheeffectivestochasticdynamics thatdeterminesthefrequencies0of0event0histories.hereisa verysimplemodel,wherethiscanbedone!

23 Effective0dynamics,0ctd.0 Inthismodel,thestatetendstopurifyafterN projective measurementsofx0withinatimeintervaloflengthδand thenevolvesunitarilyduringatimeintervaloflengthδ`δ. Inthelimitingregime,whereZirstN0tendstoandthenδ to0,amarkov0chain0withtransitionfunction (15) describestheeffective0dynamics0of0events0consistingofa jumpprocessonthespectrumofthenumberoperator. Thereisamoreinterestinglimitingregimethatcanbe analyzedwiththehelpofkurtz stochastictrotterproduct formula.butwedon thavetimetodescribeithere. 0

f@e26+)0:*26: In all my films, I have been faithful to these suspension points in

(Federico Fellini),-.*/0'#1)'+2#+0#3'"*2+)'"#)*+#9:.

#+(#'&7(+<#+(#=01*2<##.*/+(&'7#)*03'"#32<#1&+(03+#+*/&'7#+0# 3'"*2+)'"#+(>:#?

24 In all my films, I have been faithful to these suspension points in the conclusions. Besides, I have never written the word END on the screen. (Federico Fellini),-.*/0'#1)'+2#+0#3'"*2+)'"#)*+#9:.,*;,<4# 5(/#"0'6+#1#+*/#+0#3'"*2+)'"#+(#20'7#08#)# 9&*":#5(/#"0#1#;0.#+(#'&7(+<#+(#=01*2<##.*/+(&'7#)*03'"#32<#1&+(03+#+*/&'7#+0# 3'"*2+)'"#+(>:#?3+#&'#+(#@)2#08#)# A)&'B'7#91&,07)!*(!:.,*;,<<#A0A;#+(&'C#+(/# ().#+0#3'"*2+)'"4D#EF)9;0#F&@)220G# ###########$-(=!.'0!>'1!7*,)&(*(+?#

25 Appendix0to0Sect.020(Quantum0Systems )0 Recall*deTinitions*of*********,********and*******(t().*Potential properties*measureable/obs.*at*times** *t((generate****** W*7alg. ************************************************************************************************************(A1) *********************************************************************************************************************** ***************************************************************************************************************** *************************************************************************************************************(A2) ********************************************************************************************************************************************* ******************************************************************************************************************************************* *****************************************************************************# Information(Loss!

26 Appendix,0ctd.0 DeTine *************************************************************************************,***********************(A3) so*that ********************************************************************************************** ********is**a**endom;*********not((a***autom*************information(loss!!!!!!!!!!!!! **********************************************************************************entanglement! Some(fundamental(questions(to(be(answered: (1) What**is*meant*by*a* measurement *of******************?****** Around*which*time*does*it*take*place*?*A*measurement** of*********ought*to*result*in* ******having*a*value,*i.e.,*become* an* empirical/objective*property *of*s***** *******for*some*time*t.*************************************

27 Appendix,0ctd. (2)GivenastateofS,doesQM0predictwhich willbemeasuredzirst;whatdoesqm0predictaboutthe outcomeofmeasnt.of?inwhichwayisqm intrinsicallyindeterministic?Whydoesameant.of havearandom0outcome? Projective0measurements Consider (A4) Measurement/observation 0ofaroundtimet,or 0

28 Appendix,0ctd.0!!!!!!!!!!isan empirical/objective0property 0ofS0aroundtimet0: (A5)00 0 whereisthestateofs00rightbeforemeasnt.of,i.e., (A6) Information0Loss isusuallyamixedstate oneveniftheinitialstateofs0hasbeenpure,asa stateon!

29 Appendix,0ctd. Suppose,forsimplicity,thatisisomorphictosome B( t ),(i.e.,isof).then Eq.(5)(A7) DeZinition0 ismeasured/observedaroundtimet isan empirical/objective0prop. 0ofS0aroundtimet0000iff! (A8)0 andsomez0inthecenterof. (Moregenerally,belongstothe center0of0the0 centralizer0of0the0state ;useoftomita+takesaki/th.!) 0

30 Appendix,0ctd.0 Axiom0A0 Ifismeasured(i.e.,anempirical/objectiveprop.ofS) aroundtimet00thenhasavaluearound timet (A9)0 shouldbeusedforimprovedpredictionsoffutureafter timet.0 0

31 Appendix,0ctd.:0Summary0 (1) Giventheinitial0state0ofthesystemS,!time0evolution, {U(t,s)},determineswhichpot.prop.willZirst becomeempirical(objective,measureable),andaround whichtime. (2) Measnt.ofisindependent0ofanearlier0measnt.of iffbecomesempirical/objectiveafter0timeofmeasnt. of,nomatterwhattheoutcomeofmeasnt.ofwas, i.e.,for0all0states ,0j=01,0 0,0k,0withasin(8) Decoherence,0 consistent0histories.0 (3) Timeofmeasurement:00Time,t *0,ofobservationof det.byminimizingthefu. forsomefu.f0andsomez.0 (4)Generaltheoryofrepeatedmeasurements( POVM s ), etc.:anothertime! 0

32 My Manifesto I propose that, at all decent institutions of higher education, one or two days per semester will be declared to be Days of Reflection and of Protest! During these days, we will not teach or attend committee meetings, and there won t be any exercise classes. Instead, we will discuss some of the serious problems threatening our civilization, draft declarations and reach out to the media, with the aim to make it clear to all circles wielding power that we no longer accept (to mention some examples among others):

33 My Manifesto, ctd.! That internal tensions and conflicts in countries, such as the Ukraine, are solved by armed conflicts rather than by political dialogue and compromise;! that innocent people are slaughtered in ugly civil wars and by terrorist activities, such as those in Syria and Iraq;! that countries threaten other countries with warfare;! that weapons are sold to (clans) in countries plagued by civil war or other forms of unrest and conflict;! that religions are abused for purposes of power and suppression of people;! that the dignity and the rights of women are abused and offended in the name of religion;

34 My Manifesto, ctd.! that people are harassed or killed because of their race or faith;! that nothing is done against the perversions of 21 st Century Capitalism;! that the resources of Planet Earth continue to be looted shamelessly. These are but some examples of numerous problems threatening the survival of humankind in peace and dignity. Where is the Peace Movement, where are movements such as Occupy Wall Street, Survivre et Vivre? What is the Club of Rome doing? Why are the media silent about the activities of these and other groups? Why do academics not have a strong voice in political debates, anymore? *** Students and Academics raise your voices, arise!

4. A Theory of Events and Their Direct/Projective Observation. von Neumann Measurements

4. A Theory of Events and Their Direct/Projective Observation von Neumann Measurements I leave to several futures (not to all) my garden of forking paths J. L. Borges Les Diablerets, January 2017 1. Some