Naturalisms

For many philosophers and scientists, the word “naturalism” conjures up a metaphysical and a methodological thesis. Both concern objects that are out there in “nature,” meaning things that exist in space and time. The contrast is with supernatural entities; if such things exist, they exist outside of space and time:

I put an “s” subscript on the second naturalism to mark the fact that it gives advice about doing science. These two naturalisms are the ones that get cited in discussions of the conflict between evolutionary theory and creationism. Evolutionary biologists often say that their theory obeys the requirements of methodological naturalism but is silent on the metaphysical question. They further contend that creationism rejects both these naturalisms; here they are helped by creationists themselves, who often express their belief in a supernatural deity and argue that methodological naturalisms is a shackle from which science needs to break free. Although it is worth inquiring further into this interpretation of evolutionary theory and creationism, I won't do so here. Rather, I am interested in a third naturalism. Like the second, it is methodological, but it is aimed at the practice of philosophy, not of science (hence the “p” subscript that I use to label it):

There are trivial similarities linking science and philosophy that lend a superficial plausibility to this naturalistic thesis.

Classical ergodicity retains its meaning in the quantum realm when the employed measurement is protective. This unique measuring technique is re-examined in the case of post-selection, giving rise to novel insights studied in the Heisenberg rep-resentation. Quantum statistical mechanics is then briefly described in terms of two-state density operators.

Introduction

In classical statistical mechanics, the ergodic hypothesis allows us to measure position probabilities in two equivalent ways: we can either measure the appropriate particle density in the region of interest or track a single particle over a long time and calculate the proportion of time it spent there. As will be shown below, certain quantum systems also obey the ergodic hypothesis when protectively measured. Yet, since Schrödinger's wave function seems static in this case [1, 2, 3], and Bohmian trajectories were proven inappropriate for calculating time averages of the particle's position [4, 5], we will perform our analysis in the Heisenberg representation.

Indeed, quantum theory has developed along two parallel routes, namely the Schrödinger and Heisenberg representations, later shown to be equivalent. The Schrödinger representation, due to its mathematical simplicity, has become more common. Yet, the Heisenberg representation offers some important insights which emerge in a more natural way, especially when employing modular variables [6]. For example, in the context of the two-slit experiment it sheds a new light on the question of momentum exchange [7, 8, 9]. Recently studied within the Heisenberg representation are also the double Mach–Zehnder interferometer [10] and the N-slit problem [11]. As can be concluded from [11], the Heisenberg representation prevails in emphasizing the non-locality in quantum mechanics, thus providing us with insights about this aspect of quantum mechanics as well.

Equipped with the backward evolving state-vector within the framework of two-state-vector formalism (TSVF) [12], the Heisenberg representation becomes even more powerful since the time evolution of the operators includes now information from the two boundary conditions.

We generalize protective measurement for protective joint measurement of several observables. The merit of joint protective measurement is the determination of the eigenstates of an unknown Hamiltonian rather than the determination of features of an unknown quantum state. As an example, we precisely determine the two eigenstates of an unknown Hamiltonian by a single joint protective measurement of the three Pauli matrices on a qubit state.

Introduction

Protective measurement is one of the unexpected consequences of the strange structure of quantum mechanics. According to general wisdom, we cannot gain information on the unknown state ρ of a single quantum system unless we distort the state itself. In particular, we cannot learn the unknown state of a single system whatever test we apply to it. It came as a surprise that in weak measurements [1] the expectation value 〈Â〉 of an observable  can be tested on a large ensemble of identically prepared unknown states in such a way that the distortions per single systems stay arbitrarily small (see [2], too). An indirectly related surprise came with the so-called protective measurements [3, 4, 5] capable of testing 〈Â〉 at least in an unknown eigenstate of the Hamiltonian Ĥ at arbitrarily small distortion of the state itself. Interesting debates followed the proposal as to the merit of protective measurement in the interpretation of the wave function of a single system instead of a statistical ensemble (see, e.g., [6] and references therein).

My work investigates an alternative merit of protective measurement. First I construct joint protective measurements of several observables Â1, Â2, … and re-state the original equations for them in a general form. Then I show that the straightforward task that a single joint protective measurement solves on a single system is the determination of the eigenstates of an otherwise unknown Hamiltonian.

In this chapter we show how the weak values, are related to the T0µ(x, t) component of the energy–momentum tensor. This enables the local energy and momentum to be measured using weak measurement techniques. We also show how the Bohm energy and momentum are related to T0µ(x, t) and therefore it follows that these quantities can also be measured using the same methods. Thus the Bohm “trajectories” can be empirically determined, as was shown by Kocsis et al. (2011a) in the case of photons. Because of the difficulties with the notion of a photon trajectory, we argue the case for determining experimentally similar trajectories for atoms where a trajectory does not cause these particular difficulties.

Introduction

The notion of weak measurement introduced by Aharonov, Albert and Vaidman (1988) and Aharonov and Vaidman (1990) has opened up a radically new way of exploring quantum phenomena. In contrast to the strong measurement (von Neumann, 1955), which involves the collapse of the wave function, a weak measurement induces a more subtle phase change which does not involve any collapse. This phase change can then be amplified and revealed in a subsequent strong measurement of a complementary operator that does not commute with the operator being measured. This amplification explains why it is possible for the result of a weak spin measurement of a spin-1/2 atom to be magnified by a factor of 100 (Aharonov et al., 1988; Duck, Stevenson and Sudarshan, 1989). A weak measurement, then, provides a means of amplifying small signals as well as allowing us to gain new, more subtle information about quantum systems.

One of the new features that we will concentrate on in this chapter is the possible measurement of the T0µ (x, t) components of the energy–momentum tensor.

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”).

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].