Introduction

I was going to complete my series on different philosophical interpretations of quantum physics with the last post on Relational Quantum Mechanics. But that paper referenced a work by Fleming I hadn't encountered before, and a quick glance at Fleming's paper made me think that it would be worth writing a small note on it. Fleming's work is another that looks at an Aristotelian interpretation of quantum physics, joining people such as Robert Koons and Wolfgang Smith, alongside others Fleming cites in his work.

The origins of the idea

In his first section, Fleming mentions several precursors who suggested that Aristotelian potentiality is relevant to the philosophy of quantum physics.

The first of these was Kurt Riezler, writing in 1940. Riezler noted that the wavefunction is different before and after the observation. The state afterwards is not determined by the state before. The wavefunction thus only describes possibilities of actions which could take place. On the other hand, what is observed is actual. Thus we are forced to distinguish two modalities: possibility and actuality, with the possibility not merely relative to a deficient knowledge, but belongs to the reality of the thing itself.

He next mentions Heisenberg from 1958 and 1974, who likewise stated that the wavefunction contains objective statements about possibilities or tendencies, corresponding to Aristotelian potentia. This is mixed in with subjective statements about our knowledge of the system.

Abner Shimony in 1986, who again used potentiality to describe the combination of indefiniteness of value with definite probabilities of outcomes. The physical variable is actualised when it takes on a definite value. The measurement is only a special case of when systems interact.

Fleming next turns to an earlier writer, Charles Peirce, to define what is meant by potentiality. Peirce didn't consider quantum physics, but still argued for ontological chance an potentiality in nature. Peirce defined potentiality as some inherent tendency towards actuality, which, if not thwarted, leads to final completeness of being. Peirce's justification for this looks (to me at least) to be a bit weird, saying that the only way to account for the uniformity of the laws of nature is to suppose the laws as results of evolution.

General features of quantum phenomena

This section discusses the structure of quantum theory, in terms of three parts.

1. A framework representing allowed states and possible observables. The measured values associated with those observables often take on a descrete set of values, even for qualities which are continuous in classical physics. He also discusses superposition here, stating that in all circumstances most attributes of a quantum system have no definite value at all.
2. The Born rule, assigning probabilities for various outcomes to occur. He briefly mentions the uncertainty principle, relating the shapes of the wavefunctions for non-commuting observables.
3. A description of how the probabilities change over time. There are two different mechanisms, one stochastic, and one deterministic. The deterministic part is described by either the Schroedinger equation or Heisenberg's matrix mechanics. But if this were the whole story, the smoothly changing wavefunctions would not be interpretable as probabilities at all (I think he means by this that the notion of probability is only useful when there is a degree of unpredictability). So there are also stochastic changes of the probabilities, changing the probabilities for the values acquired at measurement to unity. After this , the wavefunctions continue their causal evolution. These unpredictable and causally unanalysable changes of the probabilities are State Reductions. While one cannot predict which of the State Reductions will occur, one can predict when they occur, as that is when a measurement happens. Such State Reductions also involve the loss of definite values for some observables.

This account is reasonably standard and (I think) uncontroversial.

Competing interpretations of the quantum theory

Fleming divided interpretations of quantum physics into two categories. Firstly, that the restrictions on associations with definite values to attributes represented limitations imposed on humans due to the limitations of human interaction and exploration of microscopic physics. Or secondly, they should be interpreted ontologically as implying that quantum attributes do not possess definite values, but only very briefly acquire them in particular circumstances. He placed Einstein, de Broglie, Bohm and Popper in the first category, and Heisneberg, Born and Pauli in the second, albeit not with perfect consistency. Others have belonged to both camps.

Von Neumann's theorem that one could not mathematically assign definite values simultaneously consistently with the predictions of the theory initially dampened the enthusiasm for hidden variable interpretations, until Bohm's revival of those interpretations. The Copenhagen (ontological) positions were kept respectable, and most textbooks promoted it, meaning that most physicists fell into that camp.

Bell's theorem then showed that any hidden variable scheme would have to violate various intuitive assumptions, most significantly locality. This has forced advocates of the hidden variable approach to admit extremely non-local causal connections. I'll quote Fleming directly for the next part rather than paraphrasing:

The real difficulty, not to say embarrassment, however, for Hidden Variable advocates, is the rich additional structure of their models which is rigorously prevented from manifesting itself. With the exception of the measurement of the positions of the constituent particles in the Hidden Variable models, the results of measurements, in general, cannot be identified with prior possessed properties of the system. Instead the results are created by the measurement interaction. Furthermore, those properties possessed by the system, other than particle positions, are not susceptible to measurement at all. To be so would require a deviation from the predictions of the quantum theory. Thus the additional structure of the Hidden Variable models, the wealth of possessed properties, has, at present, much in common with the epicycles of Ptolemaic astronomy!

Fleming also discusses the Kochen and Specher model, showing that it is impossible to assign simultaneously possessed values to all attributes represent by Hermitian operators so that assigned values are taken from the eigenvalues of the operator, and the assigned values satisfy all functional relationships between operators representing compatible attributes. The Bohm model evades this by completely severing the connection between possessed values and and possible measurement results. For example, in many bound states, the Bohm particles are stationary, yet still possess momentum on measurement. The angular momentum of Bohm particles is not quantised at all.

While the violation of the Bell inequalities implies some variant of locality violation, this need not be as strong as required by Bohm. For example, if we relate the quantum states to the propensities or potentialities of quantum systems to give particular results at state reduction. This violates the locality condition that states a broad wavefunction for an attribute cannot be changed to a narrow one (with fewer quantum states having a high probability) by non-local measurements. This principle is clearly violated in entanglement experiments, and it seems to be unavoidable that it is violated.

However, if probabilities are interpreted as measures of our ignorance of definite values always possessed, then the locality principle violated seems to be that a macroscopic object cannot have its state changed by altering the setting of a remote apparatus. This cannot hold in a deterministic interpretation where the attributes are always possessed. In this he follows a argument presented in Redhead's textbook.

Recent arguments on behalf of objective potentialities

Fleming next presents an argument against hidden variables systems. He relies on three pairs of two attributes, (A_i,B_i), with i=1,2,3. These are constructed so that A_i commutes with A_j and B_j for j ≠ i, and A_i anti-commutes with B_i.

He then considers a system that is an eigenstate of various products of these observables, in four different configurations:

These four operators all commute with each other, so a state can be in simultaneous eigenstates of all of them. The three operators within each Σ operator also commute with each other.

Suppose, assuming a hidden variable system, that the measurements all have defined results all the time. This means that each observable takes on a definite eigenvalue, even if we don't know what that is until decoherence and measurement. On each individual run of the experiment, the system is such that the state corresponding to a measurement value of A_i with eigenvalue a_i, and B_i corresponds to b_i, and Σ_i to σ.

Then

This implies that σ₁σ₂σ₃ = σ₀(b₁b₂b₃)². Thus σ₁σ₂σ₃ has the same sign as σ₀. However, if we consider the operators, we find that

which implies that σ₁σ₂σ₃ has the opposite sign to σ₀

The solution is to reject the assumption that the various observables have defined results, so we cannot simply replace the operators with their eigenvalues midway through the calculation. We can only replace the operators with eigenvalues after decoherence, after the system collapses into a defined state. So the second equation (in terms of operators) is valid, while the first equation (expanding in terms of the eigenvalues) is invalid.

So it seems that the system cannot have real (but hidden) values of the measurements at any given moment of time. If they cannot have real values, then the values must represent potentialities, i.e. there is no defined value except at the point of measurement.

Fleming's second example of the superiority of an objective potentiality interpretation is related to relativistic quantum theory. Suppose that the particle state is such that it is within a given sphere in an inertial reference frame K. We can make the sphere as small as we like; infinitesimally small would imply that the particle is in a location eigenstate at a given time. Then perform a Lorentz transformation in the frame K'. The particle will be at a transformed state vector, but is no longer confined to the sphere, or any bounded region of space.

In relativistic theory each particle has a variety of position-like attributes associated with it, and the position attributes are neither identical nor compatible. However, the potentiality interpretation of quantum theory seems to cope with this better than a hidden variables interpretation.

I don't fully understand this argument as he presents it. He references a few of his earlier papers, including this one. Fleming's earlier paper restricts its discussion to relativistic quantum mechanics (RQM), rather than relativistic quantum field theory (RQFT). There are certainly issues covariantly defining position states in RQM. It is one of the motivations for adopting RQFT. But I don't really care about that, since RQM is not the true theory of nature.

The discussion relating to whether this problem carries onto RQFT is ongoing. For example, one can see this paper.

Fleming concludes that an objective potentiality interpretation of quantum theory, where any given attribute does not take a definite value (even if we don't know what that value is) except at decoherence. At decoherence, potentialities for the acquired values are actualised, and alongside the actualisation of a potentiality, there is also a potentialisation of actualities. State reductions remove definite values from at least as many attributes as are actualised. If this is the case, there cannot be a system where all attributes are known.

Thoughts and reflections

I should say that I found this paper interesting. Obviously I am sympathetic to his conclusions, although his account would be more complete if it also reflected on efficient and final causality.

I thought his division of interpretations into two camps,

1. That attributes don't seem to take on definite values during quantum evolution is a reflection of limitations imposed on humans.
2. That attributes don't seem to take on definite values during quantum evolution is ontological, so the attributes only possess definite values during decoherence.
one where the theory concerned limitations on human knowledge, or one where the objects of the theory should be interpreted ontologically where quantum attributes only possess definite values at decoherence. This seems like a division I use into psi-ontic or psi-epistemic interpretations, but there are differences.
• Psi-epistemic: the wavefunction represents human knowledge of the physical system.
• Psi-ontic: the wavefunction represents the actual state of the physical system.

Fleming's first category seems to suggest that the system possesses definite values, but we don't know what they are. He places the pilot wave interpretation into this camp, because here the apparent indefinite values arises from gaps in our knowledge. I would add consistent histories (at least how I interpret it) here as well, although this is a bit different to the pilot wave. Although the quantum state has a definite state at all times in reality (albeit that we don't know what it is), still most observables are still undetermined. The observables are expressed in non-commuting states. But the probability still reflects our lack of knowledge of that state, coupled with an indeterministic evolution (meaning we can't predict which one of a small number of options the system will evolve into in the next moment of time).

In his second camp, he places Copenhagen and his own interpretation.

I'm not sure that QBism fits neatly into either of his two categories. In QBism, the quantum representation is a matter of human knowledge, but (unlike the pilot wave and consistent histories) there is no definite state of nature.

Neither do I think that Many worlds fits neatly into Fleming's categorisation. This denies that there are restrictions of definite values with attributes, but states that all terms in the superposition occur simultaneously.

In my classification, I would regard consistent histories, QBism and related interpretations as psi-epistemic, while everything else (including the pilot wave) as psi-ontic.

So one major weakness of this paper is that it only really considers the pilot wave and Copenhagen interpretations, and minor variants of them.

Does his analogy between the pilot wave and Ptolemy's epicycles stand up? The commonality is that they both introduce features into the theory which we can't directly observe. However, the unobservable objects in Pilot wave theory are said to be actual physical objects. They are a natural consequence of a particular reformulation of the equations, and (baring complications adapting the pilot wave interpretation to QFT with its spontaneous particle creation and annihilation) leads to the same results as other interpretations. The epicycles (of both Ptolemy and Copernicus -- don't forget that while Copernicus had fewer epicycles than Ptolemy, he still needed some) are mathematical abstractions introduced to try to save the idea that heavenly bodies can only travel in circles from conflicting experimental data (and they still didn't work). So I don't think the comparison is fair.

I think he is right to say that there has to be some locality violation, but this need not be as strong as in Pilot wave theory. In Pilot wave theory, the locality violations affect the equations describing the dynamical evolution from one quantum state to another. The wavefunction at one point in the universe affects the evolution of the particle somewhere else. All we need, however, is that there are non-local correlations in the outcomes of certain events. There need not be a physical structure spanning the universe whose parts in one place affect outcomes in another. Correlations between outcomes in an indeterministic evolution is a weaker breach of locality than non-local interactions between physical objects. So the argument for pilot wave theory that there is bound to be some non-locality somewhere, so we might as well put it in the equations of motion isn't sufficient.

So Fleming's argument seems to be the following.

1. The choice is between a hidden variables theory, and one where physical attributes are undetermined except at decoherence events.
2. If physical attributes are undetermined, except at decoherence events, then that implies the possible outcomes exist potentially (in some sense similar to but perhaps not quite identical to Aristotelianism).
3. There are arguments ruling out a hidden variable theory.
4. Therefore the wavefunction is an expression of potentiality, with the potential actualised at decoherence.

I have discussed premise 1 already, and think there are issues with it.

What of premise 2? Fleming's argument for this is that if we accept it, it allows for a weaker violation of locality than seen in the Pilot wave, but still sufficiently locality-violating to satisfy Bell's and other theorems. While not accepting the idea that the measurement events lead to a collapse of potential states implies that they lead to non-local collapses of actual states or non-local influences during wavefunction collapse. The violation of locality is merely that one measurement could be affected by altering settings of a remote apparatus. Thus an interpretation of the superposition as a description of potentiality allows the weakest possible violation of locality.

Although I support the conclusion that a wavefunction in superposition across multiple states is best expressed by describing those states in terms of Aristotelian potentia, I'm not convinced by Fleming's argument (as I have interpreted it). Why should we desire to have the weakest possible violation of locality? The obvious answer is because special relativity implies that the Hamiltonian describing the evolution of physical beings should be local. But Fleming allows information to be shared between remote apparatus. How, in his system, does this happen? If there is a exchange of information mediated by physical particles, that means that either some physical particles will interact with each other at a distance, or some particle needs to pass from one detector to another "instantaneously" in order to carry the information. This strikes me as being no better than the Pilot wave models. QBism provides an alternative: the information is subjective, so only actualised in an observer when that observer learns of the other measurement. I find that unconvincing; there still has to be an objective truth underlying the subjective knowledge of the quantum state. Many-worlds states that there is no sharing of data at a distance, because all outcomes occur (i.e. there is one "world" where the two particles emerge in a spin-up/spin-down state and another world where the two particles emerge in a spin-down/spin-up state, so there is always anti-correlation in the spin, even if we have no interaction with the "version" of ourselves that measured the other result), but I'm unconvinced that many worlds adequately explains the emergence of probability. The remaining alternative is that the outcome of events (and thus correlation at a distance) is determined by something immaterial, and thus not subject to the rules of special relativity.

But why insist on special relativity, beyond an accidental property of the Hamiltonian underlying wavefunction evolution? As I discussed in my post on pilot wave theory, I do think that its inability to explain why there is Lorentz symmetry in at least this equation when it is not a symmetry of the real world is a weakness of pilot wave theory. But clearly, the advocate of pilot wave theory would disagree. So I don't they would be convinced by Fleming's argument here. Neither do I think a many worlds advocate (who denies there is a need to violate locality) would be convinced.

But his argument isn't targeted at these people, but towards advocates of Copenhagen style interpretations. These clearly have an issue with wavefunction collapse, and its non-local requirements. Does Fleming's interpretation reduce these issues? I'm not sure. If they see the wavefunction as representing an delocalised actual state which collapses into a localised actual state, then this collapse is not described by different rules to the Hamiltonian governing wavefunction evolution outside of collapses. If it obeys different rules, then it need not obey the restrictions of special relativity.

That said, I still think we should identify quantum states in a superposition with potentialities, but I don't think this is the best way to argue for it.

What of point 3 in his argument? I'll just answer it from the perspective of consistent histories. Here the response is the standard one to these apparent paradoxes. The argument violates the single framework rule, and is thus invalid. For example, the construction assumes that there can simultaneously be a definite value of A₁ (as part of Σ₁) and B₁ (as part of Σ₃), but these don't commute with each other, and thus belong to different bases or frameworks. The basic rule of consistent histories is that you can only calculate a probability, expectation value, or substitute in the eigenvalue of an operator, if all the amplitudes contributing to that probability, expectation value or combined eigenvalue are in the same basis. As soon as he introduces Σ₁Σ₂Σ₃ he combines operators from a different basis, and it is invalid to substitute in the eigenvalues of the operators midway through the calculation. One can only expand the operators Σ₃ into their eigenvalues when they are applied to a given eigenstate, |ψ>. |ψ> might be an eigenstate of Σ₁, Σ₂ and Σ₃, but it is not an simultaneous eigenstate of all of A_i and B_i. Thus it is inappropriate to expand both A_i|ψ> = a_i|ψ> and B_i|ψ> = b_i|ψ>, as no simultaneous eigenstate of A_i and B_i exists. Thus he prematurely substitutes his operators with the eigenvalues of their component parts, and his argument is invalid.

I won't try to predict how an advocate of the pilot wave interpretation would resolve this dilemma, but I imagine they will have similar concerns.

Thus I don't find Fleming's argument convincing. This is unfortunate, because I agree with his introduction of Aristotelian potentialities into the interpretation of quantum physics. I just don't think he has adequately argued that case.




The Authority and Transmission of the New Testament


Reader Comments:


1. Julian


Posted at 11:29:35 Wednesday October 22 2025


Copyright request



Dear Nigel,

thank you for your interesting, interdisciplinary articles (my farorite series is on why there is something and not nothing).

Could we republish one or another article in our Catholic Journal Life has a Name?

With best regards

Julian



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