Causation and chance in quantum mechanics

The competing and complementary concepts of causality within the dispositional or powers approach is already quite intricate, as indicated by a previous post.  There are also concepts of causality that do not follow a dispositional approach and are in fact better known.

For a concise and balanced overview of the status see Anjum and Mumford Causality, Oxford 2014. My previous post provided a commentary on Chapter 3 of "What Tends to Be the Philosophy of Dispositional Modality", which examined a dispositional ontological for objective probability. In Chapter 4 of the same book Anjum, Mumford and Andersen examine how a dispositional theory of causality stands up against the ontological challenges of quantum mechanics. 

 The understanding of causality that is still the most influential can be traced back to David Hume's sceptical analysis and Emmanuel Kant's response. In A Treatise of Human Nature (1973, Book I, Part III, Section VI) Hume argued that all that can be observed in nature is a series of events. One thing happens and then another, and then another, and so on. Whether any of those events are causally connected is not itself part of experience of events. For example, a match is struck and then almost immediately that same match lights, but what is not observed is that the striking of the match caused it to light. If there is cause and effect, then it is not known through direct observation of events. But what do our physical theories tells us? In Newtonian physics one thing happens followed by another but that following is determined by the laws of physics. The state of the world at one time determines the state of the world later. That this seems not to be the case in a world described by quantum mechanics leads to claims that causality no longer holds. In standard quantum mechanics a later state does follow deterministically from an earlier state but measurement breaks the causal link.

Heisenberg and Bohr (Walter Heitler and Léon Rosenfeld in the background), in the mid 1930s 

Now let's look at those features of classical causation that were explicitly discussed by Bohr and Heisenberg: necessitation, determinism, predictability, and separability.

Classical causation

Hume was sceptical about a necessary connection between cause and effect because none can be known from experience. However, the intuition that a cause necessitates an effect is strong and led Kant to formulate causal necessity as a precondition for science.  It may seem strange that the radical empiricist, Hume, is seen to be undermining the scientific enterprise but his analysis, according to Kant, denies the possibility of scientific explanation. Kant, in the Critique of Pure Reason, states:

[T]he very concept of a cause so obviously contains the concept of a necessity of connection with an effect and a strict universality of rule that it would be entirely lost if one sought, as Hume did, to derive it from a frequent association of that which happens with that which precedes and a habit (thus a merely subjective necessity) of connecting representations arising from that association.

Kant's influence was strong, especially in Germany, and Heisenberg felt compelled to respond given the empirical success of quantum mechanics. In the lectures "Physics and Philosophy" he uses the radioactive decay of the radium atom as an example in which there is no predictability of the timing of a decay event.  There is no decay event that follows through necessity from the prior state of the atom. There is no quantum law that necessarily connects the state of the radium atom with the decay event. However, quantum theory does predict the probability of a decay event per unit time. There is no determinacy. If necessity and determinism are characteristic of a cause, then causality does not hold in quantum mechanics. 

Heisenberg, in agreement with Bohr argues that causality is needed but only as a classical concept for interpreting experiments. This is because we do not directly observe the decay but use a detector and it is taken as a rule that the decay causes a chain of events that result in what is detected or measured. 

The question remains whether necessitation and determinism are characteristic of all theories of causality. A further question is whether it is coherent to have classical causal laws holding in interpreting measurements but not in atomic physics.

Without determinism and necessitation at the fundamental level, the same event would not always follow from two identical sets of initial conditions or states. Predictions in the quantum realm, prior to detection, are probabilistic in general and this provides a reason for rejecting a classical causal interpretation of quantum mechanics.

The connection between determinism and prediction is that the former provides the metaphysical ground for the later.

In addition, the classical theory of causation has the cause and the effect as two distinct things. How distinct must these two things be? The predictions of quantum mechanics, in addition to being indeterministic, do not, according to Bohr (Causality and Complementarity, Philosophy of Science, Vol. 4, No. 3, 1937), allow the separation of the microscopic quantum system and the measuring apparatus. This gives rise to the notion that the act of measurement is in part responsible for the measured result. Cause and effect cannot be separated and so there is no clear demarcation between them. Classical causality breaks down again.

In the same paper Bohr insists that "the concept of causality underlies the very interpretation of each result of experiment, and that even in the coordination of experience one can never, in the nature of

things, have to do with well-defined breaks in the causal chain."

So, for Bohr there is a theory of atomic and sub-atomic physics, quantum mechanics, but there is no separable cause and effect in that theory. After an event is detected by the measuring apparatus then classical cause and effect come into play.

Potentiality

Heisenberg was not content to leave atomic physics inexplicable and so proposed an ontology at the atomic level with substance in isolation understood as pure potentiality

All the elementary particles are made of the same substance, which we may call energy or universal matter; they are just different forms in which matter can appear. If we compare this situation with the Aristotelian concepts of matter and form, we can say that the matter of Aristotle, which is mere ‘potentia’, should be compared to our concept of energy, which gets into ‘actuality’ by means of the form, when the elementary particle is created. (Heisenberg 1959)

Every time a potency gets actualised, causation happens and that is due to isolated substance encountering actualised matter. In mentioning particle creation Heisenberg has moved from quantum mechanics to field theory and his claim 'poentia' are energy like can only be sustained in an Aristotelian sense that has little to do with the concept in physics. As will be explained below, potentiality can be introduced within quantum mechanics using the theory outlined in the previous post. The idea is that objects behave the way they do, not because of some external laws that determine what happens to them, but because of their own intrinsic dispositions and their interactions. Let's follows Anjum and Mumford and call such a theory of causality neo-Aristotelian.

Neo-Aristotelian causality does not invoke necessitation. Instead, there are irreducible tendencies. It adopts dispositional modality rather than conditional necessity. A tendency is less than necessity, so the effect is not guaranteed by its cause.

As described in the previous post, a typical example of a probabilistic disposition is the 50:50 propensity of a fair coin to land heads or tails if tossed, while a non-probabilistic disposition could be the propensity of a vase to break if dropped onto a firm surface.

Neo-Aristotelian causation is not a relation between two separate events or objects, but is a continuous, unified process that typically takes time to unfold. One way to understand an event as stochastic is that there is some objective probabilistic element involved. In other words, a chance event. This is an ontological interpretation of probability, which contrasts with the purely epistemic notion of credence or subjective probability. An individual event could therefore still be caused, in the neo-Aristotelian sense, even if it is random to some degree and not predictable. Typically, this type of causation takes place when the possibilities exist, when the effect is enabled by the right stimulus and under the right conditions. Neo-Aristotelian causation happens once the disposition manifests itself. 

An object with the potentiality to manifest as possible events will find itself in a situation where there are enabling dispositions and interfering dispositions. For example, firewood is disposed to ignite but the process leading up to it burning requires the presence of manifestation partners: a suitable site, proper ignition, enough oxygen, and so on, and that inhibitors such as dampness are not too strong. The firewood, its enablers and its inhibitors are all active in the cause of the effect. No ontological distinction is drawn between properties belonging to the object undergoing change and the contextual properties in this process. They are all, in a general sense, causes of the specific outcome. Depending on the balance the firewood may

• Burn brightly and sustainably
• Burn but go out quickly
• Smolder and smoke
• Fail to light at all.

In summary, neo-Aristotelian causation

• Involves irreducible tendencies
• Is not deterministic, although some process can be close to deterministic
• Supports predictions of what tends to happen, not what will happen with certainty
• Is a unified process, not a relationship between two distinct events.

A sketch of quantum causation

Take the electron as a typical quantum object. We need to identify what possibilities it can potentially manifest and what is needed to enable that manifestation.

The possible manifestations of the electron are as values of position, momentum, spin, charge, and mass. At this level of description charge and mass are classical properties. The others are potential properties and from quantum theory there is structure to and constraints on how the other properties can appear. This is captured in the formulation of quantum mechanics favoured in this blog that uses the generalisation of probability to a \(\sigma\)-complex.

As indicated, at this stage I only propose a sketch of how this may work. Developing the detail and confirming whether the proposed process is correct will require much more work.

For the potentiality to manifest the electron cannot be isolated. It therefore interacts with other objects. It is proposed that the interaction selects one \(\sigma\)-algebra from the complex. That is, the context will be for certain spin values or position or momentum to manifest. This provides a preliminary selection of a \(\sigma\)-algebras from the complex and therefore standard probability description of the tendency of the electron properties to manifest. Now this manifestation takes place via a Monte-Carlo selection process from the probability distribution associated with the selected \(sigma\)-algebra. This provides a model of the causal chain from potentiality to actuality in the quantum domain.

To make this more concrete consider the analysis of the double slit experiment.  In the initial version discussed previously the slits split the quantum state to produce an interference effect. The context is already selecting position and therefore one probability distribution from those possible. At the detecting screen the position of the electron is made actual through Monte-Carlo selection from the probability distribution.

Now consider the addition of a pointer immediately after the double slit. The pointer tends to point towards the electron. The pointer description is purely quantum and is not a measurement apparatus. As my earlier analysis shows the presence of the pointer eliminates the interference effect and selects a probability distribution for the electron position that is a normalised sum of the distributions that would obtain if the electron only passed through one slit. Again, the context has selected the probability distribution over position and at the detecting screen the position of the electron is made actual with a Monte-Carlo selection from the probability distribution.

Potentiality and probability

As outlined in the previous post, Barbara Vetter (Potentiality from Dispositions to Modality Oxford University Press, 2015) developed the concept of potentiality in her theory of dispositional powers. In that theory potentials are dispositions that are responsible for the manifestation of possibilities. The possibilities then tend to become actual events or states of affairs. The concept of 'potential' in philosophy, in a sense close to that discussed here, goes back at least to Aristotle in Metaphysics Book \(\Theta\).

In contrast probabilities are weightings summing to one that describe in what proportion the possibilities tend to appear. I propose that the potential underpins the actual appearance of the possibilities while probability shapes it. This will be discussed further in this post. 

Barbara Vetter proposed a formal definition of possibility in terms of potentiality:

POSSIBILITY:  It is possible that \(p =_{def}\) Something has an iterated potentiality for it to be the case that \(p\).

 So, it is further proposed that the probabilities are the weights that can be measured through this iteration using the frequency of appearances of each possibility. Note that this indicates how probabilities can be measured but it is not a definition of probability.

In the field of disposition research there is an unfortunate proliferation of terms meaning roughly the same thing. The concept of 'power' brings out a disposition's causal role but so does 'potential'. As technical terms in the field both are dispositions. Now 'tendency' will also be introduced, and it is often used as yet another flavour of disposition.

 Tendencies

Barbara Vetter mentions tendencies in passing in her 2015 book on potentiality and, although she discusses graded dispositions, tendencies are not a major topic in that work. In "What Tends to Be the Philosophy of Dispositional Modality" Rani Lill Anjum and Stephen Mumford (2018) provide an examination of the relationship between dispositions and probabilities while developing a substantial theory of dispositional tendency. In their treatment powers are understood as disposing towards their manifestations, rather than necessitating them. This is consistent with Vetter's potentials. Tendencies are powers that do not necessitate manifestations but nonetheless the power will iteratively cause the possibility to be the case.

In common usage a contingency is something that might possibly happen in the future.  That is, it is a possibility. A more technical but still common view is that contingency is something that could be either true or false. This captures an aspect of possibility, but not completely because there is no role for potentially; something responsible for the possibilities.  There is also logical possibility in which anything that does not imply a contradiction is logically possible. This concept may be fine for logic but in this discussion, it is possibilities that can appear in the world that are under consideration. Here an actual possibility needs a potentiality to tend to produce it.

Example (adapted from Anjum and Mumford)

Struck matches tend to light. Although disposed to light when struck, we all know that there is no guarantee that they will light as there are many times a struck match fails to light. But there is an iterated potentiality for it to be the case that the match lights. The lighting of a struck match is more than a mere possibility or a logical possibility. There are many mere possibilities towards which the struck match has no disposition - that is no potential in the match towards struck matches melting, for instance.

Iterated potentiality provides the tendency for possible outcomes to show some patterns in their manifestation. In very controlled cases the number of cases of success in iterations of match striking could provide a measure of the strength of the power that is this potentiality. This would require a collection of matched that are essentially the same.

Initial discussion of probability

Anjum and Mumford introduce their discussion of probability through a simple example that builds on a familiar understanding of dispositional tendencies associated with fragility.

"The fragility of a wine glass, for instance, might be understood to be a strong disposition towards breakage with as much as 0.8 probability, whereas the fragility of a car windscreen probabilities its breaking to the lesser degree 0.3. Furthermore, it is open to a holder of such a theory to state that the probability of breakage can increase or decrease in the circumstances and, indeed, that the manifestation of the tendency occurs when and only when its probability reaches one." 

This example is merely an introduction and needs further development but already the claim that "the manifestation of the tendency occurs when and only when its probability reaches one" shows that it is not a model for objective probability. What is needed is a theory of dispositions that explains stable probability distributions. Of course, if the glass is broken then the probability of it being broken is \(1\). However, this has nothing to do with the dispositional tendency to break. What is needed is a systemic understanding of the relationship between the strength of a dispositional tendency and the values or, in the continuous case, shape of a probability distribution.

In the quoted example above each power is to be understood in terms of a probability of the occurrence of a certain effect, which is given a specific value. The fragility of a wine glass, for instance, might be understood to be a strong disposition towards breakage with as much as 0.8 probability, whereas the fragility of a car windscreen is less, and the probability of its breaking is a lesser degree 0.3. But given a wine glass or windscreen produced to certain norms and standards it would be expected that the disposition towards breakage would be quite stable. A glass with a different disposition would be a different glass.

Anjum and Mumford claim, that in some understandings the manifestation of a possibility occurs when and only when its probability reaches one (see Popper, "A World of Propensities", 1990: 13, 20). This is a misunderstanding of how probability works. Popper distinguished clearly between the mathematical probability measure and what he called the physical propensity, which is more like a force, but Popper does limit a propensity to have a strength of at most \(1\).   As I will attempt to show below, Popper in proposing propensity interpretation of objective probabilities oversimplifies the relationship between dispositions and probabilities. This confusion led Humphreys to draft a paper (The Philosophical Review, Vol. XCIV, No. 4 (October 1985)) to show that propensities cannot be probabilities. As indeed they are not. They are dispositions. That would leave open the proposition that probabilities are dispositional tendencies, but that also will turn out to be untenable.

The proposal by Anjum and Mumford that powers can over dispose does seem to be sound. Over disposing is where there is a stronger magnitude than what is minimally needed to bring about a particular possibility. This indicates that there is a difference between the notion of having a power to some degree and the probability of the power’s manifestation occurring. Among other conclusions, this also shows that the dispositional tendency does not reduce to probability, preserving its status as a potential. 

 Anjum and Mumford continue the discussion using 'propensity' as having a tendency to some degree, where degree is non-probabilistically defined.  Anjum and Mumford use the notions of ‘power’, ‘disposition’ and ‘tendency’ more or less interchangeably, whereas an object may have a power to a degree there are powers that are simply properties. In what follows I try to will eliminate the use of 'propensity', except where commenting of the usage of others, and use 'tendency' to qualify either 'power', 'potential' or 'disposition' rather than let it stand on its own.

A probability always takes a value within a bounded inclusive range between zero and one. If probability is \(1\) then probability theory stipulates that it is almost certain (occurs except for a set of cases of measure zero). In contrast to what Anjum and Mumford claim it is not natural to interpret this as necessity because there are exceptions. For cases where there are only a finite set of possibilities then probability \(1\) does mean that there are no exceptions. But as this is a special case in applied probability theory there is no justification in equating it with logical or metaphysical necessity.

A power must be strong enough to reach the threshold to make the possibilities actual.  Once the power is strong enough then the probability distribution over the possibilities may be stable or affected by other aspects of the situation. So, instead of understanding powers and their degrees of strength as probabilistic, powers and their tendencies towards certain manifestations are the underpinning grounds of probabilities.  Consider the example of tossing a coin.

A coin when tossed has the potential to fall either heads or tails. This tendency to fall either way can be made symmetric and then the coin is 'fair'. From which probability weightings of \(1/2\) for each outcome (taking account of the tossing mechanism) can be assumed and then confirmed by measuring the proportion of outcomes on iteration. The reason why the head and the tail events are equally probable statistically, when a fair coin is tossed, is that the coin is equally disposed towards those two outcomes due to its physical constitution. The probability weightings derive, in this example, from a symmetry in the potentiality, which in turn derives from the physical composition and detailed geometry of the coin.

https://en.m.wikipedia.org/wiki/Rai_stones)

Consider a society that uses very large stone discs as currency.  On examination of the disc, it would be possible to conject that if it were tossed then there would be two possible outcomes and that those outcomes are equally likely. But this disposition is not realised because of the effort required to construct the tossing mechanism, as such a stone may weigh several metric tons. The enabling disposition that would give rise to the iteration of possibilities would have been this missing tossing mechanism. It is not a property of the disc. The manifestation of the dispositional tendency of the disc to come to lie in one of two states needs an external mechanism that is disposed through design to toss the coin in a certain way. If the mechanism is constructed it may be too weak. It may tend to only flip the coin once giving a sequence such as 

... T H T H T H H T H T H T H T H T H T H ...

that would give a frequency of T close to \(1/2\) but the sequence does not exhibit the potential for random outcomes to which the coin disposed. 

 Probabilities and chance

From the above: potential and possibility are more fundamental than (or prior to) probability. Both are needed to construct and explain objective probability. The alternative, subjective probability, is based on beliefs about possibilities but that is not the same thing as what is actually possible and how things will appear independently of anyone's beliefs or judgements.

In this blog I have already referred to a dispositional tendency begin to explain objective probabilities in quantum mechanics. The term propensity has been used to describe these probabilities. I now think that was wrong. Propensity should be reserved for the dispositional tendency that is responsible for the probabilities to avoid this term merging the underpinning dispositional elements and probability structure. Anjum and Mumford claim that they have made a key contribution to clarifying the relationship between dispositional tendencies and probability through their analysis of over disposition. 

Anjum and Mumford claim "information is lost in the putative conversion of propensities to probabilities" but only if the dispositional grounding of probabilities is forgotten.  Their discussion is strongly influences by their interest in application to medical evidence where a major goal is reduction of uncertainty.  Anjum and Mumford propose two rules on how dispositions and probability relate.

1. The more something disposes towards an effect \(e\), the more probable is \(e\), ceteris paribus; and the more something over disposes \(e\), the closer we approach probability \(P(e) =1\).
2. There is a nonlinear ‘diminishing return’ in over disposing. E.g., if over disposing \(e\) by a magnitude \(x2\) produces a probability \(P(e) =0.98\), over disposing \(x3\) might increase that probability ‘only’ to \(P(e) =0.99\), and over disposing \(x4\) ‘only’ to \(P(e) =0.995\), and so on.

While these rules are fine as propositions, they miss the mark in explaining the relationship between dispositions and probability. In the coin tossing example strengthening the mechanism is not about strengthening one outcome. Over disposing does provide support for the distinction between the strength of the disposition and value of the probability but the relationship between underpinning potentials, dispositional mechanisms, and the iterated outcomes needs to be made clear.

Anjum and Mumford also discuss coin tossing and make substantially the same points as I made above. However, having clarified the distinction between propensity and probability, they revert to using the term propensity in a way that risks confusing the concepts of dispositional tendency and probabilities with random outcomes. They say "50% propensity" rather than "50% probability". They then introduce the term "chance" that they relate to outcomes in some specified situations. Propensity is then reserved by them for potential probability while chance is the probability of an outcome in a situation. This is more confusing than helpful.

Anjum and Mumford go on to a discussion of radioactive decay that is known to be described by quantum theory. They make no mention of quantum theory (this will be corrected by them in Chapter 4) and strangely claim that radioactive decay is not probabilistic.   The probability distributions derived from quantum mechanics unambiguously give the probability of decay per unit time. There are, per unit time, two possibilities "decay" or "no decay". Their error is to claim, "only one manifestation type" (decay) and from this that there is only one possibility. Ignoring quantum mechanics, they write:

 "The reason it is tempting to think of radioactive decay as probabilistic is that there is certainly a distinct tendency to decay that varies in strength for different kinds of particles, where that strength is specified in terms of a half-life (the time at which there is a 50/50 chance of decay having occurred)."

 But no, the reason to think that radioactive decay is probabilistic is that our best theory of nuclear phenomena explains it in terms of probabilities. This misunderstanding leads them to introduce the concept of indeterministic propensities. However, they have arrived at the concept it is left open as to whether there are non-probabilistic indeterminate powers, but radioactive decay is not an example.

The examples of the concept chance provided by Anjum and Mumford can be derived in their examples from a correct application of probability theory. Chance is often used as a term for 'objective probability', and I have done so in previous posts. I will continue to follow that usage and exploit the clarification obtained from the analysis above that shows that objective probability depends on the possibilities that are properties of an object. The manifestation of these possibilities may require an enabling mechanism.    The statistical regularities displayed by these manifestations on iteration are due primarily to the physical properties of the object unless the enabling mechanism is badly designed.

Thet term 'propensity' has given rise to much confusion in the literature. Now that we are reaching an explanation of objective probability the term 'propensity' might better be avoided. 

I propose that the model of objective probability is that:

OBJECTIVE PROBABILITY An object has probabilistic properties if it is physically constituted so that it has a potential to manifest possibilities that show statistical regularities.

It it possible to describe statistical regularities without invoking the term 'probability'.

Although some criticism of Anjum and Mumford is implied here, I recognise that their contribution has done much to disentangle considerations about the strength of dispositions that describe tendencies form a direct interpretation as probabilities. However, the value of the three distinctions they have identified is mixed

• Chance and probability are not fundamentally distinct and just require a correct application of probability theory
• Probabilistic dispositional tendencies are distinct from non-probabilistic dispositional tendencies: this is a real and fruitful distinction
• Deterministic and indeterministic dispositional tendencies also provide a useful distinction but it remains to be seen whether there are fundamental non-probabilistic indeterministic dispositions.

The next post will continue this theme with a discussion of dispositional tendencies in causality and quantum mechanics, engaging once more with the 2018 book by Anjum and Mumford.