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: June 6, 2014) 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 asserts 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 case of open quantum systems as well. 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 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. arXiv:1405.6755v3 [quant-ph] 5 Jun 2014 ∗ [email protected] y [email protected] 2 CONTENTS I. Introduction 2 II. Preliminary Concepts 6 III. The Minimal Modal Interpretation 13 IV. The Measurement Process 40 V. Lorentz Invariance and Locality 48 VI. Conclusion 59 Acknowledgments 65 Appendix 66 References 73 I. INTRODUCTION A. Why Do We Need a New Interpretation? 1. The Copenhagen Interpretation Any mathematical-physical theory like quantum theory1 requires an interpretation, by which we mean some asserted connection with the real world. The traditional Copenhagen interpretation, with its axiomatic Born rule for computing empirical 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 [254, 287], the Copenhagen interpretation is still the most popular interpretation today. 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 questionable demarcation, known as the Heisenberg cut (Heisenbergscher Schnitt)[198, 304], between the large classical systems that carry out measurements and the small quantum systems that they measure; this ill-defined Heisenberg cut has never been identified in any experiment to date and must be put into the interpretation by hand. (See Figure1.) An associated issue is the interpretation's assumption of wave-function collapse|known more formally as the Von Neumann-L¨uders projection postulate [214, 305]|by which we refer to the supposed 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. 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 dis- tribution,3 or is it an irreducible ingredient of reality like the state of a classical system? 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 entanglements 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 supposedly 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 1 We use the term \quantum theory" here in its broadest sense of referring to the general theoretical framework consisting of Hilbert spaces 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 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 Recent work [35, 85, 247] casts considerable doubt on assertions that state vectors are nothing more than probability distributions over more fundamental ingredients of reality. 4 We will discuss all of these thought experiments in SectionIVC. 3 ? ? ? ? Heisenberg Cut ? ? ? ? Ψ Ψ Ψ Figure 1. The Heisenberg cut. 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, 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, an alternative to this project is always available: One could instead simply insist upon instrumentalism| known in some quarters as the \shut-up-and-calculate approach" [220, 222]|meaning that one should regard the mathematical formalism of quantum theory merely as a calculational recipe or algorithm for predicting measurement results and empirical outcome probabilities obtained by the kinds of physical systems (agents) that can self-consistently act as observers, and, furthermore, that one should refuse to make definitive metaphysical claims about any underlying reality.5 We provide a more extensive description of the instrumentalist approach in Appendix1d. On the other hand, if the physics community had uniformly 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 origin to people thinking seriously about the meaning of quantum theory. 5 \Whereof one cannot speak, thereof one must be silent." [326] 4 B. Our Interpretation In this paper and in a brief companion letter [34], we present a realist interpretation of quantum theory that hews closely to the basic structure of the theory in its widely accepted current form. Our primary goal is to move beyond instrumentalism and describe an actual reality that lies behind the mathematical formalism of quantum theory. We also intend to provide new hope to those who find themselves disappointed with the pace of progress on making sense of the theory's foundations [314, 317]. Our interpretation is fully quantum in nature. However, for purposes of motivation, consider the basic theoretical structure of classical physics: A classical system has a specific state that evolves in time through the system's configuration space according to some dynamical rule that may or may not be stochastic, and this dynamical rule exists whether or not the system's state lies beneath a nontrivial evolving probability distribution on the system's configuration space; moreover, the dynamical rule for the system's underlying state is consistent with the overall evolution of the system's probability distribution, in the sense that if we consider a probabilistic ensemble over the system's initial underlying state and apply the dynamical rule to each underlying