Nonlocality and local causality in the Schrödinger Equation with time-dependent boundary conditions
NNonlocality and local causality in the Schr¨odinger Equation withtime-dependent boundary conditions
A. Matzkin
Laboratoire de Physique Th´eorique et Mod´elisation (CNRS Unit´e 8089),Universit´e de Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France
S. V. Mousavi
Department of Physics, University of Qom,Ghadir Blvd., Qom 371614-6611, Iran
M. Waegell
Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA
Abstract
We investigate the nonlocal dynamics of a single particle placed in an infinite well with movingwalls. It is shown that in this situation, the Schr¨odinger equation (SE) violates local causalityby causing instantaneous changes in the probability current everywhere inside the well. Thisviolation is formalized by designing a gedanken faster-than-light communication device which usesan ensemble of long narrow cavities and weak measurements to resolve the weak value of themomentum far away from the movable wall. Our system is free from the usual features causingnonphysical violations of local causality when using the (nonrelativistic) SE, such as instantaneouschanges in potentials or states involving arbitraily high energies or velocities. We explore in detailseveral possible artifacts that could account for the failure of the SE to respect local causality forsystems involving time-dependent boundary conditions. a r X i v : . [ qu a n t - ph ] O c t . INTRODUCTION Nonlocality is the hallmark of quantum mechanics. It is generally taken for granted thatnonlocality requires two or more particles, along the lines of the early paper by EinsteinPodolsky and Rosen [1], subsequently put into a firm footing by Bell [2]. Although ithas been suggested that a single particle could in some instances exhibit nonlocality, suchresults have been disputed. This is particularly the case of the two main candidates forsingle particle nonlocality, the Aharonov-Bohm effect [3] and the entanglement betweenspatial modes of a single photon (see [4] and Refs. therein for previous works), discussedrespectively in Refs. [5, 6] and [7, 10].The present work introduces a new “candidate” for single particle nonlocality. It is basedon the fact that the Schr¨odinger equation solved on a domain with moving boundaries givesrise to apparent violations of local causality. It appears that time-dependent boundary con-ditions can potentially induce a nonlocal change in a region located far from the locationof the moving boundary. Here we will examine the case of a particle in a box with in-finitely high but moving walls. We will see that for quantum states extended all over thebox, the moving walls generate instantaneously a current density almost everywhere in thebox. We will indicate how this effect could be in principle tested, namely by making weakmeasurements of the particle momentum in the central region of the box before light has thetime to propagate from the walls to that region. To this effect, a gedanken faster-than-lightcommunication device will be presented.Let us state right away that we are not advocating the position that it is possible tosend a signal faster than the speed of light. Nevertheless, the present problem is interestingbecause the non-relativistic Schr¨odinger equation fails to prevent superluminal signaling ina situation where relativistic considerations do not seem to play a significant role. It isindeed well-known that the Schr¨odinger equation does not bound particle velocities, nordoes it constrain instantaneous changes in potentials, but we will argue that in our systemthe nonlocal aspects do not rely on spurious violations of special relativity allowed by aemploying a nonrelativistic framework.Note that the effect reported in this work is not due to a non-dynamical phase term,such as a geometric phase (in which case we would have in the present context a non-adiabatic, non cyclic geometric phase [8, 9]). There have been in the past claims that such2on-dynamical phases in the same type of system that we will be investigating in this workcould be envisaged as a specific form of “hidden” (i.e., non-signaling) non-locality [11–13].We will see instead that the non-local aspect in our candidate system is not based on theexistence of such phases.We will start by revisiting the treatment of systems with time-dependent boundary con-ditions of the form ψ ( x ( t ) , t ) = 0, where ψ is the wavefunction. Such systems are delicate tohandle because from a formal point of view a different Hilbert space needs to be defined foreach time t , so that a simple operation like taking the time derivative ∂ t ψ is not straightfor-ward. We will introduce the system we will deal with – a particle in an expanding infinitewell – in the context of recent works [14–16] involving time dependent boundary conditionsin Sec. 2.Weak measurements were originally [17] introduced to measure an observable without sig-nificantly disturbing the system, allowing a subsequent standard (projective) measurementof a different observable. The outcome, known as a weak value, is not generally an eigenvalue(since the quantum state of the system is barely modified and no projection takes place) butstill gives some information on the weakly measured observable, provided enough statisticsare gathered by repeating the experience a certain number of times. In particular, it wasshown [18] that the weak value of the momentum is directly related to the current density.We will recall these facts in Sec. III where we will present our main results concerning theinstantaneous response of the current density to a change in the boundary conditions.We will then proceed (Sec. 4) to analyze and discuss this novel type of nonlocality.The first issue we will address is no-signaling. No-signaling stands as the major constraintpermitting the “peaceful coexistence” [19] of relativity and quantum mechanics. At firstsight it would appear that no-signaling is respected here, since a single weak measurementdoes not convey any information, but the situation is more involved, and a protocol thatwould allow us to test in principle the possibility of signaling will be presented. Giventhat this nonlocal effect appears to conflict with the no-signaling principle, we will criticallyassess the origins of nonlocality, in search of possible artifacts. We will then discuss thepresent results in the framework of the Bohmian model, where nonlocality is a built-infeature claimed to hold for individual events but is washed out at the statistical level. Asummary and our conclusions will be given in Sec. V.3 I. A PARTICLE IN AN INFINITE WELL WITH MOVING WALLS
The particle in an infinite well with moving walls was widely investigated in the contextof quantum chaos (see e.g. [20–23]). Another line of studies concerning this system involvesthe conjecture of nonlocality induced by the moving wall on a localized state [12, 24–29],that was recently disproved [16]. The Hamiltonian for a particle of mass m in an infinitewell with the left wall fixed at x = 0 and the right wall moving according to the function L ( t ) is given by H = P m + V (1) V ( x ) = ≤ x ≤ L ( t )+ ∞ otherwise. . (2)The solutions of the Schr¨odinger equation i (cid:126) ∂ t ψ ( x, t ) = Hψ ( x, t ) must obey the boundaryconditions ψ (0 , t ) = ψ ( L ( t ) , t ) = 0. The instantaneous eigenstates of H , φ n ( x, t ) = (cid:112) /L ( t ) sin [ nπx/L ( t )] (3)verify H | φ n (cid:105) = E n ( t ) | φ n (cid:105) where E n ( t ) = n (cid:126) π / mL ( t ) are the instantaneous eigenvalues,but, due to the time varying boundary conditions, the φ n are not solutions of the Schr¨odingerequation. To solve the Schr¨odinger equation different approaches have been proposed, likeintroducing a covariant time derivative [30], implementing an ad-hoc change of variables[31], or relying on a time-dependent quantum canonical transformation [14, 32]. Here wefollow the latter option, as implemented in Ref. [16]. However, rather than going throughthe transformation to derive the solutions for the general case (this is done in [16]), we willchoose from the beginning a specific function L ( t ) for which analytic basis solutions of theSchr¨odinger equation are known. Indeed, for the linearly expanding case L ( t ) = L + qt (4)it can be checked by inspection [31] that ψ n ( x, t ) = (cid:114) L + qt exp (cid:18) − iπ (cid:126) n t − iL m qx (cid:126) mL ( L + qt ) (cid:19) sin (cid:18) nπxL + qt (cid:19) (5)verifies the Schr¨odinger equation and the boundary conditions ψ (0 , t ) = ψ ( L ( t ) , t ) = 0.Here, q > ψ n ( x, t ) (with n a positive integer) form a set of orthogonal basis functionsuseful to determine the time evolution of an initial arbitrary quantum state. The simplestinitial state would be to pick a given ψ n ( x, t = 0); its evolution follows directly from Eq.(5). From a physical standpoint, it would be more realistic to start from the standard fixedwall eigenfunctions. A typical initial state woud then be an eigenstate φ n ( x, t = 0) [see Eq.(3)] or a linear combination thereof, say ψ ( x, t = 0) = ∞ (cid:88) n =1 c n φ n ( x, t = 0) (6)whose evolution is given by ψ ( x, t ) = (cid:88) k,n c n (cid:104) ψ k ( t = 0) | φ n ( t = 0) (cid:105) ψ k ( x, t ) . (7)We may want to include additional refinements, like allowing for a continuous transitionfrom the fixed walls to the linear regime by setting L ( t ) = L + qt (1 − e − γt ) . (8)This requires numerical solutions. The numerical method that will be used here is verysimilar to the one exposed in Ref. [22]; it is based on looking for numerical solutions ζ ( x, t )by using expansions over the instantaneous eigenstates of the form ζ ( x, t ) = ∞ (cid:88) k =1 a k ( t ) φ k ( x, t ) . (9)The coefficients a k ( t ) are retrieved by solving a system (arising by plugging ζ ( x, t ) in theSchr¨odinger equation) of coupled differential equations. III. CURRENT DENSITY AND MOMENTUM WEAK VALUESA. Current density evolution
We first briefly look at the standard current density j = 12 m ( ψ ∗ P ψ − ψP ψ ∗ ) , (10)where P is the momentum operator for the states in an expanding infinite well. Whenthe initial state is taken to be an eigenstate φ n ( x,
0) of the fixed walls well, given by Eq.53) with all the c k vanishing except for c n = 1, the current density is initially zero, butbecomes non-zero for t >
0. Indeed, near the wall, the initial wavefunction is nonzero, since φ n ( x ≈ L , t = 0) (cid:39) n ( L − x ), and is substantially modified when the wall moves. By thearguments given in [16] (or simply by the continuity of the logarithmic derivative noting thatthe potential remains unchanged except at x = L ( t )) we then know that at an infinitesimaltime t = ε we will have ψ n ( x, ε ) − ψ n ( x, (cid:54) = 0 at any x, although we expect this quantityto be large near x = L ( ε ) and smaller in the regions away from the moving wall.When the initial state is taken to be a basis state ψ n given by Eq. (5), the current densityis immediately computed as j ψ n ( x, t ) = 2 qx sin nπxL ( t ) L ( t ) . (11)We see that j ψ n ( x, t ) changes continuously both in the space and time variables. The changein the current density at x (cid:28) L can easily be computed. From∆ j ( x ) ≡ j ( x, ε ) − j ( x, , (12)we have ∆ j ψ n ( x ) (cid:39) x (cid:28) L π n qx (cid:18) L + qε ) − L (cid:19) (cid:39) ε → − π n q x L ε. (13)Let t S = ( L − x ) /c be the time it takes for a light signal emitted at the wall to reach thepoint x where the current density is monitored ( c is the light velocity). Then there is a rangeof times ε such that ε < t S and ∆ j ( x ) (cid:54) = 0: the current density is modified instantaneouslyby the wall’s motion. The significance of this instantaneous appearance of a current densitywill be discussed in Sec. IV. We next examine how this current density could in principlebe experimentally tested. B. Weak measurements and the current density
The underlying idea at the basis of the weak measurement (WM) framework [17] is togive an answer to the question:“ what is the value of a property (represented by an observable A ) of a quantum system while it is evolving from an initial state | ψ ( t i ) (cid:105) to a final state | b f ( t f ) (cid:105) ? ”. This is done by coupling the system observable A to a dynamical variable of anexternal pointer, say Q through an interaction Hamiltonian of the form H int = g ( t ) A Q , (14)6here g ( t ) is a smooth function nonzero during the interaction time. The effective couplingconstant g ≡ (cid:82) g ( t ) dt is chosen to be very small so that although the system and the externalpointer become entangled, the system state is minimally disturbed by H int . A standardmeasurement of another system observable, say B can then be undertaken. Assume that theeigenvalue b f corresponding to the eigenstate | b f (cid:105) is obtained – a step called postselection.It can then be shown ([17]; see e.g. Sec. II of [33] for a brief derivation) that the externalpointer that was coupled to A has shifted by the quantity gA w where A w = (cid:104) b f | A | ψ ( t i ) (cid:105)(cid:104) b f | ψ ( t i ) (cid:105) (15)is known as the weak value of A given the initial (preselected) state | ψ ( t i ) (cid:105) and the final(postselected) state | b f (cid:105) . Note that while A w is generally a complex quantity, when oneweakly measures observable A , the shift of the external pointer is proportional to the realpart of A w . Let us now specialize Eq. (15) to a weak measurement of the momentum P immediatelyfollowed by a standard measurement of the position, denoting the outcome by x . The weakvalue is then given by P w = (cid:104) x | P | ψ (cid:105)(cid:104) x | ψ (cid:105) . It is easy to see that P w can be written as [18, 34, 35] P w = mj ψ ( x, t ) | ψ ( x, t ) | − i (cid:126) ∂ x (cid:0) | ψ ( x, t ) | (cid:1) | ψ ( x, t ) | . (17)Hence the real part of the momentum weak value is the hydrodynamic velocity (well knownfrom the Bohmian model, see Sec. IV D below) v ( x, t ) given by v ( x, t ) ≡ j ψ ( x, t ) | ψ ( x, t ) | = Re P w m . (18)Our statement made above on the superluminal change in the current density following thewalls’ motion has now been couched in terms of an experimentally measurable quantity, themomentum weak value. For simplicity we have disregarded in Eq. (15) the evolution of the system between the initial preparationtime t i , the mean interaction time t w and the postselection time t f ; otherwise Eq. (15) should be replacedby A w = (cid:104) b f | U ( t f , t w ) AU ( t w , t i ) | ψ ( t i ) (cid:105)(cid:104) b f | U ( t f , t i ) | ψ ( t i ) (cid:105) (16)(see e.g. Sec. II of [33]). The imaginary part of A w is proportional to the shift of the momentum of the pointer wavefunction —for a pointer in position space. P w is shown as a functionof time. Fig. 1 shows the case of a moving wall when the system is initially prepared in agiven eigenstate φ n of the cavity at t = 0. Fig 2 shows instead the evolution of Re P w in astatic cavity when the system is initially prepared in a basis state ψ n ( x, t = 0) [Eq. (5)]. Inthe former case as noted above we will have a nonzero current density (whereas j ( x, t ) = 0for any t if the walls had remained fixed). We see indeed in Fig. 1 that Re P w changesbefore a light signal reaches the point where the weak measurement takes place; the lightcone boundary t c = ( L − x ) /c is indicated by the red-gridded plane. This is the signatureof a form of nonlocality induced by the walls’ motion.In the latter case Eqs. (11) and (17) imply that if the initial state is ψ n ( x, t = 0), thenin a moving cavity the weak value should evolve followingRe P w ( x, t ) = mqxL ( t ) . (19)This is represented in Fig. 2 by the solid black line. In a fixed cavity instead Re P w will wildlyoscillate, as shown by the blue curves in Fig. 2. Here again the behavior of a distant wall(remaining static or in motion) affects the weak value of the momentum instantaneously, i.e.before a light signal emanating from the wall reaches the point where the weak measurementis made (the light cone boundary appears as the vertical dotted line in Fig. 2). C. Weak Measurement Protocol
We now introduce the type of protocol that could in principle lead to the measurementof Re P w . An infinite square well is realized by a long and narrow cavity. The particle (sayan electron) is prepared in an initial state (a given eigenstate of the cavity at t = 0 or agiven basis function, as in Figs. 1 and 2 respectively).At time t = 0, the sender Alice is located at x = L and chooses whether to set the wallin motion (indicating a message bit 1), or to leave the wall at rest (indicating message bit0). Then, a very short time later at t , Bob, located on the opposite side of the cavity nearthe wall at x = 0 performs a weak measurement of the position of the particle at location x = x w , followed by a strong measurement of the position at x f (in the immediate vicinityof x w ) of the particle at t f , again a very short time later. This procedure is equivalent tomeasuring the weak value of the momentum. Indeed, it can be shown (see Appendix) that8
0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 0 0.0005 0.001 0.0015 0.002 | R e P w ( x , t ) | x t | R e P w ( x , t ) | FIG. 1: Time evolution of the weak value of the momentum of an electron in an expanding cavityof initial length L = 100 au . The blue curves represent | Re P w ( x, t ) | (where P w is the momentumweak value, see Eq. (17)) obtained by making a weak measurement at the corresponding value of x . The wall is initially at L and the red-gridded plane represents the boundary of the light coneoriginating from x = L at t = 0. Initially Re P w ( x, t = 0) = 0 everywhere (and would stay assuch for a static wall) but is seen to oscillate before the light cone reaches the points were P w isdetermined (the light blue curves represent P w for times inside the light cone). The dashed lineto the right schematically represents the walls motion given by L ( t ). The following parametershave been used: q = 0 . m = 1, initial wavefunction chosen to be an eigenstate of the static well φ n ( x,
0) [see Eq. (3)] with n = 11 (numbers given in atomic units (au)). with the weak value of the position given by X w = (cid:104) x f | U ( t f , t w ) X | ψ ( t w ) (cid:105)(cid:104) x f | ψ ( t f ) (cid:105) (20)where U ( t f , t w ) is the evolution operator between t w and t f , we have P w = lim t f → t w mt f − t w ( x f − X w ) . (21)9 IG. 2: Time evolution of the weak value Re P w ( x f , t ) of the momentum for x f lying near theorigin (i) in an expanding cavity (solid black line) (ii) in a static cavity (blue and light blue lines).The red-dashed line is the border of the light cone (the values of Re P w ( x f , t ) inside the light coneare in light blue). The fact that the wall moves or remains fixed is instantaneously reflected inthe behavior of the weak value. The parameters used are L = 100, x f = 2 . m = 1, initialwavefunction chosen to be a basis function ψ n ( x,
0) [see Eq. (5)] with n = 44 and in the movingcase q = 0 . Note that the protocol can also be implemented with a direct weak measurement of themomentum, rather than the two position measurements leading to Eq. (21). This type ofweak measurement of the momentum relies on a particular coupling between the particlemomentum and an external pointer (in practice, the external pointer is often another degreeof freedom of the particle). The important point is that Bob carries out the weak measure-ment procedure before a light signal sent by Alice at t = 0 reaches him, that is we musthave t f < L /c .Bob can measure whether Alice sent a bit 0 or a bit 1. When the initial state is a stationarystate φ n ( x,
0) of the cavity P w = 0 if the wall remains fixed, but takes a nonzero value (asin Fig. 1) if the wall was set in motion. When the intial state is a basis state ψ n ( x, P w is given by Eq. (19) if the wall moves, so Bob can verify by making successive weakmeasurements on the system if Re P w is consistent or departs from Eq. (19), as displayedin Fig. 2.Nevertheless, weak measurements are noisy, and it is impossible for Bob to learn Alice’schoice of signal bit in a single run of the experiment. First the postselection probability10 x w + (cid:15)x w − (cid:15) | ψ ( x, t f ) | dx is very small ( (cid:15) is the width over which the weak measurement takesplace). Second, by definition, when weakly measuring observable X , the pointer wave-function incurs small shifts [see Eq. (14)] relative to its width, so many runs of the sameexperiment will be necessary in order to extract the weak values. Third even in a weakmeasurement there is inevitably a back action of the coupling interaction on the subsequentevolution leading to the post-selection. The universal part of the back action is encoded inthe imaginary part of the weak value [36]. A large imaginary part will distort the externalpointer state and make even more difficult to extract the small shift of the pointer wave-function. We note here that for an initial state of the form given by Eq. (5), for which wecomputed Re P w ( x, t ) = mqx/L ( t ) [Eq. (19)], we haveIm P w = − (cid:126) πnL ( t ) cot (cid:18) nπxL ( t ) (cid:19) . (22)The real and imaginary parts of P w follow a different behavior, and Eq. (22) gives anindication of initial states and spatial regions minimizing the back action.However, unlike many other cases where quantum uncertainty prevents superluminalsignaling, we will see that this setup has no such limitation. This is a serious problemthat calls for a more detailed discussion of non-locality and no-signaling, and compels us toexplore possible artifacts of the model. IV. DISCUSSIONA. Relativity and non-locality
The difficulties in reconciling quantum mechanics and special relativity are well-known[37]. These difficulties stem from the global character of the state vector, defined in a mathe-matical configuration space and not in physical space. It is generally accepted that quantumcorrelations cannot lead to superluminal communication of information (no-signaling) as thiswould indeed result in an open conflict with relativity. Instead, a “peaceful coexistence” [19]between quantum mechanics and relativistic constraints is advocated: as long as one doesnot attempt to understand how the quantum correlations come about (in particular througha hidden-variable model or by endowing the state vector with physical reality), the observedstatistics predicted by quantum theory respect no-signaling. However if the state vector is11 = 0 x = L
Alice q Bob At t = 0 , Alice sets the wall speed q . q = 0q = v Message Bit SettingAt S , Bob performs measurements of the weak value of the momentum P w at x = x f . x = x f P w = p P w = p Message Bit Result
FIG. 3: A gedanken experiment enabling faster-than-light communication using a large ensembleof long narrow cavities, each containing a single particle prepared in the same state Ψ (none ofthe particles are entangled). Alice sends her message at t = 0 and Bob receives the message at0 < t < t S , where t S is the time for a light signal sent by Alice at t = 0 to reach Bob, thus violatingthe no-signaling principle. assumed to be linked to a real process, then individual events are difficult to reconcile withrelativistic invariance. This is the case for the collapse of the state vector upon measurement[38], or for sub-quantum theories such as the Bohmian model (see Sec. IV D below).The apparent non-locality seen in the infinite well with a moving wall investigated hereconflicts with this view. The reason is that the non-local effect comes about as the directresult of a change in a single-particle state vector (rather than a multi-particle entangledstate). We will first modify the weak measurement protocol given above in Sec. III C toshow how it can lead to signaling. We will then discuss the possible artifacts that couldexplain our results. B. Bypassing no-signaling with weak measurements
Again, the noisy nature of the weak measurements in the protocol of Sec. III C doesnot enable discrimination between the cases of a moving wall and fixed wall in a singlerun. However, since only Bob makes the weak measurement and the strong postselectionmeasurement, we can couch the statistical argument involving many runs in terms of a singleexperiment involving many copies of the system, as shown in Fig. 3.Consider a gedanken experiment taking the form of a very large ensemble of extremelylong and narrow cavities, all aligned together, each with a particle prepared in the samesuitably-chosen state Ψ of the cavity. At time t = 0, the sender Alice, located at x = L chooses either to set all of those walls in motion (indicating a message bit 1), or to leave them12ll at rest (indicating message bit 0). Then, as in Sec. III C, Bob, located on the oppositeside of the cavity near the wall at x = 0 performs on each cavity a weak measurement ofthe position of the particle at location x = x w , followed a very short time later by a strongmeasurement of the position at x f of the particle at t f .Next, Bob considers the weak measurement data for the sub-ensemble of cavities whereinpostselection (detection of the particle at x f ) was successful. The data allows him to inferthe real part of the weak value P w at x w . If the walls are moving, Bob detects the weakmomentum p , indicating a message bit of 1, and otherwise Bob detects the weak momentum p , indicating a message bit of 0. The values of p and p depend on the initial state Ψ inthe cavities, which is ideally chosen to make them easily distinguishable, as with φ n (Fig.1) and ψ n (Fig. 2).In principle we can make the ensemble arbitrarily large, and the cavities arbitrarily long,allowing Bob to have enough time to collect sufficient weak measurement data before a lightsignal sent from Alice at L reaches him. Hence this device enables Alice to send a signalto Bob faster than the speed of light. All of this analysis is built upon the basis solutions ψ n ( x, t ), and thus it appears that these solutions must be nonphysical if local causality is tobe respected. In the following sections we discuss possible reasons that these solutions areflawed. C. Sources of nonlocality and possible artifacts
The most obvious candidates to account for the apparent nonlocality examined in thiswork would be the action at a distance effects allowed by a nonrelativistic formalism such asthe Schr¨odinger equation. There are two types of sources that give rise to superluminal fea-tures. First, the existence of instantaneous potentials. Second, the fact that a nonrelativisticframework does not place any restriction on the energy (and hence velocity) components ofa given wavefunction. We argue why these two features can be discarded in accounting forthe results obtained here. We then examine other possible artifacts, including the fact thatunlike a wave equation of the d’Alembert type, the Schr¨odinger equation does not impose aparticular wave speed, which enables waves to propagate instantaneously.13 . Instantaneous potentials
The instantaneous propagation of potentials appears at first as irrelevant to our problem:inside the box (except in the vicinity of x = L ( t )), and in particular in the region close to x f , the potential remains zero at all times. The potential is therefore not modified so thatthe question of its instantaneous propagation appears to be moot.Nevertheless it should be mentioned that formally, the proper way of obtaining the basissolutions (see [16] and Refs. therein) given by Eq. (5) involves a time-dependent unitarytransformation mapping the moving boundaries problem to a different system with fixedboundaries. The Hamiltonian of the transformed system is h ( t ) = P L mL ( t ) − ∂ t L ( t )2 L ( t ) ( XP + P X ) (23)with vanishing boundary conditions at both ends of the interval [0 , L ]. h ( t ) can be under-stood as describing a system in a fixed wall infinite potential well with a time-dependentmass and subjected to a time and velocity dependent potential. In this mapped system –that, contrary to the original problem, is described in a single well-defined Hilbert space –the instantaneous and uniform character of the time-dependence is obvious. It is howeverunlikely that one can make valid inferences concerning the physics of the original systemfrom the physics of the mapped system (for example the issue of signaling does not evenarise in h ( t )). Quite the contrary, the global and time-dependent aspects of the mappedsystem are readily understood as unphysical features due to the dilation imposed by theunitary transformation on the original system.
2. Infinite velocities
The issue of infinite velocities arises because in the nonrelativistic framework a givenwavefunction may contain, when expanded over the energy eigenstates, high energy statesthat can account for faster than light propagation. This artifact, due to the nonrelativis-tic nature of the Schr¨odinger equation, has been known to produce apparent superluminalpropagation in several instances, in particular when the wavefunction has a discontinuouscut-off. For example in the quantum shutter problem [39], it was shown [40] that a super-luminal propagation occurs due to the high frequencies needed to account for the cutoff ofthe initial wavefunction, before the shutter is released.14n the present problem, the contribution of high energy states can be evaluated by ex-panding the solution of the Schr¨odinger equation ψ ( x, t ) as given by Eq. (7) in the instan-taneous eigenstate basis. Let us assume that a cavity is initially in the fixed wall eigenstate φ n ( x, t = 0) [cf. Eq. (3)]. In order to compute the evolution, we need to expand φ n ( x, t = 0)over the basis functions ψ k ( x, t = 0) , as in Eq. (7) but here with a single term n . Theoverlap coefficients d kn ( t ) = (cid:90) L ( t )0 ψ ∗ k ( x, t ) φ n ( x, t ) dx (24)can be readily computed in closed form as d kn ( t ) = e − iπ/ √ (cid:126) π √ M e iπ (cid:126) ( k qt + L k − n )2 ) L M ) (cid:104) erf (cid:16) e iπ/ ( π (cid:126) ( k − n )+ M ) √ (cid:126) M (cid:17) + erf (cid:16) e iπ/ ( π (cid:126) ( n − k )+ M ) √ (cid:126) M (cid:17)(cid:105) − e iπ (cid:126) ( k qt + L k + n )2 ) L M ) (cid:104) erf (cid:16) e iπ/ ( π (cid:126) ( k + n )+ M ) √ (cid:126) M (cid:17) − erf (cid:16) e iπ/ ( π (cid:126) ( k + n ) − M ) √ (cid:126) M (cid:17)(cid:105) (25)with M ≡ mqL ( t ) . We first need to determine d kn ( t = 0) . Assuming n is small, each square bracket in Eq.(25) is seen to vanish for k (cid:29) M/π (cid:126) when the error functions cancel out, so the infinite sumin Eq. (7) can actually be cut off at a value somewhat larger (depending on the mass) than¯ k = M/π (cid:126) . The contribution of an instantaneous eigenstate φ n ( x, t ) at time t is then givenby (cid:104) φ n ( t ) | ψ ( t ) (cid:105) = k cut (cid:88) k =1 d kn (0) d ∗ kn ( t ) (26)where k cut is the cutoff value. From Eq. (25) it is seen that the error functions in thebrackets will cancel each other for n (cid:29) ¯ k . The modulus of the velocity in an instantaneouseigenstate φ n ( x, t ) is given by v ( n, t ) = (cid:126) πn/mL ( t ) (27)and for n = ¯ k , this becomes v (¯ k ) = q . Therefore, depending on the mass, we can expect fromthe properties of the error function that the highest energy eigenstates that will contributewill have at most a corresponding velocity 2 or 3 orders of magnitude larger than q , thevelocity at which the wall is expanding, which can be arbitrarily smaller than c .Mathematically the tail remains as d kn falls off as 1 /n for large n and is not strictly zero.The physical effects relevant to the tail can generally be ignored, at least as quantities relateddirectly to the wavefunction are concerned (for instance when comparing the wavefunction15volution ψ ( x, t ) − ψ ( x, k cut (cid:88) k =1 | d kn =11 (0) | = 1 (28)holds with k cut = 650 if the numerical zero is set at 10 − (meaning that the states with k >
650 account for a relative part of less than 10 − in the total state vector). At somearbitrary time t , the solution ψ ( x, t ) is expanded over the instantaneous eigenstates (3).Now n cut (cid:88) n =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k cut (cid:88) k =1 d kn =11 (0) d ∗ nk ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 (29)holds again up to 10 − for n cut = 650 for, say t = t S / v ( n = 650 , t S ) /c < − while remaining far from the regime of superluminal velocities.However concerning the current density, the quantities we are looking at are very smalland convergence to the same numerical zero taken for the wavefunction is achieved onlywhen including d kn ( t ) coefficient lying in the tail. For example when the initial state is φ n ( x, t = 0), the current density in the moving cavity is initially zero, and for small x values j ( x, t ) rises slowly. The determination of j ( x, t ), which needs to be done by expandingover the basis functions, converges by including expansion coefficients lying in the tail andcorresponding to velocities with arbitrarily high energies. To be clear, j ( x, t ) is nonzeroif coefficients in the tail are excluded, but convergence is only achieved when coefficientsin the tail are included. In this sense, going into the tail (i.e., including arbitrarily highenergies into the computation) appears as a mathematical requirement to obtain a stableresult rather than giving rise to the phenomenon itself.We can therefore conclude that the nonlocal effect put into evidence above does notappear to be due to the existence of arbitrarily high velocities that would propagate thechange in the quantum state at an arbitrarily high velocity. The ambiguity in the inclusionof the tail terms (necessary to achieve convergence for the initial states we have worked withhere) can only be lifted by working within a fully relativistic framework.16 . Model artifact and wave equations Mathematically, the source of nonlocality is straightforward to pinpoint. As can be readoff from Eqs. (17) or (21), the instantaneous change of the wavefunction at every pointof space as time unfolds (due to the expansion of the cavity) is what changes the currentdensity or the momentum weak value. From a formal point of view, the peculiarity ofthe model, as mentioned in Sec. II, is that at each time t , the system is defined on adifferent Hilbert space. Following the system evolution as time unfolds implies connectingvectors belonging to different Hilbert spaces. Hence, connecting “independent” solutionsbelonging to different Hilbert spaces might result in an unphysical picture, resulting infictitious instantaneous effects. Note that such a phenomenon is expected to be ubiquitouswhen the potential changes in a specific restricted region of space: the state vector changesinstantaneously in Hilbert space, leading to a modification of the wavefunction in the regionsin which the potential was not modified.This is related to the fact that from the point of view of wave equations, we know that theSchr¨odinger equation does not impose a finite propagation velocity. It might be conjecturedthat by supplementing the Schr¨odinger equation with a propagation velocity, as is the case ofthe relativistic Klein-Gordon equation or d’Alembert equation for Maxwell fields, we wouldget rid of these nonlocal effects. We can pursue the analogy with classical electromagnetismfurther: from the point of view of Maxwell fields in an expanding cavity, we would see theinstantaneous change of standing waves as an artifact of the model. Indeed, we can rely onMaxwell’s equations to take into account the transient effects (radiation and propagation)due to the moving charges composing the wall. However in standard unitary quantummechanics there are no fundamental equations underlying the Schr¨odinger evolution onwhich we could rely to take into account this specific transient effect. Note that the basisfunctions (5) are exact solutions of the Schr¨odinger equation: they play the same role in thepresent problem as the Moshinsky function in the paradigmatic shutter problem [39, 41]. TheMoshinsky functions form the transient basis solutions of the Schr¨odinger equation in theshutter problem [41]. It thus looks like taking into account a putative transient phenomenonfor our moving wall problem would need to supplement the standard quantum formalism.17 . Signaling and the Bohmian model As we noted below Eq. (17), the real part of the momentum weak value P w is essentiallythe velocity of the particle postulated in the de Broglie-Bohm interpretation [42] (in shortdBB or Bohmian model). Recall that dBB accounts for quantum phenomena by postulatingthe existence of point-like particles guided by the wavefunction. If we write the wavefunctionin polar form as ψ ( x, t ) = ρ ( x, t ) exp( iσ ( x, t ) / (cid:126) ) , (30)then ρ and σ obey the equation ∂σ∂t + ( (cid:79) σ ) m + V + Q = 0 (31)where V is the usual potential and the term Q ( x, t ) ≡ − (cid:126) m ∂ x ρρ (32)is known as the quantum potential. The particle velocity, defined from within the Bohmianmodel by v ( x, t ) = ∂ x σ ( x, t ) /m rather than the equivalent Eq. (18) obeys a Newton lawmodified by the presence of the quantum potential: m dvdt = − ∂ x ( V + Q ) . (33)Bohmian trajectories in systems analogous to the one investigated here have been previouslycomputed [29].The Bohmian model is generally recognized as being nonlocal. The culprit is the quantumpotential, whose local value depends on the instantaneous positions of all particles in theuniverse. Hence, a Bohmian particle is instantaneously affected by the motion of all ofthose particles, including those which produce the effective barriers of a potential well.Nevertheless it is generally accepted by proponents of the model that this nonlocality cannotbe used to communicate due to the intrinsic quantum randomness (which also includes butis not limited to the ignorance of the particle’s initial condition in a given realization).There is therefore nonlocality at the individual level, but because of the random characterof quantum mechanics, no-signaling holds at the statistical level.In the Bohmian account of our device introduced in Sec. III C, the postselected particle at x f was assumed to be there even before it was detected. However its dynamics were affected18y the quantum potential, which carries the influence of the far wall’s motion. Here, thesuperluminal signaling aspect of our protocol yields a conflict with the usual notion thatdBB, despite being explicitly nonlocal, obeys the no-signaling principle. For instance inthe EPR-Bell setting involving two particles in an entangled state, the quantum potentialchanges instantaneously, thereby “contradicting the spirit of relativity” (as put nicely byHolland, cf Sec. 11.3 of [42]), although this change has no observable consequences. In oursystem, the only quantity that changes instantaneously is the quantum potential given byEq. (32), since as we remarked above, the usual potential remains constant except in thevicinity of the wall, but there are observable consequences. Note that the enforcement of afinite propagation velocity mentioned in Sec. IV C 3 in dBB would constitute a constrainton the propagation speed of the quantum potential itself. V. CONCLUSION
To summarize, we have put forward a model displaying an apparent single-particle non-locality and enabling faster-than-light communication. As discussed in Sec. IV, we believethese effects are artifacts of the fact that we are using the nonrelativistic Schr¨odinger equa-tion, and not genuine physical effects that will one day be realized as actual communicationdevices. However, even after this analysis, we still cannot claim to have pinned down ex-actly why the Schr¨odinger equation violates local causality, even in a regime where it seemsrelativistic effects should be negligible.From a physical standpoint it seems likely that there must be a transient behavior whichbegins at the moving wall and propagates through the wavefunction at or below c . Thesetransient behaviors are not given by the solutions of the standard Schr¨odinger equation, soit seems plausible to suggest that the correct dynamical evolution equation would containterms accounting for such transients. Even presuming this is the right way to impose arelativistic constraint on the non-relativistic Schr¨odinger equation, we do not presently havea suggested form for incorporating this constraint. A relativistic treatment would be helpful,despite the well-known limitations affecting the single particle relativistic wavefunctions.To conclude, the system investigated in this work raises interesting questions aboutthe general trustworthiness of any solutions to the Schr¨odinger equation involving time-dependent potentials localized in a given spatial region but affecting the entire wavefunction.19e remain open to the possibility that there could be some other explanation that wehave not considered, and we would be very pleased if a more complete resolution of thisconundrum could be found. Acknowledgments:
AM thanks the participants of a FQXI workshop in Marseille (July2017) for useful exchanges on single particle nonlocality. MW acknowledges partial supportfrom the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust.
Appendix: Weak value of the momentum in terms of position measurements
We prove here Eq. (21) expressing P w in terms of a weak and then a projective positionmeasurements. The manipulations are similar to the ones employed in Ref. [34]. With∆ t ≡ t f − t w very small, we have X w = (cid:104) x f | U ( t f , t w ) X | ψ ( t w ) (cid:105)(cid:104) x f | ψ ( t f ) (cid:105) = (cid:104) x f | (cid:0) − i (cid:126) ∆ tH (cid:1) X | ψ ( t w ) (cid:105)(cid:104) x f | exp (cid:0) − i (cid:126) ∆ tH (cid:1) | ψ ( t w ) (cid:105) . (34)Since i [ H, X ] = (cid:126) P we have X w = (cid:104) x f | (cid:0) X − ∆ tm P − i ∆ t (cid:126) XH (cid:1) | ψ ( t w ) (cid:105)(cid:104) x f | exp (cid:0) − i (cid:126) ∆ tH (cid:1) | ψ ( t w ) (cid:105) (35)= x f (cid:104) x f | ψ ( t w ) (cid:105) − ∆ tm (cid:104) x f | P | ψ ( t w ) (cid:105) − x f (cid:104) x f | (cid:0) − exp (cid:0) − i (cid:126) ∆ tH (cid:1)(cid:1) | ψ ( t w ) (cid:105)(cid:104) x f | exp (cid:0) − i (cid:126) ∆ tH (cid:1) | ψ ( t w ) (cid:105) (36)= − ∆ tm (cid:104) x f | P | ψ ( t w ) (cid:105)(cid:104) x f | exp (cid:0) − i (cid:126) ∆ tH (cid:1) | ψ ( t w ) (cid:105) + x f . (37)Therefore (cid:104) x f | P | ψ ( t w ) (cid:105)(cid:104) x f | exp (cid:0) − i (cid:126) ∆ tH (cid:1) | ψ ( t w ) (cid:105) = m ∆ t ( x f − X w ) (38)and taking the limit ∆ t → t f → t w ) we recover Eq. (21). [1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).[2] J.S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge University Press,Cambridge, 2004), Chap. 4. [Original publication in
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