This chapter will be concerned with yet another asymmetry of causation. Recently, several authors have argued that there is an asymmetry of robustness or invariance. In particular, this chapter will consider three related purported asymmetries:

The phrase “the relationship between x and y” is intentionally vague, because different theorists have focused on different relationships.

In my initial picture, events are located in space and stretch through time. Causation links them. Our ability to infer that some events have occurred or will occur – when we know that other events occurred – rests on objective causal relations among events. This chapter will articulate and modify this picture. I shall argue that causation relates events in virtue of explanatory links between simple tropes. Tropes are located values of relevant variables or located instantiations of relevant properties. Claims about causal relations among events are true in virtue of the relations that obtain among simple tropes, and so it turns out that the theory of causation is largely independent of questions about the metaphysics of events and facts.

Some philosophers take tropes to be properties that are also particulars and specifically distinguish tropes from property exemplifications (Ehring 1997), and some take tropes to be fundamental and attempt to replace an ontology of substances and properties with an ontology of tropes (Williams 1953). My use of tropes is less metaphysically ambitious. I regard a trope – intuitively something like the whiteness of a particular pebble at a particular time – as a located instantiation of a property. I do not regard tropes as particularized properties, and I do not think they are ontologically fundamental, but I also think that this chapter's claims about the relata of the causal relation are independent of these questions concerning the status of tropes. The reason why I speak of “simple tropes” is that events can be conceived of as instantiations of exceedingly complicated properties and thus as themselves tropes.

Causation apparently has several different asymmetrical features. In this book I shall say what these features are and how they are related to one another. Here is a list of many of these purported asymmetries:

There is a kind of science of everyday phenomena at which we are all experts. We can all predict what will happen when gasoline is thrown on the fire, or when a rock is thrown at the window. None of us is surprised when heated water boils, or when cooled water freezes. These everyday scientific facts come easily.

This everyday science is readily extended to the laboratory, where we learn, for example, that sodium burns yellow, or that liquid helium is very cold. With work, we can learn more complicated facts, involving delicate equipment, and complicated procedures. The result is a kind of science of laboratory phenomena, not different in kind from the science of everyday phenomena.

But what about quantum mechanics? It is, purportedly at least, not about phenomena of the sort mentioned thus far. It is, purportedly at least, not about bunsen burners and cathode ray tubes and laboratory procedures, but about much smaller things – protons, electrons, photons, and so on. What is the relation between the science of quantum mechanics and the science of everyday phenomena, or even the science of laboratory phenomena?

It is no part of my aim to answer this question. However, it will be helpful to note some possibilities.

Quantum probability and quantum mechanics

The formalism of quantum probability theory

Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory.

Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology.

In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.

Review and preview

In chapters 6 and 7, I described some links between locality and the treatment of probabilities in models of the EPR–Bohm experiment. From chapter 6, the main lesson was that, given the strict correlations of quantum mechanics, factorizability of the two-time probabilities is equivalent to two-time determinism. I also noted there that an adequate factorizable theory – one that avoids Bell's theorem – is possible only if λ-independence fails. In chapter 7, I argued that weak symmetry, local determination, and independent evolution together entail deterministic results and weak deterministic transitions.

In chapter 8, I discussed the relationships among the locality conditions of chapter 6 and 7, Lorentz-invariance, and ‘metaphysical’ locality, especially local causality. I have by no means tried to give general answers to the questions raised there. Indeed, part of my thesis is that there is no general answer to be had. Nonetheless, one can use the results of chapters 6 and 7 to investigate the status of various interpretations. Most obviously, they can be used to evaluate whether a given theory is local: if a theory can be shown to violate one or more of the types of determinism entailed by the locality conditions of chapters 6 and 7, then that theory must be non-local in some sense.

Thus far I have been primarily concerned with how an interpretation of quantum mechanics might, or might not, be local. This question is traditionally not sharply distinguished from the question of whether quantum mechanics is consistent with the theory of relativity. However, the two questions are indeed quite distinct, and should be recognized as such explicitly.

The theory of relativity

What does relativity require?

As a first approximation, we may make the distinction by noting that the requirements of the theory of relativity are themselves unclear. Minimally, relativity seems to require that there be no way to distinguish one reference frame from another – i.e., that there be no experimental procedure that can determine which of two inertial observers is ‘really’ moving, or more generally that there be no way to discover an observer's absolute velocity.

Most authors are willing to find in (special) relativity a stronger requirement: namely, Lorentz-invariance. They say that relativity requires that in fact there is no such thing as absolute velocity. This requirement goes beyond the minimum – it might be that there is an absolute rest frame (so that absolute velocities are given by motion relative to this frame) while there is no way to find it. (As I will discuss in chapter 9, exactly this situation occurs in Bohm's theory.)

We now have under our belts several interpretations of quantum mechanics, each requiring, or anyhow advocating, some understanding of quantum probabilities. In the next two chapters, I will consider some connections among various interpretations of quantum probabilities and non-locality. I do so in the context of the well-known EPR–Bohm experiment, though it is worth emphasizing at the start that non-locality is very likely the rule rather than the exception for quantum-mechanical systems. Entanglement of systems occurs not only in the confines of a laboratory, but also in the course of quite typical interactions among quantum-mechanical systems.

Nonetheless, the EPR-Bohm experiment shines a bright light on the phenomenon of non-locality, and is therefore the most useful context in which to explore the relation between probability and non-locality. In this chapter, I consider models of the EPR–Bohm experiment that deliver probabilities for the various outcomes given the initial state of the pair of particles. In the next chapter, I consider fully dynamical models, i.e., ones that provide a dynamics for the complete state of the pair of particles as well as probabilities for various outcomes based on these complete states.

The EPR–Bohm experiment

The EPR–Bohm experiment is well known, but some observations about it are important for later.

How is orthodoxy possible?

How can an interpretation maintain both the eigenstate–eigenvalue link and indeterminism? Given the former, the properties possessed by a system are completely fixed by its quantum-mechanical state, but the quantum-mechanical state evolves deterministically, as I noted at the end of chapter 1. By themselves, then, the eigenstate–eigenvalue link and the quantum-mechanical equation of motion lead to determinism. Orthodoxy must change one of these things if it wants to maintain indeterminism.

Of course, it cannot change the eigenstate–eigenvalue link, lest it no longer be orthdoxy. Hence it changes the equation of motion. In this chapter, I will discuss two ways to change the quantum-mechanical equation of motion: by ‘interupting’ it from time to time with some other (indeterminsitic) equation, or by making a wholesale replacement. The first strategy I discuss in the next section, and the second in the subsequent section.

The projection postulate

Collapse as an analogue of Lüder's rule

Thus far, we have been working in the ‘Schrödinger picture’, according to which states evolve in time (according to the Scrödinger equation) and any given observable is at all times represented by the same operator. The Heisenberg picture reverses things: the states are constant in time and the operators representing observables change.