Finally making sense of the double-slit experiment Yakir Aharonova,b,c,1, Eliahu Cohend,1,2, Fabrizio Colomboe, Tomer Landsbergerc,2, Irene Sabadinie, Daniele C. Struppaa,b, and Jeff Tollaksena,b aInstitute for Quantum Studies, Chapman University, Orange, CA 92866; bSchmid College of Science and Technology, Chapman University, Orange, CA 92866; cSchool of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel; dH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom; and eDipartimento di Matematica, Politecnico di Milano, 9 20133 Milan, Italy Contributed by Yakir Aharonov, March 20, 2017 (sent for review September 26, 2016; reviewed by Pawel Mazur and Neil Turok) Feynman stated that the double-slit experiment “. has in it the modular momentum operator will arise as particularly signifi- heart of quantum mechanics. In reality, it contains the only mys- cant in explaining interference phenomena. This approach along tery” and that “nobody can give you a deeper explanation of with the use of Heisenberg’s unitary equations of motion intro- this phenomenon than I have given; that is, a description of it” duce a notion of dynamical nonlocality. Dynamical nonlocality [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures should be distinguished from the more familiar kinematical non- on Physics]. We rise to the challenge with an alternative to the locality [implicit in entangled states (10) and previously ana- wave function-centered interpretations: instead of a quantum lyzed in the Heisenberg picture by Deutsch and Hayden (11)], wave passing through both slits, we have a localized particle with because dynamical nonlocality has observable effects on prob- nonlocal interactions with the other slit. Key to this explanation ability distributions (unlike, e.g., measurements of one of two is dynamical nonlocality, which naturally appears in the Heisen- spins in Bell states, which do not change their probability distri- berg picture as nonlocal equations of motion. This insight led us to bution). Within the Schrodinger¨ picture, dynamical nonlocality is develop an approach to quantum mechanics which relies on pre- manifest in the unique role of phases, which although unobserv- and postselection, weak measurements, deterministic, and mod- able locally, may subsequently influence interference patterns. ular variables. We consider those properties of a single particle Finally, in addition to the initial state of the particle, we will that are deterministic to be primal. The Heisenberg picture allows also need to take into account a final state to form a twofold us to specify the most complete enumeration of such determinis- set of deterministic properties (one deterministic set based on tic properties in contrast to the Schrodinger¨ wave function, which the initial state and a second based on the final state). The above remains an ensemble property. We exercise this approach by ana- amounts to a time-symmetric Heisenberg-based interpretation of lyzing a version of the double-slit experiment augmented with nonrelativistic quantum mechanics. postselection, showing that only it and not the wave function approach can be accommodated within a time-symmetric inter- Wave Function Represents an Ensemble Property pretation, where interference appears even when the particle is The question of the meaning of the wave function is central to localized. Although the Heisenberg and Schrodinger¨ pictures are many controversies concerning the interpretation of quantum equivalent formulations, nevertheless, the framework presented mechanics. We adopt neither the standard ontic nor the epis- here has led to insights, intuitions, and experiments that were temic approaches to the meaning of the wave function. Rather, missed from the old perspective. we consider the wave function to represent an ensemble prop- erty as opposed to a property of an individual system. This Heisenberg picture j two-state vector formalism j modular momentum j approach resonates with the ensemble interpretation of the wave double slit experiment Significance eginning with de Broglie (1), the physics community em- Bbraced the idea of particle-wave duality expressed, for exam- We put forth a time-symmetric interpretation of quantum ple,inthedouble-slitexperiment.Thewave-likenatureofelemen- mechanics that does not stem from the wave properties of tary particles was further enshrined in the Schrodinger¨ equation, the particle. Rather, it posits corpuscular properties along with which describes the time evolution of quantum wave packets. nonlocal properties, all of which are deterministic. This change It is often pointed out that the formal analogy between of perspective points to deterministic properties in the Heisen- Schrodinger¨ wave interference and classical wave interference berg picture as primitive instead of the wave function, which allows us to interpret quantum phenomena in terms of the famil- remains an ensemble property. This way, within a double-slit iar classical notion of a wave. Indeed, wave-particle duality was experiment, the particle goes only through one of the slits. construed by Bohr and others as the essence of the theory and, In addition, a nonlocal property originating from the other in fact, its main novelty. Even so, the foundations of quantum distant slit has been affected through the Heisenberg equa- mechanics community have consistently raised many questions tions of motion. Under the assumption of nonlocality, uncer- (2–5) centered on the physical meaning of the wave function. tainty turns out to be crucial to preserve causality. Hence, a From our perspective and consistent with ideas first expressed (qualitative) uncertainty principle can be derived rather than by Born (6) and thereafter extensively developed by Ballentine assumed. (7, 8), a wave function represents an ensemble property as opposed to a property of an individual system. Author contributions: Y.A. conceived research; Y.A., E.C., and T.L. designed research; Y.A., What then is the most thorough approach to ontological ques- E.C., F.C., T.L., I.S., D.C.S., and J.T. performed research; and E.C., T.L., and J.T. wrote the tions concerning single particles (using standard nonrelativis- paper. tic quantum mechanics)? We propose an alternative interpreta- All authors are ordered alphabetically. tion for quantum mechanics relying on the Heisenberg picture, Reviewers: P.M., University of South Carolina; and N.T., Perimeter Institute. which although mathematically equivalent to the Schrodinger¨ Conflict of interest statement: Y.A. is a visiting scholar at Perimeter Institute, and has picture, is very different conceptually. For example, within the received funding from reviewer N.T.’s institution, Perimeter Institute. Heisenberg picture, the primitive physical properties will be rep- Freely available online through the PNAS open access option. resented by deterministic operators, which are operators with 1To whom correspondence may be addressed. Email: [email protected] and measurements that (i) do not disturb individual particles and [email protected]. (ii) have deterministic outcomes (9). By way of example, the 2E.C. and T.L. contributed equally to this work. 6480–6485 j PNAS j June 20, 2017 j vol. 114 j no. 25 www.pnas.org/cgi/doi/10.1073/pnas.1704649114 Downloaded by guest on September 25, 2021 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 . 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 j i in a Hilbert space H, and any observable A^ is a “directly verified,” we mean measured to an arbitrary accuracy Hermitian operator on H. The eigenstates of A^ form a complete in an arbitrarily short time (excluding practical and relativistic orthonormal system for H. When an ideal measurement of A^ 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 j i) and corresponds to an eigenvalue within the ^ relate to single particles in statistical