Finally making sense of the double-slit experiment

Significance

We put forth a time-symmetric interpretation of quantum mechanics that does not stem from the wave properties of the particle. Rather, it posits corpuscular properties along with nonlocal properties, all of which are deterministic. This change of perspective points to deterministic properties in the Heisenberg picture as primitive instead of the wave function, which remains an ensemble property. This way, within a double-slit experiment, the particle goes only through one of the slits. In addition, a nonlocal property originating from the other distant slit has been affected through the Heisenberg equations of motion. Under the assumption of nonlocality, uncertainty turns out to be crucial to preserve causality. Hence, a (qualitative) uncertainty principle can be derived rather than assumed.

Abstract

Feynman stated that the double-slit experiment “…has in it the heart of quantum mechanics. In reality, it contains the only mystery” and that “nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it” [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures on Physics]. We rise to the challenge with an alternative to the wave function-centered interpretations: instead of a quantum wave passing through both slits, we have a localized particle with nonlocal interactions with the other slit. Key to this explanation is dynamical nonlocality, which naturally appears in the Heisenberg picture as nonlocal equations of motion. This insight led us to develop an approach to quantum mechanics which relies on pre- and postselection, weak measurements, deterministic, and modular variables. We consider those properties of a single particle that are deterministic to be primal. The Heisenberg picture allows us to specify the most complete enumeration of such deterministic properties in contrast to the Schrödinger wave function, which remains an ensemble property. We exercise this approach by analyzing a version of the double-slit experiment augmented with postselection, showing that only it and not the wave function approach can be accommodated within a time-symmetric interpretation, where interference appears even when the particle is localized. Although the Heisenberg and Schrödinger pictures are equivalent formulations, nevertheless, the framework presented here has led to insights, intuitions, and experiments that were missed from the old perspective.

Beginning with de Broglie (1), the physics community embraced the idea of particle-wave duality expressed, for example, in the double-slit experiment. The wave-like nature of elementary particles was further enshrined in the Schrödinger equation, which describes the time evolution of quantum wave packets.

It is often pointed out that the formal analogy between Schrödinger wave interference and classical wave interference allows us to interpret quantum phenomena in terms of the familiar classical notion of a wave. Indeed, wave-particle duality was construed by Bohr and others as the essence of the theory and, in fact, its main novelty. Even so, the foundations of quantum mechanics community have consistently raised many questions (2–5) centered on the physical meaning of the wave function.

From our perspective and consistent with ideas first expressed by Born (6) and thereafter extensively developed by Ballentine (7, 8), a wave function represents an ensemble property as opposed to a property of an individual system.

What then is the most thorough approach to ontological questions concerning single particles (using standard nonrelativistic quantum mechanics)? We propose an alternative interpretation for quantum mechanics relying on the Heisenberg picture, which although mathematically equivalent to the Schrödinger picture, is very different conceptually. For example, within the Heisenberg picture, the primitive physical properties will be represented by deterministic operators, which are operators with measurements that (i) do not disturb individual particles and (ii) have deterministic outcomes (9). By way of example, the modular momentum operator will arise as particularly significant in explaining interference phenomena. This approach along with the use of Heisenberg’s unitary equations of motion introduce a notion of dynamical nonlocality. Dynamical nonlocality should be distinguished from the more familiar kinematical nonlocality [implicit in entangled states (10) and previously analyzed in the Heisenberg picture by Deutsch and Hayden (11)], because dynamical nonlocality has observable effects on probability distributions (unlike, e.g., measurements of one of two spins in Bell states, which do not change their probability distribution). Within the Schrödinger picture, dynamical nonlocality is manifest in the unique role of phases, which although unobservable locally, may subsequently influence interference patterns. Finally, in addition to the initial state of the particle, we will also need to take into account a final state to form a twofold set of deterministic properties (one deterministic set based on the initial state and a second based on the final state). The above amounts to a time-symmetric Heisenberg-based interpretation of nonrelativistic quantum mechanics.

Wave Function Represents an Ensemble Property

The question of the meaning of the wave function is central to many controversies concerning the interpretation of quantum mechanics. We adopt neither the standard ontic nor the epistemic approaches to the meaning of the wave function. Rather, we consider the wave function to represent an ensemble property as opposed to a property of an individual system. This approach resonates with the ensemble interpretation of the wave function that was initiated by Born (6) and extensively developed by Ballentine (7, 8). According to this interpretation, the wave function is a statistical description of a hypothetical ensemble, from which the probabilistic nature of quantum mechanics stems directly. It does not apply to individual systems. Ballentine (7, 8) justified an adherence to this interpretation by observing that it overcomes the measurement problem—by not pretending to describe individual systems, it avoids having to account for state reduction (collapse). We concur with the conclusion by Ballentine (7, 8) but do not concur with his reasoning. Instead, we contend that the wave function is appropriate as an ontology for an ensemble rather than an ontology for an individual system. Our principle justification for this is because the wave function can only be directly verified at the ensemble level. By “directly verified,” we mean measured to an arbitrary accuracy in an arbitrarily short time (excluding practical and relativistic constraints).

Indeed, we only regard directly verifiable properties to be intrinsic. Consider, for instance, how probability distributions relate to single particles in statistical mechanics. We can measure, for example, the Boltzmann distribution, in two ways—either instantaneously on thermodynamic systems or using prolonged measurements on a single particle coupled to a heat bath. We do not attribute the distribution to single particles, because instantaneous measurements performed on single particles yield a large error. Conversely, when the system is large [containing $N\gg 1$ particles (the thermodynamic limit)], the size of the error, which scales like $\sqrt{N}$, is relatively very small. In other words, the verification procedure transitions into the category of being directly verified only as the system grows. Due to these characteristics, the distribution function is best viewed as a property of the entire thermodynamic system. On the single-particle level, it manifests itself as probabilities for the particle to be found in certain states. However, the intrinsic properties of the individual particle are those that can be verified directly, namely position and momentum, and only they constitute its real properties.

Similar to how distributions in statistical mechanics can be directly verified only on a thermodynamic system, the wave function can be directly verified only on quantum ensembles. Continuing the analogy, on a single-particle level, the wave function can only be measured by performing a prolonged measurement. This prolonged measurement is a protective measurement (12). Protective measurements can be implemented in two different ways: the first is applicable for measuring discrete nondegenerate energy eigenstates and based on the adiabatic theorem (13); the second, more general way requires an external protection in the form of the quantum Zeno effect (14). In either of these two ways, a large number of identical measurements are required to approximate the wave function of a single particle. We conclude that, analogous to how statistical mechanical distributions become properties for thermodynamic systems, the wave function is a property of a quantum ensemble.

Unlike Born (6), we do not wish to imply that the wave function description is somehow incomplete (and could become “complete” with the addition of a classical-like reality, such as with a hidden variable theory). Nor do we oppose the consequence of the Pusey–Barrett–Rudolph (PBR) theorem (15), which states that the wave function is determined uniquely by the physical state of the system. We only mean to suggest that the wave function cannot constitute the primitive ontology of a single quantum particle/system. That being said and contrary to ensemble interpretation advocates, we will not duck out of proposing a single-particle ontology. In what follows, we expound such an ontology based on deterministic operators, which are unique operators with measurement that can be carried out on a single particle without disturbing it and with predictable, definite outcomes. Because properties corresponding to these operators can be directly verified at the single-particle level, they constitute the real properties of the particle. To derive this ontology, we turn the spotlight to the Heisenberg representation.

Formalism and Ontology

In the Schrödinger picture, a system is fully described by a continuous wave function $\psi$. Its evolution is dictated by the Hamiltonian and calculated according to Schrödinger’s equation. As will be shown below, in the Heisenberg picture, a physical system can be described by a set of Hermitian deterministic operators evolving according to Heisenberg’s equation, whereas the wave function remains constant.

In the traditional Hilbert space framework for quantum mechanics along with ideal measurements, the state of a system is a vector $|\psi ⟩$ in a Hilbert space $\mathcal{H}$, and any observable $\widehat{A}$ is a Hermitian operator on $\mathcal{H}$. The eigenstates of $\widehat{A}$ form a complete orthonormal system for $\mathcal{H}$. When an ideal measurement of $\widehat{A}$ is performed, the outcome appears at random (with a probability given by initial $|\psi ⟩$) and corresponds to an eigenvalue within the range of $\widehat{A}$s allowed spectrum. Thereafter, from the perspective of the Schrödinger picture, the ideal measurement leads to the “collapse” (true or effective depending on one’s preferred interpretation) of the wave function from $|\psi ⟩$ into an eigenstate corresponding to that eigenvalue. This fact can be verified by performing subsequent ideal measurements that will yield the same eigenvalue. This collapse corresponds to a disturbance of the system.

However, one could invert the process and consider nondisturbing measurements of the “deterministic subset of operators” (DSO). This set involves measurement of only those observables for which the state of the system under investigation is already an eigenstate. Therefore, no collapse is involved. This set answers the question “what is the set of Hermitian operators ${\widehat{A}}_{\psi }$ for which $\psi$ is an eigenstate?” for any state $\psi$:

$${\widehat{A}}_{\psi }=\{{\widehat{A}}_i\mathrm{such}\mathrm{that}{\widehat{A}}_i|\psi (t)⟩=a_i|\psi (t)⟩,a_i\in ℝ\}.$$ [1]

This question is dual to the more familiar question: “what are the eigenstates of a given operator?” Clearly, ${\widehat{A}}_{\psi }$ is a subspace closed under multiplication. Moreover, $[{\widehat{A}}_i,{\widehat{A}}_j]={\widehat{A}}_k\in {\widehat{A}}_{\psi }$ is such that ${\widehat{A}}_k|\psi ⟩=0$.

Theorem.

Let $\mathcal{H}$ be a Hilbert space, $\widehat{A}$ be an operator acting on $\mathcal{H}$, and $|\psi ⟩\in \mathcal{H}$ (16). Then,

$$\widehat{A}|\psi ⟩=⟨\widehat{A}⟩|\psi ⟩+\mathrm{\Delta }A|{\psi }_{⟂}⟩,$$ [2]

where $⟨\widehat{A}⟩=⟨\psi |\widehat{A}|\psi ⟩$, $\mathrm{\Delta }{A}^2=⟨\psi |{(\widehat{A}-⟨\widehat{A}⟩)}^2|\psi ⟩$, and $|{\psi }_{⟂}⟩$ is a vector such that $⟨\psi |{\psi }_{⟂}=0$.

The physical significance of DSOs stems from the possibility to measure them without disturbing the particle (i.e., without inducing collapse). As long as only such eigenoperators are measured, they all evolve unitarily by applying Heisenberg’s equation separately to each of them. DSOs (with measurement outcomes that are completely certain) are dual to the “completely uncertain operators” with measurement outcomes that are completely uncertain. Complete uncertainty means that they satisfy the condition that all of their possible measurement outcomes are equiprobable (17). Thus, no information can be gained by measuring them. Mathematically, the two limiting cases represented by Eq. 2 are given by deterministic operators for which $\mathrm{\Delta }A|{\psi }_{⟂}⟩=0$ and completely uncertain operators for which $⟨\widehat{A}⟩=0$ (a necessary but insufficient condition as will be described below).

An important ingredient to consider of our proposed interpretation is a final state of the system. The idea that a complete description of a quantum system at a given time must take into account two boundary conditions rather than one is known from the two-state vector formalism (TSVF). This approach has its roots in the work of Aharonov et al. (18), but it has since been extensively developed (19) and led to the discovery of numerous interesting phenomena (17).

The TSVF provides an extremely useful platform for analyzing experiments involving pre- and postselected ensembles. Weak measurements enable us to explore the state of the system during intermediate times without disturbing it (20, 21). The power to explore the pre- and postselected system by using weak measurements motivates a literal reading of the formalism (that is, as more than just a mathematical tool of analysis). It motivates a view according to which future and past play equal roles in determining the quantum state at intermediate times and are, hence, equally real. Accordingly, to fully specify a system, one should not only preselect but also, postselect a certain state using a projective measurement. In the framework that we propose within this article, adding a final state is equivalent to adding a second DSO in addition to the one dictated by the initial state. This twofold set forms the basis for the primal ontology of a quantum mechanics for individual particles.

Nonlocal Dynamics and Wave-Like Behavior

Interference patterns appear in both classical and quantum grating experiments (most conveniently analyzed in a double-slit setup, which will be referred to hereinafter, although our results are completely general). We are taught that the explanation for interference phenomena is shared across both domains, the classical and quantum: a spatial wave (function) traverses the grating, one part of which goes through the first slit while the other part goes through the second slit, before the two parts later meet to create the familiar interference pattern. Although it is indeed tempting to extend the accepted classical explanation into the quantum domain, nevertheless, there are important breakdowns in the analogy. For example, in classical wave theory, one can predict what will happen when the two parts of the wave finally meet based on entirely local information available along the trajectories of the wave packets going through the two slits. However, in quantum mechanics, what tells us where the maxima and minima of the interference will be located is the relative phase of the two wave packets. Although we can measure the local phase in classical mechanics, we cannot in principle measure the individual local phases for a particle, because this would violate gauge symmetry (17). Only the phase difference is observable, but it cannot be deduced from measurements performed on the individual wave packets (until they overlap). The analogy is, therefore, only partial. For this reason, we contend that the temptation to jump on the wave function bandwagon should be resisted. Our goal now is to show how quantum interference can be understood without having to say that each particle passed through both slits at the same time as if it were a wave. For this purpose, we examine those operators that are relevant for all interference phenomenon. When we transform back to the Schrödinger picture and apply these operators, we will see that these operators are sensitive to the relative phase, which again, is the property that determines the subsequent interference pattern.

We, therefore, consider the state ${\psi }_{\varphi }(x,t)={\psi }_1(x,t)+e^{i\varphi }{\psi }_2(x,t)$, which in the Schrödinger picture, represents the wave at the double slit. We now ask which operators $\widehat{f}(x,p)$ belong to the DSO ${\widehat{A}}_{{\psi }_{\varphi }}$. In addition, we ask which operators are sensitive to the relative phase $\varphi$. It is not difficult to show that, if we limit ourselves to simple functions of position and momentum (i.e., any polynomial representation of the form

$$\widehat{f}(x,p)=\sum a_{mn}{x}^m{p}^n,$$

then any resulting operator is not sensitive to the relative phase between different “lumps” of the wave function (i.e., lumps centered around each slit). This fact suggests that simple moments of position and momentum are not the most appropriate dynamical variables to describe quantum interference phenomena. Indeed, it is easy to prove the following theorem.

Theorem.

Let ${\psi }_{\varphi }(x,t)={\psi }_1(x,t)+e^{i\varphi }{\psi }_2(x,t)$, and assume no overlap of ${\psi }_1(x,0)$ and ${\psi }_2(x,0)$ ($t=0$ is when the particle is going through the double slit) (16). If $m,n$ are integers, then for all values of $t$ and choices of phases $\alpha \beta$,

$$\int [{\psi }_{\alpha }^{*}(x,t)x^m{p}^n{\psi }_{\alpha }(x,t)-{\psi }_{\beta }^{*}(x,t)x^m{p}^n{\psi }_{\beta }(x,t)]dx=0.$$ [3]

Let us now consider operators of the form $\widehat{f}(x,p):=e^{ipL/\mathrm{\hslash }}$ (where $L$ is the distance between the slits). Evolving this through the Heisenberg equation,

$$i\mathrm{\hslash }\frac{\partial \widehat{f}(x,p)}{\partial t}=[\widehat{f},\widehat{H}],$$

where $H=p^2/2m+V(x)$ appropriate for the double slit. In this particular case, we obtain a nonlocal equation of motion:

$$\frac{\partial \widehat{f}(x,p)}{\partial t}=[e^{\frac{ipL}{\mathrm{\hslash }}},V(x)]=\frac{1}{\mathrm{\hslash }}[V(x+L)-V(x)]e^{\frac{ipL}{\mathrm{\hslash }}}$$ [4]

(that is, the value of $\dot{\widehat{f}}$ depends on the potential at not only $x$ but also, the remote $x+L$). This operator leads us naturally to realize that the variable that accounts for the effect of the double slit is not $p$ but its modular version. Indeed, because

$$e^{\frac{ipL}{\mathrm{\hslash }}}=e^{i\left(\frac{p+2\pi k\mathrm{\hslash }}{L}\right)\frac{L}{\mathrm{\hslash }}},k\in \mathbb{Z},$$

the observable of interest is the modular momentum (i.e.,

$$p_{mod}:=p\mathrm{mod}{p}_0,$$

where $p_0=2\pi \mathrm{\hslash }/L$). Eq. 4 differs considerably from the classical evolution that is given by the Poisson bracket:

$$\frac{d}{dt}{e}^{\frac{i2\pi p}{p_0}}=\{e^{\frac{i2\pi p}{p_0}},\widehat{H}\}=-i\frac{2\pi }{p_0}\frac{dV}{dx}{e}^{\frac{i2\pi p}{p_0}},$$ [5]

which involves a local derivative, suggesting that the classical modular momentum changes only if a local force $dV/dx$ is acting on the particle. We thus understand that, although commutators have a classical limit in terms of Poisson brackets, they are fundamentally different, because they entail nonlocal dynamics. The connection between nonlocal dynamics and relative phase via the modular momentum suggests the possibility of the former taking the place of the latter in the Heisenberg picture. The nonlocal equations of motion in the Heisenberg picture thus allow us to consider a particle going through only one of the slits, but it nevertheless has nonlocal information regarding the other slit.

Unlike ordinary momentum, modular momentum becomes, on detecting (or failing to detect) the particle at a particular slit, maximally uncertain. The effect of introducing a potential at a distance from the particle (i.e., of opening a slit) is equivalent to a nonlocal rotation in the space of the modular variable (22). Denote it by $\theta \in [0,2\pi )$. Suppose the amount of nonlocal exchange is given by $\delta \theta$ (i.e., $\theta \to \theta +\delta \theta$). Now “maximal uncertainty” means that the probability to find a given value of $\theta$ is independent of $\theta$ [i.e., $P(\theta )=\text{constant}=1/2\pi$]. Under these circumstances, the shift in $\theta$ to $\theta +\delta \theta$ will introduce no observable effect, because the probability to measure a given value of $\theta$, say ${\theta }_1$, will be the same before and after the shift, $P({\theta }_1)=P({\theta }_1+\delta {\theta }_1)$. We shall call a variable that satisfies this condition a “completely uncertain variable.”

Theorem (Complete Uncertainty Principle for Modular Variables).

Let $\mathrm{\Phi }$ be a periodic function, which is uniformly distributed on the unit circle (17). If $⟨e^{in\mathrm{\Phi }}⟩=0$ for any integer $n\ne 0$, then $\mathrm{\Phi }$ is completely uncertain.

When a particle is localized to within $|x|<L/2$, the expectation value of $e^{ipL/\mathrm{\hslash }}$ vanishes. This result is obvious, because $e^{ipL/\mathrm{\hslash }}$ functions as a translation operator, shifting the wave packet outside $|x|<L/2$ (i.e., outside its region of support). Accordingly, when a particle is localized near one of the slits, as in the case of either ${\psi }_1$ or ${\psi }_2$, then $⟨e^{inpL/\mathrm{\hslash }}⟩=0$ for every $n$. It then follows from the complete uncertainty principle that the modular momentum is completely uncertain. Accordingly, all information about the modular momentum is lost after we find the position of the particle. This onset of complete uncertainty is crucial to prevent signaling and preserve causality. As an example, suppose we apply a force arbitrarily far away from a localized wave packet. We thus change operators depending on the modular momentum instantly, because modular momentum relates remote points in space. If we could measure this change on the wave packet, then we could violate causality, but all such measurements are precluded by the complete uncertainty principle.

The fact that the modular momentum becomes uncertain on localization of the particle also fits well with the fact that interference is lost with localization. In the Schrödinger picture, interference loss is understood as a consequence of wave function collapse. After the superposition is reduced, there is nothing left for the remaining localized wave packet to interfere with. The Heisenberg picture, however, offers a different explanation for the loss of interference that is not in the language of collapse: if one of the slits is closed by the experimenter, a nonlocal exchange of modular momentum with the particle occurs. Consequently, the modular momentum becomes completely uncertain, thereby erasing interference and destroying the information about the relative phase.

Note also that, because $p=p_{mod}+N\mathrm{\hslash }/L$ for some integer $N$, the uncertainty of $p$ is greater than or equal to that of $p_{mod}$ (the integer part can be uncertain as well). For this reason, a complete uncertainty of the modular momentum $p_{mod}$ [which means that its distribution function is uniform in the interval $[0,\mathrm{\hslash }/L)$] sets $\mathrm{\hslash }/L$ as a lower bound for the uncertainty in $p$ (i.e., $\mathrm{\Delta }p\ge \mathrm{\hslash }/L$). This inequality parallels the Heisenberg uncertainty principle, equating it in the case of $\mathrm{\Delta }x=L$.

At first blush, it seems that, as axioms, dynamical nonlocality and relativistic causality nearly contradict each other. Nevertheless, by prohibiting the detection of nonlocal action, complete uncertainty enables one to reconcile nonlocality with relativistic causality, so that they may “peacefully coexist.” This reconciliation is why we regard this principle as very fundamental.

Measuring Nonlocal Operators

Consider a system described at time $t=0$ by a vector $|\psi ⟩$ in a Hilbert space. Fundamental properties of operator-valued functions allow us to reconstruct $|\psi ⟩$ using weak measurements of the position of the particle at various instants $t$. Indeed, if we call $\rho (x,t)$ the density of $\psi (x,t)$, namely

$$\rho (x,t)={\psi }^{*}(x,t)\psi (x,t)$$

then we can calculate its Fourier transform:

$$\mathcal{F}\rho (k,t)={\int }_{ℝ}{\psi }^{*}(x,t)\psi (x,t)e^{ikx}dx.$$ [6]

For a given operator $\widehat{A}$, we can write its expectation value as

$$\overline{{\widehat{A}}_x(t)}=⟨\psi (x,t)|\widehat{A}|\psi (x,t)⟩.$$ [7]

Therefore, Eq. 6 is nothing but the expectation value of $e^{ikx}$. Note that, in Eq. 7, we have been using the Schrödinger picture with a time-evolving state $\psi (x,t)$. Rewriting Eq. 7 in the Heisenberg picture,

$$⟨\psi (x,t)|\widehat{A}|\psi (x,t)⟩=⟨\psi (x,0)|\widehat{A}(t)|\psi (x,0)⟩.$$

We know that the two pictures are equivalent: the time evolution has simply been moved from the vector in the Hilbert space to the operator. Given that $x(t)=x(0)+p(0)t/m$, we have

$$\overline{e^{ikx(t)}}=\overline{e^{ik(x(0)+p(0)\frac{t}{m})}}.$$

If we set $\alpha =k$ and $\beta =kt/m$, we see that, as time $t$ changes,

$$\overline{e^{i(\alpha x(0)+\beta p(0))}}$$

assumes all of the possible values. Hence, nonlocal operators at $t=0$ can be measured locally at some later time. The following theorem shows that this description is exhaustive.

Theorem.

The collection for all $(\alpha ,\beta )\in {ℝ}^2$,

$$f(\alpha ,\beta )={\int }_{ℝ}{\psi }^{*}(x)e^{i(\alpha x+\beta p)}\psi (x)dx,$$

uniquely determines the state $\psi$.

Proof.

First we multiply both sides by $e^{-i\alpha \beta \mathrm{\hslash }/2}$. Integration with respect to $\alpha$ lets us find ${\psi }^{*}(0)\psi (\beta )$ for all $\beta$. This expression amounts to finding $\psi (x)$ when setting $\psi (0)=1$.

Double-Slit Experiment Revisited

Performing certain experiments involving postselection allows us to both measure interference and deduce which-path information. However, the Schrödinger picture is very awkward with such experiments, which posit both wave and particle properties at the same time. Alternatively, in the Heisenberg picture, the particle has both a definite location and a nonlocal modular momentum that can “sense” the presence of the other slit and therefore, create interference. This description thus evades difficulties present in the Schrödinger picture.

To emphasize this point, let us consider a simple 1D Gedanken experiment to mimic the double-slit experiment. In the Schrödinger picture, a particle is prepared in a superposition of two identical spatially separated wave packets moving toward one another with equal velocity (Fig. 1):

$${\mathrm{\Psi }}_i(x,t=0)=\frac{1}{\sqrt{2}}[e^{\frac{i{p}_{0}x}{\mathrm{\hslash }}}\mathrm{\Psi }\left(\frac{x+L}{2}\right)+e^{i\varphi }{e}^{\frac{-i{p}_{0}x}{\mathrm{\hslash }}}\mathrm{\Psi }\left(\frac{x-L}{2}\right)],$$ [8]

where $\mathrm{\Psi }(x)$ is a Gaussian wave function. To simplify, we assume that the spread $\mathrm{\Delta }x$ obeys $\mathrm{\hslash }/p_0\ll \mathrm{\Delta }x\ll L$; hence, the wave packet approximately maintains its shape up to the time of encounter (our results are, however, general). The relative phase $\varphi$ has no effect on the local density $\rho (x)$ or any other local feature until the two wave packets overlap. The phase $\varphi$ manifests itself by shifting the interference pattern by $\delta =\mathrm{\hslash }\varphi /p_0$.

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 the time $T$ when the wave packets completely overlap. The shift $\delta$ of the interference pattern is proportional to the relative phase $\varphi$.

This initial configuration is identical to that of the standard double-slit setup, but instead of letting the two wave packets propagate away from the grating to hit a photographic plate, we confine ourselves to one dimension and let them meet at time $T$ on the plane of the grating. On meeting, the density of the two wave packets becomes

$$\rho (x,T)\approx 4{|{\mathrm{\Psi }}_i(x)|}^2{\text{cos}}^2(p_{0}x/\mathrm{\hslash }-\varphi /2),$$ [9]

which displays interference, similar to that of a standard double-slit experiment.

We now augment the experiment with a postselection procedure, where we place a detector on the path of the wave packet moving to the right ${\mathrm{\Psi }}_f(x)=e^{i{p}_{0}x/\mathrm{\hslash }}\mathrm{\Psi }(x-L/2)$. Although the probability to find the particle there is $1/2$, let us consider an ensemble of such pre- and postselected experiments, which realizes the rare case where all of the particles are found by this detector (that is, we determine the position operator for the entire ensemble by a postselection). The two-state, which constitutes the full description of pre- and postselected systems at any intermediate time $t$, is given by $⟨{\mathrm{\Psi }}_f(t)||{\mathrm{\Psi }}_i(t)⟩$. Within the TSVF, we can define a two-times generalization of the pure-state density:

$${\rho }_{two-time}(x,T)=\frac{⟨x|{\mathrm{\Psi }}_f⟩⟨{\mathrm{\Psi }}_i|x⟩}{⟨{\mathrm{\Psi }}_f|{\mathrm{\Psi }}_i⟩}=2{|\mathrm{\Psi }(x)|}^2{e}^{i(p_{0}x/\mathrm{\hslash }-\varphi /2)}\text{cos}(p_{0}x/\mathrm{\hslash }-\varphi /2).$$ [10]

To measure this density, during intermediate times, we perform a weak measurement using $M\gg 1$ projections ${\mathrm{\Pi }}_i(x)$ with the interaction Hamiltonian $H_{int}=g(t)q{\sum }_i^M{\mathrm{\Pi }}_i(x)$, where $q$ is the pointer of the measuring device, $i$ sums over an ensemble of particles, and ${\int }_0^{\tau }g(t)dt=g$ is sufficiently small during the measurement duration $\tau$. For a large-enough ensemble, these measurements allow us to observe the two-time density while introducing almost no disturbance to the state of the particle. If we perform many such measurements in different locations within the overlap region, they will add up to a histogram tracing the two-time density in that region (Fig. 2) from which we find the parameter $\delta$ that depends on the relative phase $\varphi$. This Gedanken experiment shows a perplexing situation from the point of view of the Schrödinger picture. The real part of this density, which describes the evolution of the two-state, exhibits an interference pattern when weakly measured. However, by virtue of the postselection, we know that the particle has a determinate position described by a right-moving wave packet that went through the left slit. Interference is thus still present, despite the fact that the particle is localized around one of the slits. Recall that the interpretation of the particle as having a wave-like nature was originally devised to account for interference phenomena, and here, we have shown that this is not necessary and in fact, is inconsistent with a time-symmetric view.

Fig. 2.

Weak measurement of the interference pattern. The two wave packets are preselected in A and postselected in C. Weak measurements in B performed at $t=T$ show the usual interference pattern, despite the fact that detector $D$ detects all particles as belonging to just one (moving to the right) wave packet.

In contrast, the Heisenberg picture tells us that each particle has both a definite position and at the same time, nonlocal information in the form of DSOs, which are simple functions of the modular momentum (9).

Discussion

After the Schrödinger picture has dominated for many years, we have elaborated a Heisenberg-based interpretation for quantum mechanics. In this interpretation, individual particles possess deterministic, yet nonlocal properties that have no classical analog, whereas the Schrödinger wave can only describe an ensemble. An uncertainty principle appears not as a mathematical consequence but as a reconciler between metaphysical desiderata—causality—and the nonlocality of the dynamics. While this complete uncertainty principle (qualitatively) implies the Heisenberg uncertainty principle, the implication does not work the other way around, i.e., from Heisenberg uncertainty principle to the complete uncertainty principle. For this reason, we regard the complete uncertainty principle as more fundamental. In turn, uncertainty combined with the empirical demand for definite measurement outcomes necessitate a mechanism for choosing those outcomes. This demand is met by the inclusion of a final state. It was shown elsewhere (23) that, by considering a special final state of the kind that we had introduced but for the entire universe, the outcomes of specific measurements can be accounted for. This cosmological generalization thereby solves the measurement problem. We now understand this final state to constitute a DSO, which may be regarded as a hidden variables because of its epistemic inaccessibility in earlier times.

We contend that this interpretation conveys a powerful physical intuition. Internalizing it, one is no longer restricted to thinking in terms of the Schrödinger picture, which is a convenient tool for mathematical analysis but inconsistent with the pre- and postselection experiments. The wave function is an efficient mathematical 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 discover the Aharonov–Bohm effect. The stimulation of new discoveries 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 Schrödinger representation.

For the sake of completeness, it might be interesting to briefly address the notion of kinematic nonlocality arising from entanglement. As noted in Formalism and Ontology, a quantum system in 2D Hilbert space (e.g., a spin-1/2 particle) is described within our formalism using two DSOs. For describing a system of two entangled spin-1/2 particles (in a 4D Hilbert space), we would use a set of 10 DSOs. It is important to note that the measurements of such operators are nonlocal (26), possibly carried out in space-like separated points. Most of these operators involve simultaneous measurements of the two particles. A (nondeterministic) measurement of one particle would change the combined DSOs, thus instantaneously affecting also the ontological description of the second particle. In ref. 11, it was claimed that the information flow in the Heisenberg representation is local; however, in light of the above analysis, this analysis only refers to certain kinds of operators.

We believe that, if quantum mechanics were discovered before relativity theory, then our proposed ontology could have been the commonplace one. Before the 20th century, physicists and mathematicians were interested in studying various Hamiltonians having an arbitrary dependence on the momentum, such as $\text{cos}(p)$. In quantum mechanics, these Hamiltonians lead to nonlocal effects as discussed above. The probability current is not continuous under the resulting time evolution, which makes the wave function description less intuitive. However, those Hamiltonians were dismissed as nonphysical in the wake of relativity theory, allowing the wave function ontology to prosper. We hope that our endorsement of the Heisenberg-based ontology will promote a discussion of this somewhat neglected approach.