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Neil mechanics. . “. and quantum Mazur experiment of Pawel double-slit by heart reviewed the 2016; that 26, stated September review Feynman for (sent 2017 20, March Aharonov, Yakir by Contributed and Kingdom; United 92866; nttt o unu tde,CamnUiest,Oag,C 92866; CA Orange, University, Chapman Studies, Quantum for Institute htte stems hruhapoc ootlgclques- ontological to approach thorough most the is then What expressed first ideas with consistent and perspective our From between analogy formal the that out pointed often is It gnigwt eBole() h hsc omnt em- community exam- physics for expressed, the duality particle-wave (1), of idea Broglie the braced de with eginning dne aeitreec n lsia aeinterference wave classical and interference wave odinger ¨ c colo hsc n srnm,TlAi nvriy e vv6981 Israel; 6997801, Aviv Tel University, Aviv Tel Astronomy, and Physics of School | a,b,c,1 w-tt etrformalism vector two-state e a,b iatmnod aeaia oienc iMln,9213Mln Italy Milan, 20133 9 Milano, di Politecnico Matematica, di Dipartimento lauCohen Eliahu , dne aefnto,which function, wave odinger ¨ d,1,2 arzoColombo Fabrizio , | h ena Lectures Feynman The oua momentum modular dne itrsare pictures odinger ¨ dne equation, odinger ¨ odinger ¨ | b e cmdCleeo cec n ehooy hpa nvriy rne CA Orange, University, Chapman Technology, and Science of College Schmid oe Landsberger Tomer , 2 option. access has open 1 and PNAS the Institute, through Perimeter online at available Freely Institute. scholar Perimeter visiting institution, N.T.’s a reviewer from is funding Y.A. received statement: interest of Institute. Perimeter Conflict N.T., and Carolina; South of University P.M., Reviewers: the wrote alphabetically. J.T. ordered and are T.L., authors E.C., All and research; performed J.T. and paper. D.C.S., I.S., T.L., F.C., E.C., This system. individual an of wave the property of interpretation prop- a ensemble the ensemble with to an resonates approach opposed represent to Rather, as function function. epis- erty wave wave the the the nor of consider ontic meaning we the standard to the quantum to approaches neither central temic of adopt is interpretation function We the wave mechanics. the concerning of controversies meaning many the of question The Property Ensemble an Represents Function Wave above mechanics. The twofold quantum state). of nonrelativistic a final interpretation on Heisenberg-based the form time-symmetric on based a based will to to second set amounts state a we deterministic and final particle, state (one initial a the properties the account of deterministic into of state take set initial to the need patterns. to also interference addition influence in unobserv- subsequently Finally, although may which phases, locally, of able role unique distri- the probability in their manifest two change of Schr not the do one Within which bution). of states, Bell measurements prob- in e.g., on spins (11)], (unlike, effects Hayden observable distributions ana- and has ability Deutsch previously nonlocality by and dynamical picture (10) because Heisenberg states the in entangled lyzed in non- kinematical [implicit nonlocality familiar more locality Dynamical the nonlocality. from intro- distinguished dynamical motion be should of of equations notion unitary a Heisenberg’s duce of signifi- use along the particularly approach This with as phenomena. interference arise explaining in will cant operator momentum modular uhrcnrbtos ..cnevdrsac;YA,EC,adTL eindrsac;Y.A., research; designed T.L. and E.C., Y.A., research; conceived Y.A. contributions: Author ..adTL otiue qal oti work. and this to [email protected] equally contributed T.L. Email: and E.C. addressed. be may correspondence whom To [email protected]. anytrsott ecuilt rsrecuaiy ec,a than Hence, rather causality. derived preserve be assumed. to can crucial principle uncertainty be uncer- (qualitative) nonlocality, to of out equa- assumption turns Heisenberg the tainty other the Under motion. through the of affected from tions slits. been originating the has property slit of nonlocal distant one a through addition, only In goes double-slit a particle within the way, which experiment, This function, property. wave ensemble the an of remains instead primitive as Heisen- picture the in berg properties deterministic to change points This perspective of deterministic. of are properties which of wave with all along properties, the properties nonlocal quantum corpuscular from posits of it stem Rather, particle. interpretation not the time-symmetric does that a mechanics forth put We Significance d .H il hsc aoaoy nvriyo rso,BitlB81TL, BS8 Bristol Bristol, of University Laboratory, Physics Wills H. 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PHYSICS function that was initiated by Born (6) and extensively devel- real properties of the particle. To derive this ontology, we turn oped by Ballentine (7, 8). According to this interpretation, the the spotlight to the Heisenberg representation. wave function is a statistical description of a hypothetical ensem- ble, from which the probabilistic nature of quantum mechanics Formalism and Ontology stems directly. It does not apply to individual systems. Ballentine In the Schrodinger¨ picture, a system is fully described by a (7, 8) justified an adherence to this interpretation by observing continuous wave function ψ. Its evolution is dictated by the that it overcomes the measurement problem—by not pretend- Hamiltonian and calculated according to Schrodinger’s¨ equa- ing to describe individual systems, it avoids having to account tion. As will be shown below, in the Heisenberg picture, a phys- for state reduction (collapse). We concur with the conclusion by ical system can be described by a set of Hermitian deterministic Ballentine (7, 8) but do not concur with his reasoning. Instead, operators evolving according to Heisenberg’s equation, whereas we contend that the wave function is appropriate as an ontol- the wave function remains constant. ogy for an ensemble rather than an ontology for an individual In the traditional Hilbert space framework for quantum system. Our principle justification for this is because the wave mechanics along with ideal measurements, the state of a system function can only be directly verified at the ensemble level. By is a vector |ψi in a Hilbert space H, and any observable Aˆ is a “directly verified,” we mean measured to an arbitrary accuracy Hermitian operator on H. The eigenstates of Aˆ form a complete in an arbitrarily short time (excluding practical and relativistic orthonormal system for H. When an ideal measurement of Aˆ is constraints). Indeed, we only regard directly verifiable properties to be performed, the outcome appears at random (with a probability intrinsic. Consider, for instance, how probability distributions given by initial |ψi) and corresponds to an eigenvalue within the ˆ relate to single particles in statistical mechanics. We can mea- range of As allowed spectrum. Thereafter, from the perspective sure, for example, the Boltzmann distribution, in two ways— of the Schrodinger¨ picture, the ideal measurement leads to the either instantaneously on thermodynamic systems or using pro- “collapse” (true or effective depending on one’s preferred inter- longed measurements on a single particle coupled to a heat bath. pretation) of the wave function from |ψi into an eigenstate cor- We do not attribute the distribution to single particles, because responding to that eigenvalue. This fact can be verified by per- instantaneous measurements performed on single particles yield forming subsequent ideal measurements that will yield the same a large error. Conversely, when the system is large [containing eigenvalue. This collapse corresponds to a disturbance of the system. N 1 particles (the√ thermodynamic limit)], the size of the error, which scales like N , is relatively very small. In other words, However, one could invert the process and consider nondis- the verification procedure transitions into the category of being turbing measurements of the “deterministic subset of operators” directly verified only as the system grows. Due to these charac- (DSO). This set involves measurement of only those observables teristics, the distribution function is best viewed as a property of for which the state of the system under investigation is already an the entire thermodynamic system. On the single-particle level, eigenstate. Therefore, no collapse is involved. This set answers ˆ it manifests itself as probabilities for the particle to be found in the question “what is the set of Hermitian operators Aψ for certain states. However, the intrinsic properties of the individual which ψ is an eigenstate?” for any state ψ: particle are those that can be verified directly, namely position ˆ ˆ ˆ Aψ = {Ai such that Ai |ψ(t)i = ai |ψ(t)i, ai ∈ R}. [1] and momentum, and only they constitute its real properties. Similar to how distributions in statistical mechanics can be This question is dual to the more familiar question: “what are the directly verified only on a thermodynamic system, the wave func- eigenstates of a given operator?” Clearly, Aˆ ψ is a subspace closed tion can be directly verified only on quantum ensembles. Con- under multiplication. Moreover, [Aˆ i , Aˆ j ] = Aˆ k ∈ Aˆ ψ is such that tinuing the analogy, on a single-particle level, the wave function Aˆ |ψi = 0. can only be measured by performing a prolonged measurement. k This prolonged measurement is a protective measurement (12). Theorem. Let H be a Hilbert space, Aˆ be an operator acting on Protective measurements can be implemented in two different H, and |ψi ∈ H (16). Then, ways: the first is applicable for measuring discrete nondegener- ate energy eigenstates and based on the adiabatic theorem (13); Aˆ|ψi = hAˆi|ψi + ∆A|ψ⊥i, [2] the second, more general way requires an external protection in ˆ ˆ 2 ˆ ˆ 2 the form of the quantum Zeno effect (14). In either of these two where hAi = hψ|A|ψi, ∆A = hψ|(A − hAi) |ψi, and |ψ⊥i is a ways, a large number of identical measurements are required vector such that hψ|ψ⊥ = 0. to approximate the wave function of a single particle. We con- The physical significance of DSOs stems from the possibility clude that, analogous to how statistical mechanical distributions to measure them without disturbing the particle (i.e., without become properties for thermodynamic systems, the wave func- inducing collapse). As long as only such eigenoperators are mea- tion is a property of a quantum ensemble. sured, they all evolve unitarily by applying Heisenberg’s equa- Unlike Born (6), we do not wish to imply that the wave tion separately to each of them. DSOs (with measurement out- function description is somehow incomplete (and could become comes that are completely certain) are dual to the “completely “complete” with the addition of a classical-like reality, such as uncertain operators” with measurement outcomes that are com- with a hidden variable theory). Nor do we oppose the con- pletely uncertain. Complete uncertainty means that they sat- sequence of the Pusey–Barrett–Rudolph (PBR) theorem (15), isfy the condition that all of their possible measurement out- which states that the wave function is determined uniquely by comes are equiprobable (17). Thus, no information can be the physical state of the system. We only mean to suggest that the gained by measuring them. Mathematically, the two limiting wave function cannot constitute the primitive ontology of a single cases represented by Eq. 2 are given by deterministic operators quantum particle/system. That being said and contrary to ensem- for which ∆A|ψ⊥i = 0 and completely uncertain operators for ble interpretation advocates, we will not duck out of propos- which hAˆi = 0 (a necessary but insufficient condition as will be ing a single-particle ontology. In what follows, we expound such described below). an ontology based on deterministic operators, which are unique An important ingredient to consider of our proposed inter- operators with measurement that can be carried out on a single pretation is a final state of the system. The idea that a complete particle without disturbing it and with predictable, definite out- description of a quantum system at a given time must take into comes. Because properties corresponding to these operators can account two boundary conditions rather than one is known from be directly verified at the single-particle level, they constitute the the two-state vector formalism (TSVF). This approach has its
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The overlap). performed only they measurements therefore, (until is, from packets wave deduced individual be the on cannot it observ- is but difference would phase able, the this mea- Only because (17). particle, principle the symmetry a gauge in measure for violate cannot phases relative can local we individual the we mechanics, the Although is sure classical packets. located in maxima wave phase be the two local will where the interference of us the phase tells How- of slits. what minima two mechanics, the and quantum through going in packets tra- finally ever, wave the wave along the the can available of of information one jectories local parts theory, entirely two on wave the based when classical meet happen in will example, what For predict the analogy. into breakdowns the important explanation in are classical there nevertheless, accepted domain, indeed the quantum is extend it Although to meet other pattern. later tempting the interference parts familiar while two the the slit before create first slit, to the second the through grat- through goes the goes which traverses part of (function) part wave one spatial ing, a clas- quantum: the for and domains, explanation both sical across the shared that is taught phenomena results double-slit are interference our We although a general). hereinafter, completely in to are referred analyzed be will conveniently which setup, (most grat- quantum experiments and classical ing both in appear patterns Interference Behavior Wave-Like and Dynamics Nonlocal quantum a sec- of ontology a particles. This primal individual state. adding the for initial for mechanics to the basis by the equivalent dictated forms one is set the twofold to state addition final in within DSO a propose ond we adding that pro- article, a framework this using should the state one In certain a system, measurement. postselect jective a also, but specify preselect hence, fully only are, not to and Accordingly, times real. intermediate deter- at equally in state roles equal quantum a play the motivates past mining It and future analysis). is, which of (that to tool according formalism mathematical view a the just mea- of than weak reading more using as literal by a system postselected motivates power and surements The pre- 21). dur- the (20, system explore it the to disturbing of without state times the Weak intermediate explore ing ensembles. to us postselected enable and measurements pre- involving experiments been since has numerous it of but discovery (17). the (18), phenomena to al. interesting led et and (19) Aharonov developed of extensively work the in roots i φ e hrfr,cnie h state the consider therefore, We, analyzing for platform useful extremely an provides TSVF The ψ dne itr n pl hs prtr,w ilsethat see will we operators, these apply and picture odinger ¨ 2 (x , t hc nteSchr the in which ), A ˆ f ˆ (x ψ φ , nadto,w s hc prtr are operators which ask we addition, In . p = ) X ti o ifiutt hwta,if that, show to difficult not is It φ. dne itr,rpeet the represents picture, odinger ¨ a mn x m p n ψ , φ (x , t = ) ψ 1 (x f ˆ (x , t , + ) p ) e snwcnie prtr fteform the equation, of Heisenberg the operators consider (where now us Let atclrcs,w banannoa qaino motion: of equation nonlocal a obtain we case, particular where Z of If values (16). all slit) double the through going of overlap Theorem. theorem. following the prove to easy Indeed, is phenomena. it interference quantum dynamical describe appropriate to most the variables not are momentum and position opeeyuncertain. completely given a measure P to probability of the value because effect, in shift observable the circumstances, these (i.e., momentum modular the is interest of observable the is slit double the of effect the for accounts not that variable the that remote the also, of value the is, (that h ntcrl 1) If (17). Variables). circle unit Modular the for Principle Let Uncertainty (Complete Theorem variable.” uncertain “completely a condition value given a find to probability the of vari- that modular means the by uncertainty” given of equiv- is space is exchange the slit) by local a it in Denote opening rotation (22). of nonlocal able particular (i.e., a potential particle a to a the at introducing alent from particle of distance the effect a The at detect) uncertain. to maximally failing slit, (or detecting on nev- slit. it other but the slits, to regarding the information us of nonlocal allow one has nonlocal only thus ertheless through The picture going picture. Heisenberg particle a Heisenberg the consider in the motion in of latter equations the taking of former place the of the possibility the the suggests via momentum phase modular fun- relative and are dynamics they nonlocal The between brackets, dynamics. connection nonlocal Poisson entail of they because terms different, in damentally limit commutators classical although that, a classical understand have thus the We particle. that force the local on suggesting a if derivative, only changes local momentum modular a involves which vlto hti ie ytePisnbracket: Poisson the by given is that evolution where (θ nieodnr oetm oua oetmbecomes, momentum modular momentum, ordinary Unlike [ψ θ p 1 Φ α ∗ = ) sidpnetof independent is ∂ u t oua eso.Ide,because Indeed, version. modular its but (x H eaproi ucin hc suiomydsrbtdon distributed uniformly is which function, periodic a be p f ˆ L 0 (x , ∂ P = θ 2 = t stedsac ewe h lt) vligti through this Evolving slits). the between distance the is Let t say , )x (θ , dt d p ψ p t 1 2 m π 1 ) e /2m ψ n hie fphases of choices and (x + .Eq. ~/L). p i [e = 2π φ p θ n e x , 0 (x δθ 1 ψ 0) p ipL + ilb h aebfr n fe h shift, the after and before same the be will , ~ α + , 1 = ipL .Ti prtrlasu aual orealize to naturally us leads operator This L). (x t esalcl aibeta aifisthis satisfies that variable a call shall We ). ~ f ˆ and ˙ V = = ) i n he eed nteptnila o only not at potential the on depends , , ~ p (x V t e e θ mod θ ∂ in 4 ) ψ i ψ i ∈ f ˆ ) ( (x 2π − Φ p [i.e., ifr osdrbyfo h classical the from considerably differs 1 p (x 2 0 prpit o h obesi.I this In slit. double the for appropriate ∂ δθ [0, +2π (x i p (x ]= )] := ψ t 0 = , , L , β ∗ p , H (i.e., 2π ˆ k t 0) (x P p ) ~ + ) o ) ups h muto non- of amount the Suppose ). o n integer any for ~ 1 mod (θ [ = , θ ~ L (t = t [V = ) θ , )x to β α e f 0 = ˆ → −i i (x m , m φ p θ H constant ψ ˆ 0 , p θ , 2π + p , + n 2 swe h atceis particle the when is ], n k NSEryEdition Early PNAS 0 + (x ψ L) r nees hnfor then integers, are ∈ δθ dV dx β δθ , (x t Z, − n sueno assume and ), ilitoueno introduce will .Nw“maximal Now ). e f ˆ , 1 = dV V n t (x i 2π )] p 0 = 6 (x 0 , /2π p /dx dx p )]e , := ) then , 0 = .Under ]. sacting is ipL ~ | e x . ipL/~ f6 of 3 Φ but [4] [5] [3] is
PHYSICS When a particle is localized to within |x| < L/2, the expecta- the operator. Given that x(t) = x(0) + p(0)t/m, we have tion value of eipL/~ vanishes. This result is obvious, because eipL/~ ik(x(0)+p(0) t ) functions as a translation operator, shifting the wave packet out- eikx(t) = e m . |x| < L/2 side (i.e., outside its region of support). Accordingly, If we set α = k and β = kt/m, we see that, as time t changes, when a particle is localized near one of the slits, as in the case inpL/ of either ψ1 or ψ2, then he ~i = 0 for every n. It then fol- ei(αx(0)+βp(0)) lows from the complete uncertainty principle that the modular momentum is completely uncertain. Accordingly, all information assumes all of the possible values. Hence, nonlocal operators at about the modular momentum is lost after we find the position of t = 0 can be measured locally at some later time. The following the particle. This onset of complete uncertainty is crucial to pre- theorem shows that this description is exhaustive. vent signaling and preserve causality. As an example, suppose we 2 apply a force arbitrarily far away from a localized wave packet. Theorem. The collection for all (α, β) ∈ R , We thus change operators depending on the modular momen- Z tum instantly, because modular momentum relates remote points f (α, β) = ψ∗(x)ei(αx+βp)ψ(x) dx, in space. If we could measure this change on the wave packet, R then we could violate causality, but all such measurements are uniquely determines the state ψ. precluded by the complete uncertainty principle. The fact that the modular momentum becomes uncertain on Proof. First we multiply both sides by e−iαβ~/2. Integration with localization of the particle also fits well with the fact that inter- respect to α lets us find ψ∗(0)ψ(β) for all β. This expression ference is lost with localization. In the Schrodinger¨ picture, inter- amounts to finding ψ(x) when setting ψ(0) = 1. ference loss is understood as a consequence of wave function col- lapse. After the superposition is reduced, there is nothing left Double-Slit Experiment Revisited for the remaining localized wave packet to interfere with. The Performing certain experiments involving postselection allows us Heisenberg picture, however, offers a different explanation for to both measure interference and deduce which-path informa- the loss of interference that is not in the language of collapse: if tion. However, the Schrodinger¨ picture is very awkward with such one of the slits is closed by the experimenter, a nonlocal exchange experiments, which posit both wave and particle properties at the of modular momentum with the particle occurs. Consequently, same time. Alternatively, in the Heisenberg picture, the particle the modular momentum becomes completely uncertain, thereby has both a definite location and a nonlocal modular momentum erasing interference and destroying the information about the that can “sense” the presence of the other slit and therefore, cre- relative phase. ate interference. This description thus evades difficulties present Note also that, because p = pmod + N ~/L for some integer N , in the Schrodinger¨ picture. the uncertainty of p is greater than or equal to that of pmod (the To emphasize this point, let us consider a simple 1D integer part can be uncertain as well). For this reason, a complete Gedanken experiment to mimic the double-slit experiment. In uncertainty of the modular momentum pmod [which means that the Schrodinger¨ picture, a particle is prepared in a superposition its distribution function is uniform in the interval [0, ~/L)] sets of two identical spatially separated wave packets moving toward ~/L as a lower bound for the uncertainty in p (i.e., ∆p ≥ ~/L). one another with equal velocity (Fig. 1): This inequality parallels the Heisenberg uncertainty principle,
equating it in the case of ∆x = L. ip x −ip x 1 0 x + L iφ 0 x−L At first blush, it seems that, as axioms, dynamical nonlocality Ψi (x, t=0)= √ e ~ Ψ + e e ~ Ψ , 2 2 and relativistic causality nearly contradict each other. Neverthe- 2 less, by prohibiting the detection of nonlocal action, complete [8] uncertainty enables one to reconcile nonlocality with relativistic where Ψ(x) is a Gaussian wave function. To simplify, we assume causality, so that they may “peacefully coexist.” This reconcilia- that the spread ∆x obeys ~/p0 ∆x L; hence, the wave tion is why we regard this principle as very fundamental. packet approximately maintains its shape up to the time of Measuring Nonlocal Operators encounter (our results are, however, general). The relative phase φ has no effect on the local density ρ(x) or any other local fea- Consider a system described at time t = 0 by a vector |ψi in a ture until the two wave packets overlap. The phase φ manifests Hilbert space. Fundamental properties of operator-valued func- itself by shifting the interference pattern by δ = ~φ/p0. tions allow us to reconstruct |ψi using weak measurements of the position of the particle at various instants t. Indeed, if we call ρ(x, t) the density of ψ(x, t), namely ρ(x, t) = ψ∗(x, t)ψ(x, t) then we can calculate its Fourier transform: Z Fρ(k, t) = ψ∗(x, t)ψ(x, t)eikx dx. [6] R For a given operator Aˆ, we can write its expectation value as
Aˆ x (t) = hψ(x, t)|Aˆ|ψ(x, t)i. [7] Therefore, Eq. 6 is nothing but the expectation value of eikx . Note that, in Eq. 7, we have been using the Schrodinger¨ pic- ture with a time-evolving state ψ(x, t). Rewriting Eq. 7 in the Heisenberg picture, hψ(x, t)|Aˆ|ψ(x, t)i = hψ(x, 0)|Aˆ(t)|ψ(x, 0)i. Fig. 1. Interference of two wave packets. (A) The density of the initial superposition (8) of the two wave packets. (B) The interference pattern at We know that the two pictures are equivalent: the time evolution the time T when the wave packets completely overlap. The shift δ of the has simply been moved from the vector in the Hilbert space to interference pattern is proportional to the relative phase φ.
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T T = ) )dt hwteuulitreec atr,dsietefact the despite pattern, interference usual the show 2 = ) ≈ = Ψ hx |Ψ(x A g 4|Ψ t f | sgvnby given is , hΨ (x Ψ n oteetdin postselected and ssfcetysaldrn h measure- the during small sufficiently is H f = ) i int f )| ihΨ (x |Ψ 2 M = )| e e i i i i ip 2 g ( |x i cos p 0 (t 0 x usoe nesml fpar- of ensemble an over sums i 1 x /~ )q /~−φ/2) 2 hΨ projections (p Ψ(x P hsGdne exper- Gedanken This φ. 0 f x i M (t e scnie an consider us let 1/2, /~ − ekmaueet in measurements Weak C. | )| Π cos(p lhuhthe Although L/2). − i Ψ (x φ/2) i where ), (t 0 Π x ihnthe Within )i. i /~ , (x ) − ihthe with φ/2). q sthe is [10] [9] T s e f1 Ss ti motn ont httemeasure- out the carried that possibly note (26), to nonlocal important are is would operators It we such DSOs. space), of 10 ments Hilbert two of of 4D set system a a a use (in describing particles within For spin-1/2 described DSOs. is entangled two particle) using spin-1/2 formalism a (e.g., our space Hilbert entan- 2D in from arising in nonlocality noted As kinematic glement. of notion the address the over helpful advantage computational very a a has it also Schr (25), is cases view Moreover, several (24). in of computation quantum point discussing for foundational framework a from here interpretation. dis- an judge new to of metric ultimate stimulation dis- the The to is Y.A. effect. coveries led Aharonov–Bohm have the the ontology Importantly, cover this not are. to are themselves pertaining functions DSOs considerations wave objects—only DSOs, physical for of real useful are although dynamics Indeed, that the ontology. potentials an calculating fix from to sufficient which derived a by not criterion efficient, fields is usefulness mathematically the mathematical Hence, also real. only physically is is functions it potential although of However, statistics. use mathe- experimental efficient the of an calculations for is tool function matical wave post- The and experiments. pre- the selection with inconsistent but analysis mathematical think- for to Schr restricted the of longer terms no in is ing one it, hidden Internalizing a intuition. cal as times. earlier regarded in be inaccessibility may epistemic its which of DSO, because final variables a this constitute understand to now thereby We state problem. generalization but measurement cosmological measurements introduced the This specific had solves for. of we consider- accounted outcomes that be by the kind can that, universe, the (23) entire of the elsewhere state for shown final inclusion was special the It a by ing state. met final is a demand demand of This empirical outcomes. for mechanism those the a choosing with necessitate outcomes combined measurement definite uncertainty for turn, fundamen- In more as reason, tal. uncertainty principle this uncertainty Heisenberg For complete the principle. from regard uncertainty we i.e., complete the around, not to way does principle other implication the the principle, work uncertainty implies Heisenberg (qualitatively) the principle uncertainty dynamics. metaphysi- complete the between this of math- While reconciler nonlocality a the a as desiderata—causality—and as not cal but appears clas- consequence principle no uncertainty ematical An have Schr ensemble. that the an whereas properties analog, pos- nonlocal particles sical yet individual deterministic, interpretation, this sess In quan- for mechanics. interpretation tum Heisenberg-based a elaborated have we Schr the After the of Discussion functions simple are which DSOs, (9). infor- momentum of nonlocal modular time, form same the the in at and mation position definite a both inconsis- has is fact, in and view. necessary time-symmetric not a is with here, this tent and origi- that phenomena, shown was interference have nature for we wave-like account a to inter- having devised the as that nally particle Recall slits. the the the of that of pretation fact one the around the despite localized through present, is still went particle thus that position is packet Interference determinate slit. wave a left right-moving has a particle post- by the the described that of virtue know by we However, selection, interference measured. an weakly exhibits when two-state, pattern the of evolution the describes Schr the o h aeo opeees tmgtb neetn obriefly to interesting be might it completeness, of sake the For discussed was that representation Heisenberg the Intriguingly, physi- powerful a conveys interpretation this that contend We particle each that us tells picture Heisenberg the contrast, In dne representation. odinger ¨ dne itr.Tera ato hsdniy which density, this of part real The picture. odinger ¨ dne itr a oiae o ayyears, many for dominated has picture odinger ¨ unu system quantum a Ontology, and Formalism dne itr,wihi ovnettool convenient a is which picture, odinger ¨ dne aecnol describe only can wave odinger ¨ NSEryEdition Early PNAS | f6 of 5
PHYSICS in space-like separated points. Most of these operators involve function description less intuitive. However, those Hamiltonians simultaneous measurements of the two particles. A (nondeter- were dismissed as nonphysical in the wake of relativity theory, ministic) measurement of one particle would change the com- allowing the wave function ontology to prosper. We hope that bined DSOs, thus instantaneously affecting also the ontological our endorsement of the Heisenberg-based ontology will promote description of the second particle. In ref. 11, it was claimed that a discussion of this somewhat neglected approach. the information flow in the Heisenberg representation is local; however, in light of the above analysis, this analysis only refers to ACKNOWLEDGMENTS. Y.A., D.C.S., and J.T. acknowledge support, in part, certain kinds of operators. from the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust. Y.A. We believe that, if quantum mechanics were discovered before acknowledges support from Israel Science Foundation Grant 1311/14, Israeli Centers of Research Excellence “Circle of Light,” and the German–Israeli relativity theory, then our proposed ontology could have been the Project Cooperation (DIP). E.C. was supported by the European Research commonplace one. Before the 20th century, physicists and math- Council Advanced Grant Nonlocality in Space and Time. Funding for this ematicians were interested in studying various Hamiltonians hav- research was provided by the Institute for Quantum Studies at Chapman ing an arbitrary dependence on the momentum, such as cos(p). University. This research was also supported, in part, by the Perimeter Insti- tute for Theoretical Physics. Research at the Perimeter Institute for Theoret- In quantum mechanics, these Hamiltonians lead to nonlocal ical Physics is supported by the Government of Canada through the Depart- effects as discussed above. The probability current is not contin- ment of Innovation, Science and Economic Development and the Province uous under the resulting time evolution, which makes the wave of Ontario through the Ministry of Research and Innovation.
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