Introduction

Preamble

In physics, any complete scientific theory consists of two important components: a formalism and an interpretation. The formalism, which takes the form of a set of equations and some computational rules, is essential for the empirical adequacy of the theory and for a formal explanation. The interpretation provides a clear understanding of the ontological structure underlying the theory, since it answers the question of what there is at the fundamental level of reality.1 Therefore, we can say that every scientific theory in physics is supposed to fulfill three different roles:

1. it is expected to successfully predict certain events;

2. it is expected to provide a mathematical explanation of why they occur, by appealing to mathematical rules or mathematical objects;

3. it is expected to provide an understanding of the ontology, i. e. of what exists in the world according to the theory.

While physicists are more concerned with the empirical adequacy of the theory and the formal explanation, philosophers of physics are more inclined towards the ontological interpretation.

The aim of this book is to present and elaborate on the diverse ontological interpretations of Bohmian mechanics, which have been developed to answer the question of how the two main elements that constitute the Bohmian theory, the Bohmian particles and the Bohmian wave-function, should be ontologically interpreted. What are the properties of the Bohmian particles? How can we characterize their individuality? And what is the wave-function?

Why Bohmian Mechanics?

Bohmian mechanics is not the standard quantum theory normally endorsed by the scientific community. It is therefore important to briefly alert the reader of the difference between the orthodox theory of quantum mechanics, also called the Copenhagen interpretation, and how it differs from Bohmian mechanics. The Copenhagen interpretation is based on two elements: a wave-function ? and some macroscopic variables ? corresponding to the outcomes of certain experiments. A few important points need to be made about this theory:

• The microscopic level is described only by the mathematical object called the wave-function. The wave-function provides epistemic and subjective knowledge of the microscopic domain and successfully demonstrates predictive power and usefulness. However, this object also presents some mathematical features that, if regarded as revealing the nature of the microscopic ontology, would imply the existence of a very vague world. In such a world, microscopic entities are in a superposition of incompatible states, particles do not have continuous trajectories and so on. The consequence of this is that most defenders of the Copenhagen interpretation claim that we should not commit to a quantum reality that goes beyond the quantum description, maintaining that there is nothing there beyond the wave-function. But if this is so - if there is no microscopic world that we can be knowledgeable about - then it is not exactly clear what quantum mechanics describes at the microscopic level.

• The macroscopic level consists of our familiar objects and the outcomes of experiments (identified with variables ?) that we perform in our laboratories. In such a world, our experimental results are definite (for instance, cats are either dead or alive) and macroscopic objects always have precise positions. The wave-function is salient in our macroscopic world only because it gives us the right probabilities of the outcomes of our experiments.

• The key problem that the Copenhagen interpretation has to face is how it can bridge the fuzziness of the microscopic 'domain' or 'world' and the definite character of our classical world. In order to do this, many philosophers of physics introduced the role of observers, which have the power to create definite outcomes. This is often presented with the simplistic expression that, in quantum mechanics, 'observers create reality'.

All of this is murky and obscure. The Copenhagen interpretation, in sum, faces a dilemma: either it gives up the role of providing an understanding of what exists at the fundamental level of reality and remains only as an epistemic (and incomplete) theory, or it takes up this role but provides a very problematic and unsatisfactory ontological picture.

I take it that many may be happy with the belief that observers create reality or that at the most fundamental level physical entities are in a superposition of states. But this book aims to discuss another proposal, the theory called 'Bohmian mechanics', which I consider to be much clearer and more straightforward. Bohmian mechanics offers a formalism that is based on two elements: the wave-function ? (which is the same as the one in the Copenhagen interpretation) and the position of particles Q. The theory consists of the following:

• At the microscopic level, we have particles that follow continuous trajectories that are determined by the wave-function via the guiding law of motion for the particles.

• At the macroscopic level, all the objects consist of conglomerates of these particles, and all the measurement outcomes of our laboratories consist in particles' positions.

Bohmian mechanics takes the most natural ontological elements we know of - the particles - and, with them, it implements standard quantum mechanics in order to answer the question of what exists in the world. In other words, it completes standard quantum mechanics by presupposing an 'obvious ontology evolving in an obvious way'2 (which the Bohmians refer to with the acronym OOEOW) at the microscopic level.3

This characterization of Bohmian mechanics, however, is not enough for us to acquire a clear ontological interpretation. Indeed, it is only the first step. In order to fully understand the theory, we also need to be able to metaphysically characterize this ontology. The fact that we have particles and a wave-function in Bohmian mechanics offers us a stable ground on which we can build a sound discussion of what exists in the world, but several questions are still at stake. First of all, we need to investigate the status of the particles - that is, whether they instantiate properties, and how they can be metaphysically characterized. Secondly, and most importantly, we need to understand the ontological status of the wave-function. We said that in Bohmian mechanics the wave-function has a dynamical role: it determines the dynamics of the particles. But it is not clear how this determination occurs, and, consequently, it is not clear how the wave-function should be interpreted ontologically. Moreover, what I aim to show is that ontological interpretations are always intertwined with metaphysical considerations about the kinds of entities that our ontology deals with. In this book, there will be plenty of examples: for instance, it will not be enough to characterize the wave-function as a field or as a law, for we also need to spell out what being a 'field' or a 'law' metaphysically amounts to. Indeed, questions arise concerning the nature of fields (i. e. whether fields are only a collection of intrinsic properties of space or are concrete particulars) and of laws (i. e. whether laws are only descriptions of the world or have modal power). Another example is concerned with the status of particles and what we mean when we define particles as 'point-like' objects. In most cases, it seems that we need to endorse a particular metaphysical position regarding how to characterize the ontology of the world.

The Methodology

While developing an ontological interpretation of these two elements, the problem of the relation between formalism and interpretation will inevitably arise. In the literature, there are two main approaches to understanding the constitution of the Bohmian ontology. Sometimes, the characterization of the ontology is mainly determined by our metaphysical presuppositions of what the world should be like. For example, in the discussion of the particles, we will see that it is usually presupposed that Bohmian particles are impenetrable. But this presupposition is more determined by our metaphysical concerns and not by the formalism, which in fact allows particles to share the same position at the same time, even though this, contingently, never happens. Other times, the metaphysical characterization of the ontology is mainly determined by the formalism. For example, we will see that the realists interpret the wave-function as a physical field in configuration space because they rely on a process of reification of mathematical entities.

In this book, I discuss and evaluate both approaches. Most of the time, my approach will navigate between the 'Scylla' of our metaphysical presuppositions of what exists in the world and the 'Charybdis' of the mathematical formalism. My methodology will involve a back-and-forth approach between mathematical and metaphysical...