It is shown that the components of the wave function of a measuring device, each of which represents a definite measurement result, do not correspond to many worlds as assumed by the many-worlds interpretation, because all components of the superposed wave function can be observed in our world by protective measurement, and they all exist in one world. According to the many-worlds interpretation, each component of the wave function of a measuring device that represents a definite measurement result corresponds to each world among the many worlds (Barrett 2011). This means that in one world there is only one component of the superposed wave function and the other components do not exist, and thus these components that cor-respond to the other worlds cannot be observed in this world. As a result, in every world the whole superposed wave function of the measuring device cannot be measured. It is unsurprising that the existence of such many worlds may be consistent with the results of conventional impulse measurements1, as the many-worlds interpretation is just invented to explain the emergence of these results, e.g. the definite measurement result in each world always denotes the result of a conventional impulse measurement. However, this does not guarantee consis-tency for all types of measurements. It has been known that there exists an-other type of measurement, the protective measurement (Aharonov and Vaid-
It is shown that the components of the wave function of a measuring device, each of which represents a definite measurement result, do not correspond to many worlds as assumed by the many-worlds interpretation, because all components of the superposed wave function can be observed in our world by protective measurement, and they all exist in one world. According to the many-worlds interpretation, each component of the wave function of a measuring device that represents a definite measurement result corresponds to each world among the many worlds (Barrett 2011). This means that in one world there is only one component of the superposed wave function and the other components do not exist, and thus these components that cor-respond to the other worlds cannot be observed in this world. As a result, in every world the whole superposed wave function of the measuring device cannot be measured. It is unsurprising that the existence of such many worlds may be consistent with the results of conventional impulse measurements1, as the many-worlds interpretation is just invented to explain the emergence of these results, e.g. the definite measurement result in each world always denotes the result of a conventional impulse measurement. However, this does not guarantee consis-tency for all types of measurements. It has been known that there exists an-other type of measurement, the protective measurement (Aharonov and Vaid-
It is a fundamental and widely accepted assumption that a measure-ment result exists universally, and in particular, it exists for every ob-server, independently of whether the observer makes the measurement or knows the result. In this paper, we will argue that, based on an analysis of protective measurements, this assumption is rejected by the many-worlds interpretation of quantum mechanics, and worlds, if they indeed exist ac-cording to the interpretation, can only exist relative to systems which are decoherent with respect to the measurement result. In standard quantum mechanics, it is postulated that when the wave func-tion of a quantum system is measured by a macroscopic device, it no longer follows the linear Schrödinger equation, but instantaneously collapses to one of the wave functions that correspond to definite measurement results. However, this collapse postulate is ad hoc, and the theory does not tell us why and how a definite measurement result appears. This measurement problem is widely
My view on the meaning of the quantum wave function and its connection to protective measurements is described. The wave function and only the wave function is the ontology of the quantum theory. Protective measurements support this view although they do not provide a decisive proof. A brief review of the discovery and the criticism of protective measurement is presented. Protective measurements with postselection are discussed. Introduction In the first graduate course of quantum mechanics I remember asking the question: “Can we consider the wave function as a description of a single quantum system?” I got no answer. Twelve years later, in South Carolina, after I completed my Ph.D. studies at Tel Aviv University under the supervision of Yakir Aharonov in which we developed the theory of weak measurements [1], I asked Aharonov: Can we use weak measurement to observe the wave function of a single particle? At that time I had already become a strong believer in the many-worlds interpretation (MWI) of quantum mechanics [2] and had no doubt that a single system is described by the wave function. Yakir Aharonov never shared with me the belief in the MWI. When we realized that using what is called now protective measurement , we can, under certain conditions, observe the wave function of a single quantum system, he was really excited by the result. At 1992 I was invited to a conference on the Foundations of Quantum Mechanics in Japan where I presented this result: “The Schrödinger wave is observable after all!”[3]. Then I went home to Tel Aviv where I finished writing a letter which received mixed reviews in Phys. Rev. Lett. , while Jeeva Anandan, working on the topic with Aharonov in South Carolina, wrote a paper accepted in Phys. Rev. A [4]. After acceptance of the PRA paper it was hard to fight the referees in PRL, but PLA accepted it immediately [5].
In this book, I have argued that the wave function in quantum mechanics is real, and it represents the state of random discontinuous motion (RDM) of particles in three-dimensional space. Moreover, this picture of quantum ontology is complete in accounting for our definite experience, but it requires that the quantum dynamics be revised to include a stochastic, nonlinear evolution term resulting from the RDM of particles. Obviously I took a road less traveled by researchers in the foundations of quantum mechanics. In order to convince more readers that this road is deserved to be taken, I will review the main results of this book and discuss them in a broader context. The starting point of my road is protective measurements (Chapter 1). In 1993, this new method of measurement in quantum mechanics was discovered (Aharonov and Vaidman, 1993; Aharonov, Anandan, and Vaidman, 1993). Distinct from conventional projective measurements, protective measurement is a method for measuring the expectation value of an observable on a single quantum system. By a series of protective measurements, one can even measure the wave function of a single quantum system. Therefore, besides the Born rule for projective measurements, protective measurements provide another, more direct connection between the wave function and results of measurements, which is not probabilistic but definite. It can be expected that a definite connection between the wave function and results of measurements is more important for understanding the meaning of the wave function and searching for the ontology of quantum mechanics. However, it seems that this connection provided by protective measurements is still less well known today, and its significance has not been fully realized by most researchers, either. The first stop on my road is the reality of the wave function (Chapter 4). It is correct to say that protective measurements alone do not imply the reality of the wave function. An additional connection between the state of reality and results of measurements is needed.
Introduction One of the crucial issues about protective measurements is the extent to which they provide (additional) grounds for some realist understanding of the wave function. This issue is notoriously subject to debate, which is not the aim of this chapter. Rather, this chapter aims at clarifying the further issue of the ontological picture corresponding to the realist understanding of the wave function possibly suggested by protective measurements. Oddly enough, proponents of a realist reading of protective measurements remain in general rather vague about ontology (see for instance Aharonov et al. (1993, 4624), who assimilate the wave function to an “extended object” without further details, or the brief and somewhat unprecise discussion in Dickson (1995, 135–136) of the interpretation of the wave function within Bohmian mechanics (BM) and within the theoretical framework elaborated by Ghirardi, Rimini and Weber (GRW); in this volume, Gao does propose a more elaborated ontological model for quantum mechanics (QM) based on protective measurements). Despite this vagueness (or maybe because of it), it seems that the most obvious ontological intuition resulting from protective measurements points towards some form of straightforward realism about the wave function understood in the sense of a real, physical field: indeed, protective measurements are claimed to allow measuring expectation values of observables on a single quantum system, thereby providing the possibility of reconstructing (i.e. “measuring” in some sense) the wave function of a single quantum system. Aharonov et al. (1996, 125) clearly express this intuition: “We can observe the expectations values of operators, and we can observe the density and the current of the Schrödinger wave. We can “see” in some sense the Schrödinger wave. This leads us to believe that it has physical reality.” Therefore, this kind of intuition seems in line with a realist conception of the wave function, according to which the wave function is considered as a real substantial, material in some sense) entity on its own (this view is often called “wave function realism”).
Introduction It is a prima facie reasonable assumption that if a physical quantity is measurable, then it corresponds to a genuine physical property of the measured system. You can measure a person's mass because human beings have such a property. You can measure the average mass of a group of people because groups of people have such a collective property. And so on. Now it would be truly surprising – miraculous, perhaps – if you could determine the average mass of a group of people by making measurements on just one of them. To ascribe such a statistical property to an individual looks like a category mistake. At first glance, protective measurements seem to pull off just such a miracle, determining, for example, the expectation value of position for an ensemble of particles via a measurement performed on one of them. The lesson we are supposed to draw, of course, is that expectation values are not statistical properties at all, despite their name. Rather than being an average over an ensemble of systems, the expectation value of position for a particle is a physical property of the individual system, and the wave function, as the bearer of these properties, is a physical entity (Aharonov, Anandan and Vaidman, 1993). The protective measurement procedure has been challenged (Uffink, 1999; Gao, 2013; Uffink, 2013), but for present purposes I will assume that protective measurements exist, at least in principle, that are capable of revealing “statistical” properties like expectation values in a single measurement. My aim here is not to challenge the existence of such a physical procedure, but rather to explore the arguments that connect the existence of protective measurements with conclusions concerning the nature of physical reality. What protective measurements are sup- posed to show is that “epistemological” interpretations of the quantum state are untenable – that the wave function of a system must instead be interpreted “onto- logically” (Aharonov, Anandan and Vaidman, 1993: 4617).