1The philosophical debate over the meaning of the notion of individuality in quantum mechanics has been raging for decades. Given that virtually all positions and arguments have been thoroughly examined and re-examined, what rationale can be offered for yet another paper on such a well-researched topic? The main goal of this article is modest: I would like to revisit the notion of a permutation of objects which is a key but somewhat neglected concept. My suggestion is that if we acknowledge the fact that there is a conceptual gap between the unique mathematical notion of a permutation and its physical realizations, we can notice that there may be more than one acceptable interpretation of the latter. This is hardly a new and surprising idea; however to my knowledge no serious attempt to classify and examine possible interpretations of physical permutations of objects has been made in the context of quantum mechanics.
2The starting point of this article is a discussion of four possible interpretations of the notion of physical exchange, of which I select two that seem best suited for an analysis of permutation invariance in quantum mechanics: exchange of essences and exchange of haecceities. A deeper investigation of both concepts reveals that they lead to radically different conclusions regarding the problem of discernibility of quantum particles. I argue that the essentialist interpretation of exchange invalidates the standard argument in favor of the Indiscernibility Thesis given by [French & Redhead 1988]. The proof of the Indiscernibility Thesis goes through only under the alternative, haecceitist interpretation. Moreover, I claim that essentialism actually strongly suggests that identical fermions and bosons can be absolutely discerned in some states by their quantum-mechanical properties. The formal proof of this fact which I present in the article is prefaced by a discussion of how to represent the properties of individual components of many-particle systems when meaningful operators are restricted to the symmetric ones. I end the article with an appeal for further study of the essentialist view, which is a relatively new and potentially fruitful approach.
3Formally, a permutation of an n-element set X is defined as a bijection mapping this set onto itself σ : X → X. In the context of mathematical physics the most typical permutations are those of the indices in a compound mathematical object Ψ(1, 2, …, n) (it can be a real-valued function, a vector, a density matrix, or any other object) representing a particular physical situation. Permutations applied to Ψ can be interpreted as mappings connecting it with objects arising as a result of permuting its indices: σ(Ψ(1, 2, …, n)) = Ψ(σ(1), σ(2), …, σ(n)). In the simplest case of an object containing just two indices the only non-trivial permutation leads from Ψ(1, 2) to Ψ(2, 1). The idea is of course that indices 1 and 2 are supposed to refer to physical entities (particles, properties, states of affairs, etc.), so the mathematical object Ψ(2, 1) should correspond to the situation obtained from the one described by Ψ(1, 2) by exchanging the required physical counterparts.
- 1 Tim Maudlin makes a similar observation in the context of the debate on the ontological status of s (...)
4But the notion of swapping physical objects is not so clear-cut. We have to remember that mathematical concepts do not always perfectly match physical reality. Sometimes mathematical language creates artifacts (so-called surplus structures) not corresponding to anything real, and sometimes one physical situation receives many non-equivalent mathematical representations. In other cases one and the same mathematical concept can be interpreted physically in many different ways. Oftentimes failure to realize that there is no one-to-one correspondence between mathematical and physical concepts leads to serious misunderstandings.1
5I suggest that at least four independent interpretations of the notion of exchange of physical objects can be given. The most natural way of thinking about exchanging physical objects is in terms of their location. If I asked you to swap this chair with that table, you would most probably move the chair to the place where the table stood, while simultaneously bringing the table to the location previously occupied by the chair. Hence the first interpretation presents itself naturally:
Exchange No1 (exchange of locations). To exchange an object A located in rA with an object B located in rB is to create a situation in which in rA there is an object which possesses all the non-relational (intrinsic) properties of B, while in rB there is an object which possesses all the non-relational properties of A.
6The second interpretation of exchange requires an introduction of an important notion of essential properties. As is standard in the literature, I will understand the essence of an individual object as the set of properties which this object possesses in all possible worlds in which it exists. My presupposition is that all objects have essences; however, I do not assume that each essence is unique. In fact, objects in the actual world can share their essences. One important example of essential properties is what quantum physicists call “intrinsic” (or state-independent) properties of elementary particles (among state-independent properties of an electron are its mass, charge, and total spin). It is of crucial importance to acknowledge that all particles of the same type (what physicists call, confusingly, “identical” particles, such as electrons, protons, muons) have the same essence.
Exchange No2 (exchange of essences). The result of an exchange of an object A, whose essence is EA, with an object B, whose essence is EB, is a situation in which there is an object A′ possessing properties EA and all non-essential properties of B (relational and non-relational), and an object B′ possessing properties EB and all non-essential properties of A.
7If we consider for instance an electron in a particular quantum state |u⟩ and a positron in a different state |v⟩, then after their exchange-of-essences we will have a new situation in which some electron is in state |v⟩, and some positron is in state |u⟩.
- 2 An excellent philosophical discussion of the notion of haecceity in the context of quantum mechanic (...)
8The third interpretation requires an introduction of a new and controversial metaphysical notion. This is the notion of the haecceity of an object (also called primitive thisness). It is sometimes claimed that apart from its “ordinary” properties, essential or not, each object comes equipped with a special property, which is simply defined as being identical with itself and nothing else. Haecceity is well known to be offensive to any genuine empiricist. It cannot be characterized in a qualitative way, nor can it be directly observed or detected. And yet some philosophers feel that haecceity is necessary in order to speak about the relation of numerical distinctness and identity that is conceptually independent from qualitative identity. For now I do not wish to enter the philosophical debate on the nature and admissibility of the concept of haecceity.2 Instead, I am simply going to introduce my third concept of exchange of objects.
Exchange No3 (exchange of haecceities). The result of an exchange of an object A possessing haecceity HA with an object B possessing haecceity HB is a situation in which an object possessing haecceity HA has all the properties (relational and non-relational) of B, and an object possessing HB has all the properties of A.
9It is characteristic that the process of a type 3 exchange results in a situation which is qualitatively (and hence empirically) indiscernible from the initial one, and yet we assume this situation to be ontologically different (it is supposed to be a genuinely new state of affairs). It is a scenario in which this table becomes qualitatively indistinguishable from that chair (and also assumes the location of the chair) without actually ceasing to be itself. Of course for this notion of exchange to be consistent we have to assume that objects do not possess any essential properties except their haecceities.
10Finally, we may want to introduce an even thinner concept of exchange. In the exchange of the third type there was no epistemological difference between the initial and the final states, but an ontological one. Now we consider a case in which there is no ontological difference but a mere difference in language.
Exchange No4 (exchange of labels). An exchange of an object A which bears a label LA with an object B which bears a label LB results in a situation in which there are objects A′ and B′ qualitatively and otherwise identical with A and B and such that A′ bears the label LB while B′ bears the label LA.
11Given that the idea of an exchange of objects somehow involves the notion of retaining numerical identity in spite of undergoing superficial changes, we may note that each of the four introduced concepts of exchange corresponds to a slightly different intuition of what it takes for an object to remain the same entity. Exchange No1 presupposes that an object’s location is irrelevant to its identity, and that retaining all other properties is sufficient for it to be itself. Definition 2 implies that for an object to remain itself it is necessary that it should keep a particular subset of the set of its properties. According to the third notion, an object’s identity is defined by its haecceity. The fourth option seems to be based on the rather absurd idea that the identity of an object can be somehow associated with its name.
12In the next step we will address the question of which of the four available notions of exchange should be used as a physical interpretation of the mathematical notion of permutation. As a first example, let us consider the classical state of two particles at time t given by their position and velocity as follows:
r1(t) = r, v1(t) = v,
r2(t) = r′,v2(t) = v′.
13The result of the permutation of indices is the set of the following functions:
r2(t) = r, v2(t) = v,
r1(t) = r′ ,v1(t) = v′.
14From this it clearly follows that the corresponding physical exchange of two particles cannot be interpreted as exchange No1, since in that case the resulting functions would be
r1(t) = r′, v1(t) = v,
r2(t) = r,v2(t) = v′.
15The idea of an exchange of position obviously does not take into account the fact that in physics position is treated as no different from any other variable characterizing a particle (velocity, momentum, angular momentum, etc.). Of the remaining three notions of a physical exchange, the exchange of labels is the least interesting because it is essentially a redescription of the same physical situation. Thus it can be claimed that there are only two interesting notions of exchange available: the exchange of essences and of haecceities.
16The difference between the two concepts is clearly visible when we consider a permutation of two particles of different types. If we interpreted a permutation of indices in the description of the state of a positron and an electron as representing exchange of essences, then the permuted function would describe a situation in which the electron is in the state initially occupied by the positron, and vice versa. In contrast, the exchange of haecceities leads to the state in which the object possessing the haecceity of the electron now has all the properties (state-dependent and state-independent) of the positron, and likewise for the positron. Thus the permuted and non-permuted functions describe ontologically distinct states which are nevertheless empirically indistinguishable.
- 3 For an extended discussion, see e.g., [Cohen-Tannoudji, Diu et al. 1977, 1370–1408].
17Let us now apply our selected interpretations of physical exchange to the analysis of the fundamental symmetrization/antisymmetrization postulate of quantum mechanics. The textbook way to introduce this postulate is through the concept of exchange degeneracy.3 Considering the joint state of two particles of the same type such that one of them occupies state |u⟩ whereas the other one is in a different state |v⟩, we should observe that the two permuted states |u⟩1|v⟩2 and |v⟩1|u⟩2 are empirically indistinguishable. According to the essentialist approach this indistinguishability comes from the fact that both bi-partite states represent one and the same physical state of affairs. On the other hand, the haecceitist approach admits that there is a difference between the permuted and non-permuted states, but this difference cannot give rise to any observational effects, as haecceities are not empirically accessible.
18In order to avoid the degeneracy problem, we adopt the symmetrization postulate, which narrows down the admissible states to the symmetric (occupied by bosons) and antisymmetric ones (applicable to fermions). Thus the only vector that can properly represent the above-discussed state of two electrons is the antisymmetric superposition
19In the case of bosons, the state has to be symmetric
20Given that in the quantum-mechanical formalism the sign of a vector has no physical meaning, it is commonly accepted that both types of vectors display the required permutation invariance which follows from the indistinguishability postulate regarding particles of the same type.
- 4 See for instance a comprehensive logical analysis of various grades of discernibility in [Ladyman, (...)
21The standard view is that this permutation invariance has dramatic consequences regarding the ontological status of quantum particles of the same type. Most famously, it is argued that fermions and bosons of the same type are indiscernible by their properties and relations, and hence they violate the Principle of the Identity of Indiscernibles (PII). Recent foundational work on the notion of discernibility has revealed that there are many non-equivalent ways of interpreting this concept,4 so we have to be precise about what particular type of discernibility is claimed to be violated by quantum particles. The logically strongest grade of discernibility is known as absolute discernibility, and it can be roughly defined as follows: two objects a and b are absolutely discernible iff there is an open formula in one variable consisting of predicates representing admissible properties or relations which is satisfied by a but not by b. The Indiscernibility Thesis applied to quantum particles of the same type can be formulated as the negation of their absolute discernibility:
(IT) Distinct fermions (bosons) of the same type are never absolutely discernible by their properties or relations.
- 5 The list of publications analyzing the violation of PII in quantum mechanics is long, and it contai (...)
22Although suggestions that quantum particles may violate PII were made quite early on in the history of quantum mechanics, the first rigorous proof of this fact was given in [French & Redhead 1988] and was subsequently generalized and improved upon in other publications.5 French & Redhead’s proof is based on the assumption that when we consider a set of n particles of the same type, any property of the ith particle can be represented by an operator of the form Oi = I(1) ⊗ I(2) ⊗ … ⊗ O(i) ⊗ … ⊗ I(n), where O is a Hermitian operator acting on the single-particle Hilbert space ℋ. Now it is easy to prove that the expectation values of two such operators Oi and Oj calculated for symmetric and antisymmetric states are identical. Similarly, it can be proved that the probabilities of revealing any value of observables of the above type conditional upon any measurement outcome previously revealed are the same for all n particles.
23In what follows I will argue that the Indiscernibility Thesis is actually contingent upon the selection of one of the two available interpretations of exchange of particles that we have discussed in the previous section. More specifically, I will try to show that the argument in favour of IT goes through only if we accept the exchange of haecceities interpretation. However, under the alternative essentialist interpretation, it can actually be argued that fermions and bosons are absolutely discernible in certain typical configurations.
24My first argument is based on the observation that the exchange-of-essences interpretation leads to the symmetrization postulate regarding admissible observables, which effectively excludes non-symmetric observables used by French & Redhead in their proof of IT. To see that this is the case, let us recall that if we interpret the permutation P12 as exchanging essences of particles 1 and 2, the mathematical vectors |ψ⟩ and P12|ψ⟩ actually represent one and the same physical state (under the condition that |ψ⟩ describes a state of two indistinguishable particles having the same essences). Hence no physically meaningful observable can discriminate between the two permuted vectors. To put it more precisely, the only admissible operators are those whose expectation values are identical in “both” states |ψ⟩ and P12|ψ⟩ : ⟨ψ|O|ψ⟩ = ⟨ψ|P12OP12|ψ⟩. But of course this equation must hold regardless of the choice of the state |ψ⟩, and this means that the operator O commutes with the permutation operator P12. Yet clearly the operators Oi introduced by French & Redhead do not commute with permutation operators, as can be seen in the following commutation relation: PijOiPij = Oj.
25French & Redhead are aware of the problem . Their response to it is based on the distinction between observable and unobservable properties. But I believe that this reply has no force in the context of the exchange-of-essences interpretation. The permuted states |ψ⟩ and P12|ψ⟩ are not merely observationally indistinguishable—they are metaphysically identical. An operator which “sees” a difference between numerically identical physical states merely because of their (or rather “its”) different mathematical representations cannot possibly represent any physically meaningful property, whether observable or not.
- 6 Nick Huggett generalizes French & Redhead’s proof of IT in a way which may seem to threaten my argu (...)
26In my mind French & Redhead’s response makes sense only under the alternative haecceity interpretation. Here the vectors |ψ⟩ and P12|ψ⟩ represent observationally indistinguishable but numerically distinct states of affairs. Thus, it can be claimed that operators Oi and Oj, whose expectation values in states |ψ⟩ and P12|ψ⟩ are different, represent some “hidden” properties of the entire system, reflecting the ontological distinctness between the permuted states. Speaking loosely, each operator Oi is “attached” to a different haecceity via its label i, so when we swap haecceities between particles i and j, clearly the result should be “registered” by Oi. But no physically meaningful operator can register any difference between a situation and itself, regardless of how we decide to represent it mathematically.6
27But this is not the end of the story. The fact that one particular argument in favor of IT turns out to be incorrect does not show that the thesis itself is false. We need a direct proof that the exchange-of-essences interpretation leads to the conclusion that some particles of the same type are indeed absolutely discernible. Finding just one physical property such that only one particle occupying a joint symmetric/antisymmetric state possesses it is all we need to reach our goal. But before we can do that, we must address the question of how to properly represent measurable characteristics of individual particles when the admissible operators are restricted to symmetric ones.
28Let us consider any one-dimensional projection operator P acting on a single-particle Hilbert space. As is well-known, such an operator is taken to represent a specific quantum-mechanical property of an individual particle. Moreover, using the whole family of such projection operators we can in principle describe any physical property of a particle. But our goal now is to construct a new projector acting on the tensor product of two Hilbert spaces which could represent the statement that (at least) one of two indistinguishable particles possesses property P. It is relatively straightforward to notice that such an operator Ω should satisfy the following desiderata:
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Ω should be Hermitian,
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Ω should be symmetric,
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Ω should be a projector (and therefore idempotent),
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Ω should be the sum of tensor products of one-particle operators involving only P and I (the identity).
29From conditions (2) and (4) it follows that the most general form Ω can have is the following:
Ω = aP ⊗ I + aI ⊗ P + bP ⊗ P.
30Given that Ω is assumed to be Hermitian, coefficients a and b have to be real. Now we can apply requirement (3):
Ω2 = Ω.
31Let us calculate the square of Ω (using the fact that P2 = P):
Ω2 = a2P ⊗ I + a2I ⊗ P + (2a2 + 4ab + b2)P ⊗ P.
32Comparing formulas for Ω and Ω2 we can first derive a2 = a. This equation obviously has two solutions in real numbers (0 and 1), but we can discard the value 0, as the operator P ⊗ P clearly represents the situation in which both particles have the same property. If we put a = 1, we can easily solve the quadratic equation in b which arises as the result of equating the coefficients of the component P ⊗ P in the expansions of Ω and Ω2. Thus the only two solutions are as follows:
Ω1 = P ⊗ I + I ⊗ P − P ⊗ P,
Ω2 = P ⊗ I + I ⊗ P − 2P ⊗ P.
33However, we don’t have to make a choice between Ω1 and Ω2 in order to prove the following theorem. It is not difficult to observe that Ω1 represents the question “Does at least one particle possess property P?” while Ω2 the question “Is it true that one particle possesses property P while the other does not possess P?” See an extensive analysis given in , .
Theorem 1. Let Ψ be a normalized vector a|u⟩1|v⟩2 + b|v⟩1|u⟩2 where |u⟩ and |v⟩ are mutually orthogonal unit vectors, and let P = |u⟩⟨u|. Then the expectation value of both operators Ω1 and Ω2 in state Ψ is 1.
34Here is a sketch of the calculation confirming this fact:
⟨Ψ|P ⊗ I|Ψ⟩ = ⟨a * uv + b * vu|P ⊗ I|auv + bvu⟩ = a * a⟨u|P|u⟩ + b * b⟨v|P|v⟩ = |a|2.
35Analogously, it can be showed that
⟨Ψ|I ⊗ P|Ψ⟩ = |b|2.
36And because the expectation value of P ⊗ P in Ψ vanishes due to the orthogonality relation between |u⟩ and |v⟩, we finally arrive at the sought-after result:
⟨Ψ|Ω1|Ψ⟩ = ⟨Ψ|Ω2|Ψ⟩ = |a|2 + |b|2 = 1
37What is the meaning of this formal derivation? It can be unpacked as follows: given the only available mathematical representation of the statement “At least one particle possesses property P”, if the system is prepared in a superposition of the product of two orthogonal states |u⟩1|v⟩2 and its permuted form |v⟩1|u⟩2, at least one of the two particles possesses the property associated with state |u⟩. But exactly the same can be proved with respect to the state |v⟩. Consequently, we have to admit that at least one particle has the property associated with |u⟩, and at least one particle has the property associated with |v⟩. But clearly one particle cannot be both in state |u⟩ and state |v⟩. Thus we have proved that the particles prepared in state Ψ are discernible by their properties P = |u⟩⟨u| and Q = |v⟩⟨v|. This result obviously applies to bosons and fermions, as symmetric and antisymmetric states are just special cases of the superposition Ψ.
38It may be instructive to see why this conclusion is avoidable under the alternative, haecceistic interpretation of permutation. Of course, the formal result of Theorem 1 still stands, as it is a mathematical fact, but its physical interpretation changes. Haecceitism implies that there is a meaningful difference between labels, as they refer to numerically distinct entities with different primitive identities. Consequently, we can conceptually (although not observationally) distinguish between statements “Particle 1 has property P” and “Particle 2 has property P”. The first statement is deemed true if and only if the operator P ⊗ I receives expectation value 1 in state Ψ, whereas the second one corresponds to the operator I ⊗ P and its expectation value. For the haecceitist the correct interpretation of the statement “At least one particle possesses property P” is just the classical disjunction of the above-mentioned individual statements: “Particle 1 has property P or particle 2 has property P”. And in the case in which none of the disjuncts receives the value “true” the entire disjunction cannot be true.
39The haecceitist interprets the fact that the operator Ω1 has its expectation value equal 1 in Ψ as a mere indication of the fact that when we decide to measure P on both particles, one measurement will reveal value 1 with certainty. But this doesn’t mean that any particle possesses the corresponding property P before the measurement. So Ω1 can be construed as referring to whatever property of the entire system is responsible for the predicted behavior (most likely this property has a fundamental dispositional character). But I would like to stress that without the “thick” metaphysics of primitive identities the disjunctive interpretation of the statement “At least one particle has property P” would not be available. Without haecceities we have only two options: either to accept Ω1 as a formal representation of this property, or to admit that the property in question is not expressible at all in our impoverished symmetric language.
40I have laid down two main philosophical positions regarding the meaning of permutation invariance in quantum mechanics: the essentialist view and the haecceitistic view. I have argued that both views come in whole packages, including a lot more than the mere philosophical interpretations of the notion of permutation. Essentialism leads to the strong symmetrization postulate with respect to admissible observables, which prevents us from representing properties of individual particles with the help of non-symmetric, label-bearing operators. Generally, essentialism repudiates the use of labels (indices) as names with fixed reference, and instead treats them only as formal devices enabling us to consider certain mathematical symmetries. Finally, it can be argued that insofar as essentialism is capable of formulating and solving the problem of absolute discernibility of particles of the same type at all, its answer is that both fermions and bosons can be actually discerned by their properties.
- 7 This point, as I believe, must have somehow escaped the notice of main experts in the field. For in (...)
41On the other side of the divide, haecceitism has to make the distinction between physically meaningful Hermitian operators and operators corresponding to observable properties. Operators which represent properties of individual particles are meaningful but, strangely enough, they are not literally observables. Labels used in the formal description of many-particle states are to be treated literally: they follow the primitive identity of individual objects. The Indiscernibility Thesis follows under this view from the fact that the expectation values for all single-particle operators are the same in antisymmetric/symmetric states. One surprising feature of haecceitism is that it is actually a necessary component of IT. Without haecceitism the argument for the indiscernibility claim could not even get off the ground.7
42Due to the lack of space I can’t discuss in detail what I consider the greatest challenge to the essentialist interpretation and the associated claim that absolute discernibility is attainable for quantum particles of the same type. This challenge is a consequence of the fact that in the state resulting from the antisymmetrization of the product state |u⟩1|v⟩2 there are infinitely many projectors representing single-particle properties other than |u⟩ and |v⟩ whose expectation values equal 1. As a result, it seems that we would have to admit that individual particles possess mutually incompatible properties. This problem requires an extensive and thorough evaluation which has to be saved for another occasion.
43I would like to end this survey with a plea on behalf of essentialism. The fact that the essentialist approach admits the possibility of discerning quantum particles by their properties fits well the everyday practice of experimental physicists who have no qualms about talking of the electron in a bubble chamber as being an entity different from an electron in the Andromeda galaxy. Although essentialism entails that in some cases quantum particles may occupy states which render them indiscernible, this does not necessarily rob them of the status of individuals, if we follow Dieks & Versteegh and define individuals as objects for which it is possible to be in a state in which they possess different properties [Dieks & Versteegh 2008]. I am aware of the conceptual difficulties afflicting this position, and I admit that at this point I can’t offer a satisfactory solution to all of them. But I believe that the advantages of the essentialist interpretation merit further investigation into this new approach to the problem of identity and individuality in quantum mechanics.