1. Introduction

Muller and Saunders (Reference Muller and Saunders2008) argue that identical particles are weakly discerned by having opposite spin. This is in response to the long-standing concern that identical particles, like fermions, violate Leibniz’s Principle of the Identity of Indiscernibles (PII). However, if “identical” particles are discernible by their spins, then they do not differ merely numerically and thus do not violate the PII. If Muller and Saunders are correct, then they will have successfully demonstrated that one form of the PII is immune from challenges posed by identical particles. The first half of this article lays out the relevant issues as well as Muller and Saunders’s position. In the second half of this article, I argue that Muller and Saunders’s account fails since they do not make use of quantum observables, and what they do make use of we are not justified in interpreting as a physical relation.Footnote ¹ Without a physically pertinent relation, one cannot even begin the process of weak discernibility.

2. The Challenge from Identical Particles and Weak Discernibility

Assuming some familiarity with both the PII and with the challenge raised by identical particles, I will be brief in my retelling of this story. Leibniz posited seven principles that he largely took as being self-evident, the PII being one of them. This principle states that if two objects are indiscernible then they are identical or, a in slightly more interesting formulation, “that it is not true that two substances may be exactly alike and differ merely numerically, solo numero” (Leibniz Reference Leibniz1686/1974, 417). I say “more interesting” because this quote makes clear that if there are two numerically distinct things then they must be discernible; however, the quote also claims that if two things are indiscernible then they must not be even numerically distinct: there must be only one thing. The second conditional will play an important role at end of this article. Leibniz’s principle can be written in the following logically equivalent ways. The first most faithfully captures the words of Leibniz, but the third is the most common way of thinking of the principle:

where {F} represents the set of monadic properties for the objects in our domain.Footnote ² According to this formulation, we define that two objects, a and b, are strongly discerned if and only if (iff)

If strong discernibility (4) is met, then the inference in (3) cannot go through. In such a case, one could then use Leibniz’s Principle of the Indiscernibility of Identicals to infer that the “substances” must not be identical. However, in the quantum case, it seems that (4) fails, and so the inference in (3) does go through.

The literature recognizes other means of discerning objects including that used by Muller and Saunders (Reference Muller and Saunders2008): two objects, a and b, are defined to be weakly discerned iff there is a physically relevant dyadic predicate that is symmetric and nonreflexive when applied to the two objects.Footnote ³

The idea behind weak discernibility is that if two objects are identical then F(a, b) is equivalent to F(a, a) and the above condition fails; thus, if it does hold then a and b cannot be identical. Although the inclusion of weak discernibility shifts the debate away from the exact form of Leibniz’s principle, the spirit of the debate remains very much the same.

Quantum mechanics challenges the truth of Leibniz’s principle in that identical particles in states like the singlet state,

are thought to be distinct particles and yet have no property to discern them. The term ‘identical particles’ is unfortunate given the current context. To be clear, in the context of quantum mechanics, all that is entailed by two or more particles being identical is that each has the same value for their nondynamical properties such as their mass and charge. Identical particles are not necessarily identical in a logical or ontological sense. For the sake of sticking closely to the terms of Leibniz’s project, I will assume that we have two objects that do differ at least numerically, but there is an important sense in which entangled states may deny this and thereby trivialize the question of discernibility.Footnote ⁴

The singlet state in (6) describes an entangled two-particle system expressed in terms of -spin, while the particles’ location has been suppressed. In order to be relevant for the PII, we insist that the implicit portion of the wave function be identical and that the particles follow the same path through space (this is why it has been suppressed). This state is physically realizable as two entangled electrons in the same orbital of some atom. In such a situation, for all the properties that are specified by quantum mechanics, the particles share said properties. In other words, there is no property held by one of the particles and not by the other, yet we do not think that these particles are ontologically one—as would be required by the PII. Consequently, many have taken the existence of such particles as being counterexamples to the PII.

Although identical particles fail to be strongly discernible, Muller and Saunders (Reference Muller and Saunders2008) claim that they are weakly discerned by the relation ‘has opposite spin’. This relation is symmetric and irreflexive, and yet in order to justify its application to the singlet state we need an ‘opposite spin’ observable that when applied to the singlet state yields an appropriate eigenvalue. A common assumption, although far from obvious, is that there is a one-to-one mapping from physical properties to quantum mechanical Hermitian operators (observables), such that if a state Ψ is the ith eigenvector of the observable with eigenvalue a_i then the state is said to have the property associated with this eigenvalue. This association between properties and eigenvalues is often referred to as the “eigenvector, eigenvalue link” (EE-link) and serves to coordinate Hermitian operators with properties of a system. In order for us to have a mathematical framework complete enough to describe quantum phenomena, we require that our mathematical machinery be able to represent every physical property that might be possessed by our system. Thus, if we want to play the game of modeling quantum mechanics at all, we must assume that the aforementioned mapping is at least injective.Footnote ⁵

Muller and Saunders (Reference Muller and Saunders2008, 533) use the following generalized operator to build the relation of having or failing to have opposite spin:

Without getting into too many of the details, note that the superscripts on the single-particle operators

pick out which slot of the tensor product the operators act on, and the subscripts refer to vectors of a given eigenbasis. For example,

, where the P _↑ projects onto the ↑

-spin eigenvector and P _↓ onto the ↓. Muller and Saunders (Reference Muller and Saunders2008, 533–35) define the following two relations and claim that both are able to weakly discern the identical particles:

In the case considered by Muller and Saunders, the

operators each reduce to the Pauli spin matrix σ_z, and we get the following refinement:

For a = 1, b = 2 (a ↔ b) the relation holds:

For a = b = 1 the relation does not hold:

For a = b = 2 the relation does not hold:

In other words Z ₋₂ holds iff a ≠ b. The relation Z ₂ holds under the exact opposite conditions. Thus, the particles are weakly discerned by the relation Z ₋₂, and the PII is immune from challenges posed by identical particles. For a more detailed account, see Muller and Saunders (Reference Muller and Saunders2008).

Although Muller and Saunders’s account is straightforward, it is far from unproblematic. Equation (10) captures three different equations, each with its own observable. As we select particular values for a and b, we construct new mathematical objects (see eq. [15]). Consequently, the formal relation Z ₋₂, used to weakly discern, is not a Hermitian operator on the system, and we cannot apply the EE-link to it. Since the EE-link is not applicable to Z ₋₂, we have no reason for thinking that it represents a physical relation. In the following section I develop a series of criticisms that stem from this fact.

3. Challenges

Muller and Saunders are aware that traditionally the EE-link (roughly what they refer to as StrPP) is used to pick out monadic predicates.Footnote ⁶ They nonetheless insist that it can also pick out polyadic predicates:

According to Muller and Saunders, the conditions for ascribing a physical relation to a formal relation as in (10) are as follows: “When the projectors under consideration belong to the spectral family of magnitude-operator A, assumed to be physically meaningful, they are themselves physically meaningful; by the WkPP, when the system is in the state W, so is relation Rt, which is defined in terms of them (Req1)” (Muller and Saunders Reference Muller and Saunders2008, 532), where the condition (Req1) is “all properties and relations should be transparently defined in terms of physical states and operators that correspond to physical magnitudes, as in WkPP, in order for the properties and relations to be physically meaningful” (527). Here, Muller and Saunders use the WkPP, which is simply a weaker version of the StrPP (514) and will make no difference to my argument. In summary, according to Muller and Saunders, what justifies assigning a physical relation, ‘opposite spin’, to Z ₋₂ is merely that such a relation gains physical transparency by piggybacking off the physical transparency of the observables that show up in (10). Before examining this claim, I want to make clear that I am in no way questioning whether the observables that show up in (10) have physical meaning and am only questioning how physical content is forced onto a formal relation built out of them.

First, Z ₋₂ is not an observable itself but is map from

to the following set of observables (acting on Ψ):Footnote ⁷

where Z ₋₂ is associated with three different observables, each of which seems to be “manifestly” physical and yet each of which could bear a distinct physical meaning. What then is the relationship between this set of observables and Z ₋₂ whereby the map becomes associated with a single physical relation? There are three questions: (1) why must the map represent any physical property of the system or of the particles that make up the system, (2) why is this map associated with a dyadic property (opposite spin) and not a property of some different arity, and (3) even if the mapping is associated with a dyadic property, why must we think that this relation is symmetric and irreflexive? Surely the formal relation

is symmetric and irreflexive on slot indices a and b, but why must the observables in (10) bind together to produce a physical relation that is symmetric and irreflexive on the particles? The criterion we are in search of needs to be more than that the objects mapped onto are physically meaningful or are individually a relation of a certain kind since in general it is false that maps have the same properties as the objects in their image.

What Muller and Saunders need to supply is an EE-link for relations that identifies when n-valued functions onto sets of observables pick out m-ary properties. Such a link would unpack what it means for a relation to be “transparently defined” by selecting which formally defined relations represent physical properties and what the relevant physical property is. However, an EE-link for relations is not provided for by Muller and Saunders, nor is it part of standard quantum mechanics. As things stand, we cannot do much with Muller and Saunders’s project until we are given more reason for thinking that Z ₋₂ represents something physical. However, for the sake of argument let us assume that Z ₋₂ is physically meaningful and consider questions 2 and 3. If Z ₋₂ does represent something physical, are there reasons for associating it with the relation ‘opposite spin’? Not clearly. Analyzing (12)–(14) and how the operators act on the singlet state does not leave one with any assurance that any weakly discerning relation is being described by them. For instance, there is nothing ruling out the possibility that equation (12) tells us that the state has total spin equal to zero, while (13) and (14) tell us that the total spin fails to be either greater or less than zero. If these are the proper interpretations suggested by the individual equations (12), (13), and (14), then it would be hard to argue that Z ₋₂ represents a relation since the individual observables are monadic properties of the system. Similarly, in order for Z ₋₂ to represent a weakly discerning relation on the particles, equations (12)–(14) must not bind together to suggest some three-place relation between the particles and their environment and so on. Somehow we must use the information in (12)–(14) to rule out all contrary interpretations and leave only an interpretation attached to some weakly discerning relation. Without this relation, Muller and Saunders (Reference Muller and Saunders2008) have only a physically salient map (since we have granted them this) and no means to weakly discern. That equations (12)–(14) do select such a relation is not argued for by Muller and Saunders, nor is it even clear what such an argument would look like.

Before exploring other motivations for why we ought to think that ‘opposite spin’ is true of the singlet state, I want to highlight one further worry about Z ₋₂. If Muller and Saunders are correct in their claim that represents a physical relation since it is built out of physically salient Hermitian operators, then it must also be true that represents a physically relevant monadic property since it too is built out of physically salient Hermitian operators.Footnote ⁸ Moreover, since it is not the case that both x = 1 and x = 2 satisfy Z(x), the particles are strongly discerned by it. However, we had already concluded that the particles are not strongly discerned.

We are able to bypass our previous conclusion by using non-Hermitian operators to represent monadic properties of the particles. If we think that we have made a mistake in getting to this conclusion (and we have), then I suggest that we have also made a mistake in the case of weak discernibility. Perhaps in filling out or adding to Muller and Saunders’s (Reference Muller and Saunders2008) conditions for why represents a physical relation we will discover conditions that fail when applied to . However, until these conditions are given, we should have no confidence that is any more physical than .

Thus far we have not found anything in the formal structure of Z ₋₂ acting on Ψ that suggests ‘opposite spin’ or any other symmetric and irreflexive physical relation. Why then think that it does? With respect to their claim that represents ‘opposite spin’, Muller and Saunders claim “Relation Z ₋₂ is the one in footnote 5 of (Saunders [Reference Saunders2003a], p. 294): ‘has opposite direction of each component of spin to’” (Reference Muller and Saunders2008, 535). Or in other words, the physical property represented by Z ₋₂ is that provided by Saunders in Reference Saunders2003: Z ₋₂ represents the property ‘has opposite direction of each component of spin to’. However, the referred-to passage (in Saunders Reference Saunders2003) does nothing to elucidate the justification for interpreting Z ₋₂ in this way. Sauders (2003) argues merely that “the most general anitsymmetrized two-particle state is where ϕ and ψ are orthogonal. Analogues of operators for components of spin can be defined as , where P _ϕ, P _ψ are projections on the states ϕ, ψ. Each of the two particles in the state Ψ, has opposite value of S. But no particle can have opposite value of S” to itself (Saunders Reference Saunders2003, 294 n. 5).Footnote ⁹

What does the heavy lifting in predicating opposite S-value (opposite spin) to the particles is apparently Saunders’s assertion that each of the two particles in the state Ψ happen to have opposite values of spin. This is surely troubling; if we are not going to reason in a circle, we must have some way of verifying that the particles do in fact give opposite values of S. Yet given that the state is entangled, we have no way of isolating any single particle to determine its spin. Saunders claims that the observable S does pick out opposite eigenvalues when applied to the particles in the singlet state; however, in order to make this claim one would need to first identify which part of the singlet state represents some single particle. Although we know that such talk is nonsense, perhaps if we were to look at the actual terms of the

-spin expansion of the singlet state, we might be able to find some motivation for thinking that ‘opposite spin’ has something to do with the singlet state. It sure seems like the particles in (16) have opposite spin.

How does one dissect this state to apprehend one particle at a time and thus come to believe that each particle has the opposite S-value from the other? If we were to assume that the slots of each term of the singlet state represent the individual particles, then we might conclude that the particles have opposite spin. For instance, let

Then according to our assumption, the first slot in ↑↓ represents the first particle and its properties and the second the second particle and its properties, and similarly for ↓↑. Since the first particle in ↑↓ has the opposite value of spin as the second particle, we conclude that the particles have opposite spin. Not forgetting that there are two terms, we check the second term ↓↑ and note that the particles represented here also have opposite spin. Therefore by the linearity of (16), it is true of the singlet state that the particles have opposite spin.

The mistake of course is that ↑↓ does not represent anything physical: no identical particles are represented by ↑↓. The eigenvector ↑↓ is no more essential to the physical description of the state than the minus sign connecting the other nonphysical pieces of syntax. As a whole, the state has physical content, but none of its parts do. To treat the terms ↑↓ or ↓↑ as individually giving us physical information of the state is to treat the singlet state as akin to a statistical mixture rather than a genuine entanglement.Footnote ¹⁰ This line of reasoning corrects an assumption we may have had regarding the EE-link: we may have assumed that since observables are linear in the wave function the properties associated with them are too. This of course is false. Physical properties only hold on physical states. The EE-link associates properties with physical states and not with nonphysical terms of a state’s decomposition. Moreover, implicit in my treatment is a denial that slot indices always represent single particles. Rather, only those slots of factorized states represent single particles: slots on nonphysical syntax cannot represent physical particles.

In summary of these arguments: the singlet state is Ψ, and focusing too much on the expansion (↑↓ − ↓↑) may tempt us into thinking that its pattern of arrows is revealing hidden relations between the single particles. We must conclude that there is nothing in the singlet state’s description that warrants attributing opposite spin to its particles. Moreover, since, as we have seen, there is nothing in the formal structure of Z ₋₂ that warrants our association of it to the relation ‘opposite spin’, quantum mechanics simply does not provide us with a description of the particles’ spins.Footnote ¹¹ If quantum mechanics does not provide a description, then why do we think there is one?

In conclusion: I have provided three challenges to predicating ‘opposite spin’ to states like the singlet state. The first challenge stems from Muller and Saunders’s (Reference Muller and Saunders2008) emphasis that their formal relation Z ₋₂ should be identified with a physical relation. I have argued that there is nothing in the formal structure of Z ₋₂ for thinking that it represents any physical property of the state, or that it is a physical relation, or that the physical relation is suitably structured to weakly discern. I close this article with the following consideration that others have also noted: Why must we think of the singlet as representing two things to begin with? As I have argued throughout, it is exceedingly difficult to justify applying relational properties to the singlet state since an entangled state’s description “hides” what is true of any single particle. If we are to extract information about single-particle properties, it is not clear how we are to do so. Given the extent to which entangled states hide the supposed particles that they represent, why not interpret such states as representing only emergent unities?

Whether or not we think the PII is a logical principle (as Leibniz did), for most of us the principle remains highly intuitive; moreover, since the principle comes under attack only to the extent that entangled states represent multiple particles, what is keeping us from flipping the argument around and using the PII to deny the composite nature of entangled states? I will not attempt to address this issue and only note that if we did deny the composite nature of entangled states, we would simultaneously block the PII from attack as well as make it unreasonable to apply relational properties to them. For if there are not two “numerically distinct” things to challenge the PII, then there are not two things that have opposite spin (for instance).