The Minimal Modal Interpretation of Quantum Theory Jacob A. Barandes1, ∗ and David Kagan2, y 1Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 2Department of Physics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747 (Dated: May 5, 2017) We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory permits assuming that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the more realistic case of open quantum systems despite the generic development of entanglement. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure|which we call subsystem space|that we believe merits further study. We conclude with a summary, a list of criteria for future work on quantum foundations, and further research directions. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text. ∗ [email protected] y [email protected] 2 CONTENTS I. Introduction 2 II. Preliminary Concepts 6 III. The Minimal Modal Interpretation 16 IV. The Measurement Process 45 V. Lorentz Invariance and Locality 53 VI. Conclusion 61 Acknowledgments 75 Appendix 76 References 84 I. INTRODUCTION A. Why Do We Need a New Interpretation? 1. The Copenhagen Interpretation Any mathematical-physical framework like quantum theory1 requires an interpretation, by which we mean some asserted connection between the theory's abstract formalism and the real world's physical, conceptual, logical, or empirical features. The textbook formulation of the Copenhagen interpretation, with its axiomatic Born rule for computing empir- ical outcome probabilities and its notion of wave-function collapse for establishing the persistence of measurement outcomes, works quite well in most practical circumstances.2 At least according to some surveys [271, 305], the Copenhagen interpretation is still the most popular interpretation today, despite the absence of a clear consensus on all the interpretation's precise metaphysical commitments. Unfortunately, the Copenhagen interpretation also suffers from a number of serious drawbacks. Most significantly, the definition of a measurement according to the Copenhagen interpretation relies on a physically questionable de- marcation, known as the Heisenberg cut (Heisenbergscher Schnitt)[211, 322], between the large classical systems that carry out measurements and the small quantum systems that they measure. (See Figure1.) This ill-defined Heisenberg cut has never been identified in any experiment to date and must be put into the interpretation by hand. Attempting to reframe the Heisenberg cut as being nothing more than an arbitrary quantum-classical boundary chosen for convenience conflicts with allowing quantum state vectors to represent objective features of reality. An associated issue is the Copenhagen interpretation's assumption of wave-function collapse, known more formally as the Von Neumann-L¨udersprojection postulate [227, 323]. By our use of the term \wave-function collapse" here, we are referring to the supposedly instantaneous, discontinuous change in a quantum system immediately following a measurement by a classical system, in stark contrast to the smooth time evolution that governs dynamically closed systems. 1 We use the term \quantum theory" here in its broadest sense of referring to the general theoretical framework consisting of Hilbert spaces, operator algebras of observables, and state vectors that encompasses models as diverse as nonrelativistic point particles and quantum field theories. We are not referring specifically to the nonrelativistic models of quantum-mechanical point particles that dominated the subject in its early days. 2 See Appendix1 for a detailed treatment of the textbook definition of the Copenhagen interpretation, as well as a description of the famous measurement problem of quantum theory and a systematic classification of attempts to solve it according to the various prominent interpretations of the theory. 3 The Copenhagen interpretation is also unclear as to the ultimate meaning of the state vector of a system: Does a system's state vector merely represent the experimenter's knowledge, is it some sort of objective probability distri- bution,3 or is it an irreducible ingredient of reality like the state of a classical system [172]? For that matter, what constitutes an observer, and can we meaningfully talk about the state of an observer within the formalism of quantum theory? Given that no realistic system is ever perfectly free of quantum entanglement with other systems, and thus no realistic system can ever truly be assigned a specific state vector in the first place, what becomes of the popular depiction of quantum theory in which every particle is purportedly described by a specific wave function propagating in three-dimensional space? The Copenhagen interpretation leads to additional trouble when trying to make sense of thought experiments like Schr¨odinger'scat, Wigner's friend, and the quantum Zeno paradox.4 2. An Ideal Interpretation Physicists and philosophers have expended much effort over many decades on the search for an alternative inter- pretation of quantum theory that could resolve these problems. Ideally, such an interpretation would eliminate the need for an ad hoc Heisenberg cut, thereby demoting measurements to an ordinary kind of interaction and allowing quantum theory to be a complete theory that seamlessly encompasses all systems in Nature, including observers as physical systems with quantum states of their own. Moreover, an acceptable interpretation should fundamentally (even if not always superficially) be consistent with all experimental data and other reliably known features of Nature, including relativity, and should be general enough to accommodate the large variety of both presently known and hypothetical physical systems. Such an interpretation should also address the key no-go theorems developed over the years by researchers working on the foundations of quantum theory; should not depend on concepts or quantities whose definitions require a physically unrealistic measure-zero level of sharpness in a sense that we'll develop in this paper; and should be insensitive to potentially unknowable features of reality, such as whether we can sensibly define \the universe as a whole" as being a closed or open system. 3. Instrumentalism In principle, a straightforward solution to this project is always available: One could instead simply insist upon instrumentalism, known in some quarters as the \shut-up-and-calculate approach" [234, 236]. Instrumentalism is a rudimentary interpretation in which one regards the mathematical formalism of quantum theory merely as being a calculational recipe or algorithm for predicting real-world measurement results and empirical outcome probabilities ob- tained by the kinds of physical systems (agents) that can self-consistently act as observers. Moreover, instrumentalism refrains from making further definitive metaphysical claims about any underlying reality.5 On the other hand, if the physics community had unquestioningly accepted instrumentalism from the very beginning of the history of quantum theory, then we might have missed out on the many important spin-offs from the search for a better interpretation: Decoherence, quantum information, quantum computing, quantum cryptography, and the black-hole information paradox are just a few of the far-reaching ideas that ultimately owe their origins to people thinking seriously about the meaning of quantum theory. 4. The Overdetermination Problem in Interpreting Quantum Theory There now exists a large variety of candidate interpretations of quantum theory, and, indeed, it has been argued that quantum theory suffers from an interpretational underdetermination problem [189]. However, precisely because all these candidate interpretations suffer from serious flaws and shortcomings|some of which we'll examine in detail in this paper|we argue that the real risk is that the nontrivial conceptual structure of quantum theory actually leads to an overdetermination problem that prevents the theory from admitting any complete, self-consistent, fully general, realist interpretation. One of our chief goals in presenting a new realist interpretation of quantum theory is, at a minimum, to provide a proof of principle that such an overdetermination problem doesn't exist. 3 Recent work [37, 87, 264] casts considerable doubt