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Home , Ensemble interpretation, Wave interference

Finally making sense of the double-slit experiment

Yakir Aharonova,b,c,1, Eliahu Cohend,1,2, Fabrizio Colomboe, Tomer Landsbergerc,2, Irene Sabadinie, Daniele C. Struppaa,b, and Jeff Tollaksena,b

aInstitute for Studies, Chapman University, Orange, CA 92866; bSchmid College of Science and Technology, Chapman University, Orange, CA 92866; cSchool of and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel; dH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom; and eDipartimento di Matematica, Politecnico di Milano, 9 20133 Milan, Italy

Contributed by , March 20, 2017 (sent for review September 26, 2016; reviewed by Pawel Mazur and Neil Turok) Feynman stated that the double-slit experiment “. . . has in it the modular momentum will arise as particularly signifi- heart of quantum . In reality, it contains the only mys- cant in explaining interference phenomena. This approach along tery” and that “nobody can give you a deeper explanation of with the use of Heisenberg’s unitary equations of intro- this phenomenon than I have given; that is, a description of it” duce a notion of dynamical nonlocality. Dynamical nonlocality [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures should be distinguished from the more familiar kinematical non- on Physics]. We rise to the challenge with an alternative to the locality [implicit in entangled states (10) and previously ana- function-centered interpretations: instead of a quantum lyzed in the by Deutsch and Hayden (11)], wave passing through both slits, we have a localized particle with because dynamical nonlocality has effects on prob- nonlocal interactions with the other slit. Key to this explanation ability distributions (unlike, e.g., measurements of one of two is dynamical nonlocality, which naturally appears in the Heisen- spins in Bell states, which do not change their distri- berg picture as nonlocal equations of motion. This insight led us to bution). Within the Schrodinger¨ picture, dynamical nonlocality is develop an approach to which relies on pre- manifest in the unique role of phases, which although unobserv- and postselection, weak measurements, deterministic, and mod- able locally, may subsequently influence interference patterns. ular variables. We consider those properties of a single particle Finally, in addition to the initial state of the particle, we will that are deterministic to be primal. The Heisenberg picture allows also need to take into account a final state to form a twofold us to specify the most complete enumeration of such determinis- set of deterministic properties (one deterministic set based on tic properties in contrast to the Schrodinger¨ , which the initial state and a second based on the final state). The above remains an ensemble property. We exercise this approach by ana- amounts to a time-symmetric Heisenberg-based interpretation of lyzing a version of the double-slit experiment augmented with nonrelativistic quantum mechanics. postselection, showing that only it and not the wave function approach can be accommodated within a time-symmetric inter- Wave Function Represents an Ensemble Property pretation, where interference appears even when the particle is The question of the meaning of the wave function is central to localized. Although the Heisenberg and Schrodinger¨ pictures are many controversies concerning the interpretation of quantum equivalent formulations, nevertheless, the framework presented mechanics. We adopt neither the standard ontic nor the epis- here has led to insights, intuitions, and experiments that were temic approaches to the meaning of the wave function. Rather, missed from the old perspective. we consider the wave function to represent an ensemble prop- erty as opposed to a property of an individual system. This Heisenberg picture | two-state vector formalism | modular momentum | approach resonates with the of the wave double slit experiment Significance eginning with de Broglie (1), the physics community em- Bbraced the idea of particle-wave duality expressed, for exam- We put forth a time-symmetric interpretation of quantum ple,inthedouble-slitexperiment.Thewave-likenatureofelemen- mechanics that does not stem from the wave properties of tary particles was further enshrined in the Schrodinger¨ equation, the particle. Rather, it posits corpuscular properties along with which describes the time evolution of quantum wave packets. nonlocal properties, all of which are deterministic. This change It is often pointed out that the formal analogy between of perspective points to deterministic properties in the Heisen- Schrodinger¨ and classical wave interference berg picture as primitive instead of the wave function, which allows us to interpret quantum phenomena in terms of the famil- remains an ensemble property. This way, within a double-slit iar classical notion of a wave. Indeed, wave-particle duality was experiment, the particle goes only through one of the slits. construed by Bohr and others as the essence of the theory and, In addition, a nonlocal property originating from the other in fact, its main novelty. Even so, the foundations of quantum distant slit has been affected through the Heisenberg equa- mechanics community have consistently raised many questions tions of motion. Under the assumption of nonlocality, uncer- (2–5) centered on the physical meaning of the wave function. tainty turns out to be crucial to preserve causality. Hence, a From our perspective and consistent with ideas first expressed (qualitative) can be derived rather than by Born (6) and thereafter extensively developed by Ballentine assumed. (7, 8), a wave function represents an ensemble property as opposed to a property of an individual system. Author contributions: Y.A. conceived research; Y.A., E.C., and T.L. designed research; Y.A., What then is the most thorough approach to ontological ques- E.C., F.C., T.L., I.S., D.C.S., and J.T. performed research; and E.C., T.L., and J.T. wrote the tions concerning single particles (using standard nonrelativis- paper. tic quantum mechanics)? We propose an alternative interpreta- All authors are ordered alphabetically. tion for quantum mechanics relying on the Heisenberg picture, Reviewers: P.M., University of South Carolina; and N.T., Perimeter Institute. which although mathematically equivalent to the Schrodinger¨ Conflict of interest statement: Y.A. is a visiting scholar at Perimeter Institute, and has picture, is very different conceptually. For example, within the received funding from reviewer N.T.’s institution, Perimeter Institute. Heisenberg picture, the primitive physical properties will be rep- Freely available online through the PNAS open access option. resented by deterministic operators, which are operators with 1To whom correspondence may be addressed. Email: [email protected] and measurements that (i) do not disturb individual particles and [email protected]. (ii) have deterministic outcomes (9). By way of example, the 2E.C. and T.L. contributed equally to this work.

6480–6485 | PNAS | June 20, 2017 | vol. 114 | no. 25 www.pnas.org/cgi/doi/10.1073/pnas.1704649114 Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 hrnve al. et Aharonov hc clslike scales which edrcl eie ttesnl-atcelvl hycnttt the constitute they level, single-particle the at can verified operators out- directly these be definite to predictable, corresponding single properties with a Because and on comes. out it carried disturbing be without unique can particle are that which propos- measurement operators, such with of expound deterministic operators we on out follows, based duck what ontology In an not ontology. will single-particle a we ing advocates, ensem- interpretation to contrary ble and said single being a That the of particle/system. by ontology that quantum primitive suggest uniquely the to constitute con- mean determined cannot only function We is the wave system. the function oppose of wave state physical we (15), the the theorem do that (PBR) states Nor Pusey–Barrett–Rudolph which as theory). the such variable of reality, sequence classical-like hidden a a of with addition become the could with (and “complete” incomplete somehow is description function ensemble. quantum func- a wave of property the a systems, is con- tion thermodynamic distributions We for mechanical particle. properties statistical single required how become a to are of analogous measurements that, function clude wave identical two the of these approximate of number either to In large (14). a effect in Zeno ways, protection quantum external the an of requires form (13); the way theorem general adiabatic more the nondegener- second, on discrete the based measuring and eigenstates for different energy applicable ate two is in first implemented the ways: be (12). can measurement measurements protective a Protective measurement. is prolonged measurement a function prolonged performing wave This by Con- the measured ensembles. level, be single-particle quantum only a can on on only analogy, verified the tinuing directly func- wave be the can system, thermodynamic tion a on only verified directly properties. position real namely its directly, constitute they verified only be and can momentum, and in that individual found those the be of are to properties particle particle intrinsic the the However, level, for states. single-particle certain as the itself of On manifests property system. it a charac- thermodynamic as these viewed entire to best Due the is grows. function distribution system the being the teristics, of as category only the verified into directly transitions procedure verification the [containing large is system the when N Conversely, error. because yield large particles, particles a single single on to performed distribution measurements the instantaneous bath. attribute heat a not to do ways— coupled We particle pro- two single using a in or on measurements systems distribution, mea- longed thermodynamic can Boltzmann on We instantaneously the mechanics. either example, statistical distributions for in probability particles sure, single how to instance, relate for Consider, intrinsic. relativistic and practical (excluding accuracy time arbitrary By short constraints). an level. arbitrarily to ensemble an measured the wave in mean at the we verified because individual verified,” directly is “directly an be this for only ontol- for ontology can an justification function an as principle than appropriate Our rather is system. ensemble Instead, function an reasoning. wave for his with the ogy concur that not contend account do by we conclusion but to the 8) having with (7, concur avoids Ballentine We it (collapse). reduction systems, pretend- state individual not for describe problem—by observing by measurement to interpretation the ing this overcomes to it adherence that an Ballentine systems. justified individual mechanics 8) to quantum apply (7, not of does nature It directly. probabilistic stems the ensem- the hypothetical which interpretation, a devel- from of this description extensively ble, to statistical and a According is (6) 8). function wave (7, Born Ballentine by by initiated oped was that function nieBr 6,w ontws oipyta h wave the that imply to wish not do we (6), Born Unlike be can mechanics statistical in distributions how to Similar be to properties verifiable directly regard only we Indeed,  1 atce tetemdnmclmt] h ieo h error, the of size the limit)], thermodynamic (the particles √ N srltvl eysal nohrwords, other In small. very relatively is , rtto)o h aefnto from function inter- wave preferred the one’s of on pretation) depending effective or (true “collapse” Schr the of of probability range a (with random initial by at given appears outcome the performed, for system orthonormal on operator Hermitian system a of vector state a the is measurements, ideal with along mechanics constant. remains deterministic whereas function Hermitian equation, wave of the Heisenberg’s phys- set to a a according picture, by evolving Heisenberg described operators the be in can below, system Schr shown ical be to will As according tion. calculated and Hamiltonian function wave continuous Schr the In Ontology and turn Formalism we ontology, representation. this Heisenberg derive the To to spotlight particle. the the of properties real h usin“hti h e fHriinoperators Hermitian of set the answers is which set “what This question involved. is the an collapse already is no investigation under Therefore, system eigenstate. the of state those the only which of for measurement involves operators” set of This subset (DSO). “deterministic the of measurements the turbing of disturbance a to corresponds system. per- same collapse by the yield This verified will be eigenvalue. that can measurements fact ideal subsequent This forming eigenvalue. that to responding A Moreover, multiplication. under the Clearly, operator?” are given “what a question: of familiar eigenstates more the to dual is question This where H Theorem. h w-tt etrfraim(SF.Ti prahhsits has approach This (TSVF). into formalism from known take vector is must one two-state time than complete the given rather a a conditions that boundary at idea two system The account quantum system. a the of of state description final a is pretation below). described limiting which be two can which the information for Eq. Mathematically, by no out- them. represented measurement Thus, cases measuring possible (17). by their sat- equiprobable gained of they all are that that comes condition means the uncertainty isfy Complete “completely com- uncertain. the are that to pletely outcomes dual measurement out- are with measurement operators” certain) uncertain (with completely equa- DSOs are them. Heisenberg’s that of applying comes each by to unitarily separately evolve without tion mea- all are (i.e., eigenoperators they particle such only sured, the as long disturbing As collapse). without inducing them measure to that such vector ˆ k and , nipratigein ocnie forpooe inter- proposed our of consider to ingredient important An possibility the from stems DSOs of significance physical The ntetaiinlHletsaefaeokfrquantum for framework space Hilbert traditional the In oee,oecudivr h rcs n osdrnondis- consider and process the invert could one However, |ψ i A ψ ˆ 0 = h h |ψ Ai ˆ Ai ψ ˆ sa iesae”fraystate any for eigenstate?” an is loe pcrm hratr rmteperspective the from Thereafter, spectrum. allowed As ˆ Let H ∈ i . = ∆A|ψ 0 = = |ψ dne itr,teielmaueetlast the to leads measurement ideal the picture, odinger ¨ dne itr,asse sflydsrbdb a by described fully is system a picture, odinger ¨ { hψ i A H ˆ ancsaybtisfcetcniina ilbe will as condition insufficient but necessary (a |ψ naHletspace Hilbert a in i 1) Then, (16). | hψ ⊥ A| eaHletspace, Hilbert a be ˆ uhthat such n orsod oa ievlewti the within eigenvalue an to corresponds and i) i A|ψ ˆ |ψ ψ 0 = PNAS i, ⊥ i ∆ 0 = H n opeeyucranoeaosfor operators uncertain completely and = H A h iesae of eigenstates The . | 2 hna da esrmn of measurement ideal an When . h . 2 Ai|ψ r ie ydtriitcoperators deterministic by given are ψ ue2,2017 20, June ˆ A ˆ = t vlto sdcae ythe by dictated is evolution Its . i |ψ hψ i (t [ |( A ˆ ∆A|ψ + H )i A ˆ i , n n observable any and , A ˆ h − A = |ψ ˆ ea prtratn on acting operator an be j ψ i | a = ] Ai) ˆ A i ˆ : noa iesaecor- eigenstate an into |ψ o.114 vol. ⊥ ψ A ˆ 2 i, (t sasbpc closed subspace a is | k A ˆ ψ )i, ∈ and i, omacomplete a form dne’ equa- odinger’s ¨ A ˆ a | i ψ o 25 no. ∈ ssc that such is R}. | ψ A ⊥ ˆ A | ˆ i ψ A 6481 ˆ sa is sa is [1] [2] for is

PHYSICS roots in the work of Aharonov et al. (18), but it has since been position and momentum are not the most appropriate dynamical extensively developed (19) and led to the discovery of numerous variables to describe quantum interference phenomena. Indeed, interesting phenomena (17). it is easy to prove the following theorem. The TSVF provides an extremely useful platform for analyzing iφ experiments involving pre- and postselected ensembles. Weak Theorem. Let ψφ(x, t) = ψ1(x, t) + e ψ2(x, t), and assume no measurements enable us to explore the state of the system dur- overlap of ψ1(x, 0) and ψ2(x, 0) (t = 0 is when the particle is ing intermediate times without disturbing it (20, 21). The power going through the double slit) (16). If m, n are integers, then for to explore the pre- and postselected system by using weak mea- all values of t and choices of phases α β, surements motivates a literal reading of the formalism (that is, Z ∗ m n ∗ m n as more than just a mathematical tool of analysis). It motivates a [ψα(x, t)x p ψα(x, t) − ψβ (x, t)x p ψβ (x, t)] dx = 0. [3] view according to which future and past play equal roles in deter- mining the at intermediate times and are, hence, Let us now consider operators of the form fˆ(x, p) := eipL/~ equally real. Accordingly, to fully specify a system, one should (where L is the distance between the slits). Evolving this through not only preselect but also, postselect a certain state using a pro- the Heisenberg equation, jective measurement. In the framework that we propose within this article, adding a final state is equivalent to adding a sec- ∂fˆ(x, p) i = [fˆ, Hˆ ], ond DSO in addition to the one dictated by the initial state. This ~ ∂t twofold set forms the basis for the primal ontology of a quantum 2 mechanics for individual particles. where H = p /2m + V (x) appropriate for the double slit. In this particular case, we obtain a nonlocal equation of motion: Nonlocal Dynamics and Wave-Like Behavior ∂fˆ(x, p) ipL 1 ipL Interference patterns appear in both classical and quantum grat- = [e ~ , V (x)] = [V (x + L) − V (x)]e ~ [4] ∂t ing experiments (most conveniently analyzed in a double-slit ~ ˙ setup, which will be referred to hereinafter, although our results (that is, the value of fˆ depends on the potential at not only x but are completely general). We are taught that the explanation for also, the remote x +L). This operator leads us naturally to realize interference phenomena is shared across both domains, the clas- that the variable that accounts for the effect of the double slit is sical and quantum: a spatial wave (function) traverses the grat- not p but its modular version. Indeed, because ing, one part of which goes through the first slit while the other ipL i(p+2πk~) L part goes through the second slit, before the two parts later meet e ~ = e L ~ , k ∈ Z, to create the familiar interference pattern. Although it is indeed tempting to extend the accepted classical explanation into the the observable of interest is the modular momentum (i.e.,

quantum domain, nevertheless, there are important breakdowns pmod := p mod p0, in the analogy. For example, in classical wave theory, one can predict what will happen when the two parts of the wave finally where p0 = 2π~/L). Eq. 4 differs considerably from the classical meet based on entirely local information available along the tra- evolution that is given by the Poisson bracket: jectories of the wave packets going through the two slits. How- d i2πp n i2πp o 2π dV i2πp p p ˆ p ever, in quantum mechanics, what tells us where the maxima e 0 = e 0 , H = −i e 0 , [5] dt p0 dx and minima of the interference will be located is the relative of the two wave packets. Although we can measure the which involves a local derivative, suggesting that the classical local phase in , we cannot in principle mea- modular momentum changes only if a local force dV /dx is acting sure the individual local phases for a particle, because this would on the particle. We thus understand that, although commutators violate gauge symmetry (17). Only the phase difference is observ- have a in terms of Poisson brackets, they are fun- able, but it cannot be deduced from measurements performed damentally different, because they entail nonlocal dynamics. The on the individual wave packets (until they overlap). The analogy connection between nonlocal dynamics and relative phase via the is, therefore, only partial. For this reason, we contend that the modular momentum suggests the possibility of the former taking temptation to jump on the wave function bandwagon should be the place of the latter in the Heisenberg picture. The nonlocal resisted. Our goal now is to show how quantum interference can equations of motion in the Heisenberg picture thus allow us to be understood without having to say that each particle passed consider a particle going through only one of the slits, but it nev- through both slits at the same time as if it were a wave. For ertheless has nonlocal information regarding the other slit. this purpose, we examine those operators that are relevant for Unlike ordinary momentum, modular momentum becomes, all interference phenomenon. When we transform back to the on detecting (or failing to detect) the particle at a particular Schrodinger¨ picture and apply these operators, we will see that slit, maximally uncertain. The effect of introducing a potential these operators are sensitive to the relative phase, which again, at a distance from the particle (i.e., of opening a slit) is equiv- is the property that determines the subsequent interference alent to a nonlocal rotation in the space of the modular vari- pattern. able (22). Denote it by θ ∈ [0, 2π). Suppose the amount of non- We, therefore, consider the state ψφ(x, t) = ψ1(x, t) + local exchange is given by δθ (i.e., θ → θ + δθ). Now “maximal iφ e ψ2(x, t), which in the Schrodinger¨ picture, represents the uncertainty” means that the probability to find a given value of θ is independent of θ [i.e., P(θ) = constant = 1/2π]. Under wave at the double slit. We now ask which operators fˆ(x, p) these circumstances, the shift in θ to θ + δθ will introduce no Aˆ belong to the DSO ψφ . In addition, we ask which operators are observable effect, because the probability to measure a given sensitive to the relative phase φ. It is not difficult to show that, if value of θ, say θ1, will be the same before and after the shift, we limit ourselves to simple functions of position and momentum P(θ1) = P(θ1 + δθ1). We shall call a variable that satisfies this (i.e., any polynomial representation of the form condition a “completely uncertain variable.” X m n fˆ(x, p) = amn x p , Theorem (Complete Uncertainty Principle for Modular Variables). then any resulting operator is not sensitive to the relative phase Let Φ be a periodic function, which is uniformly distributed on between different “lumps” of the wave function (i.e., lumps cen- the unit circle (17). If heinΦi = 0 for any integer n 6= 0, then Φ is tered around each slit). This fact suggests that simple moments of completely uncertain.

6482 | www.pnas.org/cgi/doi/10.1073/pnas.1704649114 Aharonov et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 a ipybe oe rmtevco nteHletsaeto space Hilbert the in vector the from evolution moved time been the equivalent: simply are has pictures two the that know We uewt ieeovn state picture, time-evolving Heisenberg a with Eq. ture in that, Note Eq. Therefore, o ie operator given a For the about information the phase. destroying relative and thereby interference Consequently, uncertain, erasing completely occurs. becomes particle momentum the modular the if with exchange nonlocal collapse: momentum a of experimenter, modular language the by of the closed in is slits not the is of for one that The explanation interference with. different of interfere loss a left the to offers nothing packet however, is wave picture, there Heisenberg localized reduced, remaining is the superposition for col- the function wave After of consequence inter- lapse. a that as understood fact is Schr the loss the ference with In localization. well with fits lost is also ference particle the of localization are measurements packet, such principle. wave uncertainty all the complete but on the causality, by change precluded violate this could measure we could then we If momen- space. modular points remote in packet. the relates momentum wave on modular because localized depending instantly, a tum operators from change away thus far We arbitrarily we suppose force example, pre- a an to apply As crucial causality. is preserve and uncertainty signaling complete of vent of position onset the find This we particle. after the lost is momentum modular modular the the information about all that Accordingly, principle uncertain. completely uncertainty is momentum complete the from lows hrnve al. et Aharonov transform: Fourier its calculate can we then instants various at ρ(x particle the reconstruct of position to the us func- allow operator-valued tions of properties time Fundamental at space. Hilbert described system a Consider Operators Nonlocal fundamental. very Measuring as principle reconcilia- this This regard we coexist.” why “peacefully is may tion they relativistic complete with that , nonlocality so nonlocal reconcile causality, to of one detection enables the uncertainty Neverthe- prohibiting other. by each less, contradict nearly causality relativistic and principle, of case uncertainty the Heisenberg in it the equating parallels inequality interval This the in uniform ~/L is function distribution momentum its complete modular a reason, the this of For well). uncertainty as uncertain be can part integer of uncertainty the case the in as slits, the of one near either localized of is particle a out- when packet wave the shifting side operator, translation a as functions of value tion oeas ht because that, also Note on uncertain becomes momentum modular the that fact The within to localized is particle a When tfis ls,i em ht saim,dnmclnonlocality dynamical axioms, as that, seems it blush, first At , t salwrbudfrteucranyin uncertainty the for bound lower a as |x ) | h est of density the < hψ ψ L/2 1 (x e or F ipL/~ , ie,otieisrgo fspot.Accordingly, support). of region its outside (i.e., ρ(k t 6 ψ )| A 2 ˆ A|ψ sntigbtteepcainvleof value expectation the but nothing is ˆ p , ehv enuigteSchr the using been have we 7, aihs hsrsl sovos because obvious, is result This vanishes. then , x ρ(x t sgetrta reult htof that to equal or than greater is (t = ) ecnwieisepcainvleas value expectation its write can we A, ψ ˆ (x = ) (x , t , Z ∆x = ) , t he p t R )i hψ namely ), = ψ inpL/~ = = (x ψ ∗ p |ψ (x ∗ L. mod , hψ (x ψ i t , )| i (x t (x sn ekmaueet of measurements weak using , )ψ 0 = A|ψ t + ˆ , )ψ , t 0)| N (x t ertn Eq. Rewriting ). (x (x 0 = o every for ~/L p , A(t ˆ |x t mod , , dne itr,inter- picture, odinger )e ¨ t t | ) yavector a by )i. < )|ψ ikx o oeinteger some for t p wihmasthat means [which ned fw call we if Indeed, . h expecta- the L/2, (i.e., dx (x n , . 0)i. [0, tte fol- then It . dne pic- odinger ¨ ∆ sets ~/L)] p p mod 7 |ψ ≥ e nthe in i ~/L). ipL/~ e (the na in [7] [6] ikx N . , h prtr ie that Given operator. the tefb hfigteitreec atr by phase pattern The interference overlap. the shifting packets of by wave itself two time the the until ture to phase relative up The φ general). shape however, are, its results (our encounter maintains approximately packet spread the that where 1): (Fig. toward moving equal packets with another wave one separated spatially identical In two experiment. of double-slit the Schr mimic the to experiment Gedanken present Schr difficulties cre- the evades therefore, in thus and slit description other This momentum the interference. of modular ate presence nonlocal the a “sense” and can location particle that definite the picture, a Heisenberg both the has in the Alternatively, at properties time. particle same and wave both posit informa- which experiments, which-path deduce Schr the and However, tion. interference measure both us to allows postselection involving experiments certain Performing Revisited Experiment Double-Slit to respect Proof. state the determines uniquely at operators nonlocal Hence, values. possible t the of all assumes set we If nefrnepteni rprinlt h eaiephase relative the at to pattern proportional is interference pattern The interference (B) packets. wave time two the the of (8) superposition 1. 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(x /m −ip β = ) ~ hsexpression This . nerto with Integration . 0 ~φ/p dx ehave we , x φ. | Ψ , t o 25 no. changes,  φ 0 . x manifests −L 2 δ | fthe of  6483 [8] ,

PHYSICS This initial configuration is identical to that of the standard the Schrodinger¨ picture. The real part of this density, which double-slit setup, but instead of letting the two wave packets describes the evolution of the two-state, exhibits an interference propagate away from the grating to hit a photographic plate, we pattern when weakly measured. However, by virtue of the post- confine ourselves to one dimension and let them meet at time T selection, we know that the particle has a determinate position on the plane of the grating. On meeting, the density of the two described by a right-moving wave packet that went through the wave packets becomes left slit. Interference is thus still present, despite the fact that the 2 2 particle is localized around one of the slits. Recall that the inter- ρ(x, T ) ≈ 4|Ψi (x)| cos (p0x/~ − φ/2), [9] pretation of the particle as having a wave-like nature was origi- which displays interference, similar to that of a standard double- nally devised to account for interference phenomena, and here, slit experiment. we have shown that this is not necessary and in fact, is inconsis- We now augment the experiment with a postselection proce- tent with a time-symmetric view. dure, where we place a detector on the path of the wave packet In contrast, the Heisenberg picture tells us that each particle ip0x/~ has both a definite position and at the same time, nonlocal infor- moving to the right Ψf (x) = e Ψ(x − L/2). Although the probability to find the particle there is 1/2, let us consider an mation in the form of DSOs, which are simple functions of the ensemble of such pre- and postselected experiments, which real- modular momentum (9). izes the rare case where all of the particles are found by this Discussion detector (that is, we determine the position operator for the entire ensemble by a postselection). The two-state, which con- After the Schrodinger¨ picture has dominated for many years, stitutes the full description of pre- and postselected systems at we have elaborated a Heisenberg-based interpretation for quan- tum mechanics. In this interpretation, individual particles pos- any intermediate time t, is given by hΨf (t)| |Ψi (t)i. Within the TSVF, we can define a two-times generalization of the pure-state sess deterministic, yet nonlocal properties that have no clas- density: sical analog, whereas the Schrodinger¨ wave can only describe an ensemble. An uncertainty principle appears not as a math- hx|Ψf ihΨi |xi ematical consequence but as a reconciler between metaphysi- ρtwo−time (x, T ) = hΨf |Ψi i cal desiderata—causality—and the nonlocality of the dynamics. While this complete uncertainty principle (qualitatively) implies = 2|Ψ(x)|2ei(p0x/~−φ/2) (p x/ − φ/2). cos 0 ~ the Heisenberg uncertainty principle, the implication does not [10] work the other way around, i.e., from Heisenberg uncertainty To measure this density, during intermediate times, we perform principle to the complete uncertainty principle. For this reason, we regard the complete uncertainty principle as more fundamen- a using M  1 projections Πi (x) with the tal. In turn, uncertainty combined with the empirical demand H = g(t)q PM Π (x) q interaction Hamiltonian int i i , where is the for definite measurement outcomes necessitate a mechanism for pointer of the measuring device, i sums over an ensemble of par- τ choosing those outcomes. This demand is met by the inclusion ticles, and R g(t)dt = g is sufficiently small during the measure- 0 of a final state. It was shown elsewhere (23) that, by consider- τ ment duration . For a large-enough ensemble, these measure- ing a special final state of the kind that we had introduced but ments allow us to observe the two-time density while introducing for the entire universe, the outcomes of specific measurements almost no disturbance to the state of the particle. If we perform can be accounted for. This cosmological generalization thereby many such measurements in different locations within the over- solves the . We now understand this final lap region, they will add up to a histogram tracing the two-time state to constitute a DSO, which may be regarded as a hidden density in that region (Fig. 2) from which we find the parameter variables because of its epistemic inaccessibility in earlier times. δ that depends on the relative phase φ. This Gedanken exper- We contend that this interpretation conveys a powerful physi- iment shows a perplexing situation from the point of view of cal intuition. Internalizing it, one is no longer restricted to think- ing in terms of the Schrodinger¨ picture, which is a convenient tool for mathematical analysis but inconsistent with the pre- and post- selection experiments. The wave function is an efficient mathe- matical tool for calculations of experimental statistics. However, the use of potential functions is also mathematically efficient, although it is only the fields derived from potentials that are physically real. Hence, mathematical usefulness is not a sufficient criterion by which to fix an ontology. Indeed, although useful for calculating the dynamics of DSOs, wave functions are not the real physical objects—only DSOs themselves are. Importantly, considerations pertaining to this ontology have led Y.A. to dis- cover the Aharonov–Bohm effect. The stimulation of new dis- coveries is the ultimate metric to judge an interpretation. Intriguingly, the Heisenberg representation that was discussed here from a foundational point of view is also a very helpful framework for discussing quantum computation (24). Moreover, in several cases (25), it has a computational advantage over the Schrodinger¨ representation. For the sake of completeness, it might be interesting to briefly address the notion of kinematic nonlocality arising from entan- glement. As noted in Formalism and Ontology, a quantum system Fig. 2. Weak measurement of the interference pattern. The two wave in 2D Hilbert space (e.g., a -1/2 particle) is described within packets are preselected in A and postselected in C. Weak measurements in our formalism using two DSOs. For describing a system of two B performed at t = T show the usual interference pattern, despite the fact entangled spin-1/2 particles (in a 4D Hilbert space), we would that detector D detects all particles as belonging to just one (moving to the use a set of 10 DSOs. It is important to note that the measure- right) wave packet. ments of such operators are nonlocal (26), possibly carried out

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Province and the Research of and Ministry Development the Depart- Economic through the Ontario and through of Science Canada of Innovation, Government Theoret- of the for ment by Institute Insti- supported Chapman Perimeter Perimeter is the the at Physics at by ical Research Studies this part, Physics. Quantum in Theoretical for supported, for for Funding tute also Institute was Time. Research the research and This European by University. Space the provided German–Israeli in was by the Nonlocality research supported and Grant was Light,” Advanced Y.A. E.C. of Council Trust. (DIP). Memorial “Circle Cooperation Fetzer Excellence Project E. Israeli Research 1311/14, John of Grant Foundation the Centers Science of Israel from Fund support Franklin acknowledges Fetzer the from ACKNOWLEDGMENTS. approach. neglected that somewhat this hope of promote We discussion will a theory, prosper. ontology Heisenberg-based relativity to the of of ontology endorsement wake our function the wave in the nonphysical allowing as Hamiltonians dismissed those were However, intuitive. less description function h w-tt-etrformalism. two-state-vector the Phys Theor J Int applications. and Basics values: 316. weak 100. quantum be to standing out turn can particle spin-1/2 a of 60:1351–1354. spin 369–412. pp the Berlin), (Springer, of al. component et JG, Muga eds Mechanics, Quantum in Time measurement. of plexed 90:010402. inputs. Fock-state with puting arXiv:quant-ph/9807006. measurements. weak and variables 013023. modular experiments, ference 8:475–478. WlyVH enem Germany). Weinheim, (Wiley-VCH, 2:213–230. hsRvB Rev Phys .. ... n ..akoldespot npart, in support, acknowledge J.T. and D.C.S., Y.A., PNAS 134:1410–1416. hsRvA Rev Phys | unu aaoe:QatmTer o h Per- the for Theory Quantum Paradoxes: Quantum unu tdMt Found Math Stud Quantum ue2,2017 20, June 89:022328. | o.114 vol. 1:133–146. e o Phys Mod Rev | o 25 no. e Phys J New hsRvLett Rev Phys hsRvLett Rev Phys a Phys Nat | 86:307– 6485 12:

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