1. Introduction.

According to the founding father of wave mechanics, Erwin Schrödinger, one of the ontological lessons that quantum mechanics (QM) has taught us is that the elementary building blocks of the physical world are entirely indiscernible: “I beg to emphasize this and I beg you to believe it: it is not a question of our being able to ascertain the identity in some instances and not being able to do so in others. It is beyond doubt that the question of the ‘sameness’, of identity, really and truly has no meaning” (Schrödinger Reference Schrödinger1952, 18). Similar elementary particles have no ‘identity’; there is nothing that discerns one particle from another. Neither properties nor relations can tell them apart: they are not individuals. Thus Schrödinger famously compared the elementary particles to “the shillings and pennies in your bank account” (Reference Schrödinger1995, 103), in contrast to the coins in your piggy bank. Hermann Weyl preceded Schrödinger when he wrote that “even in principle one cannot demand an alibi from an electron” (Reference Weyl1931, 241).

Over the past decades, several philosophers have scrutinized this ‘Indiscernibility Thesis’ (IT) by providing various rigorous arguments in favor of it: similar elementary particles (same mass, charge, spin, etc.)—when forming a composite physical system—are indiscernible by quantum mechanical means. Leibniz's metaphysical Principle of the Identity of Indiscernibles (PII) is thus refuted by QM—and perhaps is therefore not so metaphysical after all. This does not rule out conclusively that particles really are discernible, but if they are, they have to be discerned by means that go above and beyond QM, such as by ascribing Scotusian haecceitas to the particles, or ascribing sibling attributes to them from scholastic and neoscholastic metaphysics. Nevertheless, few philosophers have considered this move to save the discernibility of the elementary particles to be attractive—if this move is mentioned at all, and then usually only as a possibility and rarely as a plausibility. Mild naturalistic inclinations seem sufficient to accommodate IT in our general metaphysical view of the world. We ought to let well-established scientific knowledge inform our metaphysical view of the world whenever possible and appropriate, and this is exactly what Schrödinger begged us to do. The only respectable metaphysics is naturalized metaphysics (see further Ladyman and Ross Reference Ladyman and Ross2007, 1–38).

Prominent defenders of IT include Margenau (Reference Margenau1944); Cortes (Reference Cortes1976), who brandished PII ‘a false principle’; Barnette (Reference Barnette1978); French and Redhead (Reference French and Redhead1988); Giuntini and Mittelstaedt (Reference Giuntini and Mittelstaedt1989), who argued that although demonstrably valid in classical logic, in quantum logic the validity of PII cannot be established; French (Reference French1989b), who assured us in the title that PII ‘is not contingently true either’; French (Reference French1989a, Reference French and Castellani1998, Reference French2006); Redhead and Teller (Reference Redhead and Teller1992); Butterfield (Reference Butterfield1993); Castellani and Mittelstaedt (Reference Castellani and Mittelstaedt1998); Massimi (Reference Massimi2001); Teller (Reference Teller and Castellani1998); French and Rickles (Reference French, Rickles, Brading and Castellani2003); Huggett (Reference Huggett, Brading and Castellani2003); French and Krause (Reference French and Krause2006, Chapter 4).

There have, however, been dissenters. B. C. van Fraassen (Reference van Fraassen1991) is one of them (see Muller and Saunders Reference Muller and Saunders2008, 517–518, for an analysis of his arguments). We follow the other dissenters: S. W. Saunders and one of us (Saunders Reference Saunders2006; Muller and Saunders Reference Muller and Saunders2008). They neither claim that fermions are individuals nor do they rely on a particular interpretation of QM. On the basis of standard mathematics (standard set theory and classical predicate logic) and only uncontroversial postulates of QM (notably leaving out the projection postulate, the strong property postulate, and the quantum mechanical probabilities), they demonstrate that similar fermions are weakly discernible, that is, they are discerned by relations that are irreflexive and symmetric, in every admissible state of the composite system. So according to Muller and Saunders, the elementary building blocks of matter (fermions) are not indiscernibles after all, contra IT. They prove this, however, only for finite-dimensional Hilbert spaces (their Theorem 1), which is a rather serious restriction because most applications of QM to physical systems employ complex wave functions and these live in the infinite-dimensional Hilbert space $L^{2}( \mathstrut{\Bbb R} ^{3N}) $; nonetheless they confidently conjecture that their result will hold good for infinite-dimensional Hilbert spaces as well (their Conjecture 1). Furthermore, Muller and Saunders (Reference Muller and Saunders2008, 534–535) need to assume for their proof there is a maximal self-adjoint operator acting on finite-dimensional Hilbert spaces that is physically significant. In the case of dimension 2 of a single-fermion Hilbert space, Pauli's spin-1/2 operator qualifies as such a maximal self-adjoint operator, but for higher dimensions, spin is degenerate. They gloss over what this maximal operator corresponds to in those cases.

When it comes to the elementary quanta of interaction (bosons), Muller and Saunders (Reference Muller and Saunders2008, 537–540) claim that bosons are also weak discernibles, but of a probabilistic kind (the discerning relation involves quantum mechanical probabilities and therefore their proof needs the probability postulate of QM), whereas the discerning relation of the fermions is of a categorical kind (no probabilities involved). More precisely, the categorical discernibility of bosons turns out to be a contingent matter: in some states they are categorically discernible, in others, for example, direct-product states, they are not; this prevents one to conclude that bosons are categorical discernibles simpliciter. But the boson's probabilistic discernibility is a quantum mechanical necessity; Theorem 3 (Muller and Saunders Reference Muller and Saunders2008) establishes it for two bosons, with no restrictions on the dimensionality of Hilbert space but conditional on whether a particular sort of operator can be found (again, a maximal self-adjoint one of physical significance). The fermions are also probabilistic discernibles; their Theorem 2 states it for finite-dimensional Hilbert spaces only and is therefore equally restrictive as their Theorem 1.

The central aim of this article is the completion of the project initiated and developed in Muller and Saunders Reference Muller and Saunders2008, by demonstrating that all restrictions in their discernibility theorems can be removed by proving more general theorems and proving them differently, employing only quantum mechanical operators that have obvious physical significance. We shall then be in a position to conclude in utter generality that all kinds of similar particles in all their physical states, pure and mixed, in all infinite-dimensional or finite-dimensional Hilbert spaces can be categorically discerned on the basis of quantum mechanical postulates. This result, then, should be the death knell for IT, and by implication, it establishes the universal reign of Leibniz's PII in QM.

In Sections 3 and 4, we prove the theorems that establish the general result. First we introduce some terminology, state explicitly what we need of QM, and address the issue of what has the license to discern elementary particles in the next section.

2. Preliminaries.

For the motivation and further elaboration of the terminology we are about to introduce, we refer to Muller and Saunders (Reference Muller and Saunders2008, 503–505) because we follow them closely (readers of that paper can jump to the next section of this article). Here we mention only what is necessary in order to keep this article comparatively self-contained.

We call physical objects in a set ‘absolutely discernible’, or ‘individuals’, iff for every object there is some physical property that it has but all others lack; and ‘relationally discernible’ iff for every object there is some physical relation that discerns it from all others (see below). An object is ‘indiscernible’ iff it is both absolutely and relationally indiscernible, and hence ‘discernible’ iff it is discernible either way or both ways. Objects that are not individuals but are relationally discernible from all other objects we call ‘relationals’; then ‘indiscernibles’ are objects that are neither individuals nor relationals. Quine (Reference Quine1981, 129–133) was the first to inquire into different kinds of discernibility; he discovered there are only two distinct logical categories of relational discernibility (by means of a binary relation): either the relation is irreflexive and asymmetric, in which case we speak of ‘relative discernibility’; or the relation is irreflexive and symmetric, in which case we speak of ‘weak discernibility’. We call attention to the logical fact that if relation R discerns particles 1 and 2 relatively, then its ‘complement relation’, defined as $\neg R$, is also asymmetric but reflexive; and if R discerns particles 1 and 2 weakly, then its complement relation $\neg R$ is reflexive and symmetric but does not hold for $\boldsymbol{a}\neq \boldsymbol{b}$ whenever R holds for $\boldsymbol{a}\neq \boldsymbol{b}$.

Leibniz's Principle of the Identity of Indiscernibles (PII) for physical objects states that no two physical objects are absolutely and relationally indiscernible; or synonymously, two physical objects are numerically discernible only if they are qualitatively discernible. One can further distinguish principles for absolute and for relational indiscernibles and then inquire into the logical relations between these and PII; see Muller and Saunders Reference Muller and Saunders2008, 504–505. Similarly one can distinguish three indiscernibility theses as the corresponding negations of the Leibnizian principles. We restrict ourselves to the Indiscernibility Thesis (IT): there are composite systems of similar physical objects that consist of absolutely and relationally indiscernible physical objects. Then either it is a theorem of logic that PII holds and IT fails, or conversely,

Next we rehearse the postulates of QM that we shall use in our discernibility theorems.

The state postulate (StateP) associates some super-selected sector Hilbert space $\mathstrut{\cal H} $ to every given physical system $\mathstrut{\cal S} $ and represents every physical state of $\mathstrut{\cal S} $ by a statistical operator $W\in \mathstrut{\cal S} ( \mathstrut{\cal H} ) $; the pure states lie on the boundary of this convex set $\mathstrut{\cal S} ( \mathstrut{\cal H} ) $ of all statistical operators and the mixed states lie inside. If $S$ consists of $N$ similar elementary particles, then the associated Hilbert space is a direct-product Hilbert space $\mathstrut{\cal H} ^{N}=\mathstrut{\cal H} \otimes \cdots \otimes \mathstrut{\cal H} $ of $N$ identical single-particle Hilbert spaces.

The weak magnitude postulate (WkMP) says that every physical magnitude is represented by an operator that acts on $\mathstrut{\cal H} $. Stronger magnitude postulates are not needed, because they all imply the logically weaker WkMP, which is sufficient for our purposes.

In order to state the symmetrization postulate, we need to define first the orthogonal projectors $\Pi ^{\pm }_{N}$ of the lattice $\mathstrut{\cal P} ( \mathstrut{\cal H} ^{N}) $ of all projectors, defined as

where $\mathrm{sign}\,( \pi ) \in \{ \pm 1\} $ is the sign of the permutation $\pi \in \mathstrut{\Bbb P} _{N}$ on $\{ 1,\, 2,\,\ldots ,\, N\} $ (+1 if it is even, −1 if odd), and where $U_{\pi }$ is a unitary operator acting on $\mathstrut{\cal H} ^{N}$ corresponding to permutation $\pi $ (these $U_{\pi }$ form a unitary representation on $\mathstrut{\cal H} ^{N}$ of the permutation group $\mathstrut{\Bbb P} _{N}$). The projectors lead to the following permutation-invariant orthogonal subspaces:

which are called the BE-symmetric (Bose-Einstein) and the FD-symmetric (Fermi-Dirac) subspaces of $\mathstrut{\cal H} ^{N}$, respectively. These subspaces can, alternatively, be seen as generated by the symmetrized and antisymmetrized versions of the products of basis-vectors in $\mathstrut{\cal H} ^{N}$. Only for $N=2$, we have that $\mathstrut{\cal H} ^{N}_{-}\oplus \mathstrut{\cal H} ^{N}_{+}=\mathstrut{\cal H} ^{N}$.

The symmetrization postulate (SymP) states for a composite system of $N\geq 2$ similar particles with $N$-fold direct-product Hilbert space $\mathstrut{\cal H} ^{N}$ the following: (i) the projectors $\Pi ^{\pm }_{N}$ are super-selection operators; (ii) integer and half-integer spin particles are confined to the BE- and the FD-symmetric subspaces, respectively; and (iii) all composite systems of similar particles consist of particles that have all either integer spin or half-integer spin (dichotomy).

We represent a ‘quantitative physical property’ associated with physical magnitude $\mathstrut{\cal A} $ mathematically by ordered pair $\langle A,\, a\rangle $, where $A$ is the operator representing $\mathstrut{\cal A} $ and $a\in \mathstrut{\Bbb C} $. The weak property postulate (WkPP) says that if the physical state of physical system $S$ is an eigenstate of $A$ having eigenvalue $a$, then it has property $\langle A,\, a\rangle $; the strong property postulate (StrPP) adds the converse conditional to WkPP. (We mention that eigenstates can be mixed, so that physical systems in mixed states can posses properties too [by WkPP]; see Muller and Saunders Reference Muller and Saunders2008, 513, for details.) WkPP implies that every physical system $S$ always has the same quantitative properties associated with all super-selected physical magnitudes because $S$ always is in the same common eigenstate of the super-selected operators. We call these possessed quantitative physical properties ‘super-selected’ and we call physical systems (e.g., particles) that have the same super-selected quantitative physical properties ‘similar’ (this is the precise definition of ‘similar’, a word that we have been using loosely until now, following Dirac).

We also adopt the following semantic condition (SemC). When talking of a physical system at a given time, we ascribe to it at most one quantitative physical property associated with physical magnitude $\mathstrut{\cal A} $:

For example, particles cannot possess two different masses at the same time and (4) is the generalization of this in the language of QM. In other words: if $S$ possesses quantitative physical property $\langle A,\, a\rangle $, then $S$ does not possess property $\langle A,\, a^{\prime }\rangle $ for every $a^{\prime }\neq a$. Statement (4) is neither a tautology nor a theorem of logic, but we agree with Muller and Saunders in that “it seems absurd to deny it all the same” (Reference Muller and Saunders2008, 515). Notice there is mention neither of measurements nor of probabilities in the postulates mentioned above, let alone interpretational glosses such as dispositions.

For an outline of the elementary language of QM, we refer to Muller and Saunders (Reference Muller and Saunders2008, 520–521). In this language, the proper formulation of PII is that “physically indiscernible physical systems are identical” (Reference Muller and Saunders2008, 521–523):

where a and b are physical-system variables, ranging over all physical systems, and where $\mathsf{PhysInd}( \boldsymbol{a},\,\boldsymbol{b}) $ comprises everything that is in principle permitted to discern the particles, roughly, all physical relations and all physical properties. The properties and the relations may involve, in their definition, probabilities, in which case we call them ‘probabilistic’; otherwise, in the absence of probabilities, we call them ‘categorical’. So the three logical kinds of discernibility—(a) absolute and (r) relational, which further branches in (r.w) weak and (r.r) relative discernibility—come in a probabilistic and a categorical variety. In their analysis of the traditional arguments in favor of IT, Muller and Saunders (Reference Muller and Saunders2008, 524–526) make the case that, setting conditional probabilities aside, (r) relational discernibility has been largely overlooked by the tradition. (Leibniz also included relations in his PII because he held that all relations reduce to properties and thus could make do with an explicit formulation of PII that only mentions properties; give up his reducibility thesis of relations to properties and one can no longer make do with his formulation; see Muller and Saunders Reference Muller and Saunders2008, 504–505.) What in particular has been overlooked, and is employed by Muller and Saunders, are properties of wholes that are relations between their constitutive parts: the distance between the sun and the earth is a property of the solar system; the Coulomb-interaction between the electron and the proton is a property of the hydrogen atom, and so on.

Muller and Saunders (Reference Muller and Saunders2008, 524–528) argue at length that only those properties and relations that meet the following two requirements are permitted to occur in $\textsf{PhysInd}$ (5).

All proponents of IT have considered quantum mechanical means of discerning similar particles that obey these two requirements (see the references listed in the introduction)—and have found them all to fail. They were correct in this. They were not correct in not considering categorical relations.

To close this section, we want address another distinction from the recent flourishing literature on indiscernibility and inquire briefly whether this motivates a third requirement, call it Req3. One easily shows that absolute discernibles are relational discernibles by defining a relation (expressed by dyadic predicate R _M ) in terms of the discerning properties (expressed by monadic predicate M; see Muller and Saunders Reference Muller and Saunders2008, 529). One could submit that this is not a case of ‘genuine’ but of ‘fake’ relational discernibility, because there is nothing inherently relational about the way this relational discernibility is achieved: R _M is completely reducible to property M, which already discerns the particles absolutely. Similarly, one may also object that a case of absolute discernibility implied by relational discernibility by means of a monadic predicate M _R that is defined in terms of the discerning relation R is not a case of ‘genuine’ but of ‘fake’ absolute discernibility (the terminology of ‘genuine’ and ‘fake’ is not Ladyman's (Reference Ladyman2007, 36), who calls ‘fake’ and ‘genuine’ more neutrally ‘contextual’ and ‘intrinsic’, respectively). Definitions: physical systems a and b are ‘genuine’ relationals, or ‘genuine’ (weak, relative) relational discernibles, iff they are discerned by some dyadic predicate, that is not reducible to monadic predicates of which some discern a and b absolutely; a and b are ‘genuine individuals, or ‘genuine’ absolute discernibles, iff they are discerned by some monadic predicate that is not reducible to dyadic predicates of which some discern a and b relationally; discernibles are ‘fake’ iff they are not genuine. Hence there is a prima facie case for adding a third requirement that excludes fake discernibility:

In turn, a fake property or relation can be defined rigorously as its undefinability in terms of the predicates in the language of QM that meet Req1 and Req2. In order to inquire logically into genuineness and fakeness, thus defined, at an appreciable level of rigor, the entire formal language must be spelled out and all axioms of QM must be spelled out in that formal language. Such a logical inquiry is however far beyond the scope of this article. Nonetheless we shall see that our discerning relations plausibly are genuine.

But besides ‘formalese phobia’, there is a respectable reason for not adding Req3 to our list. To see why, consider the following two cases—(a) indiscernibles and (b) discernibles.

(a) Suppose particles turn out to be indiscernibles in that they are indiscernible by all genuine relations and all genuine properties. Then they are also indiscernible by all properties and all relations that are defined in terms of these, which one can presumably prove by induction over the complexity of the defined predicates. So indiscernibles remain indiscernibles, regardless of whether we require the candidate properties and relations to be genuine.

(b) Suppose next that the particles turn out to be discernibles. (i) If they are discerned by a relation that turns out to be definable in terms of genuine properties one of which discerns the particles absolutely, then the relationals become individuals—good news for admirers of PII. But the important point to notice is that discernibles remain discernibles. (ii) If the particles are discerned by a property that turns turns out to be definable in terms of genuine relations one of which discerns the particles relationally, then the individuals lose their individuality and become relationals. They had a fake identity and are now exposed as metaphysical imposters. But again, the important point to notice is that discernibles remain discernibles.

To conclude, adding Req3 will not have any consequences for crossing the border between discernibles and indiscernibles. This seems a respectable reason not to add Req3 to our list of two requirements.

3. To Discern in Infinite-Dimensional Hilbert Spaces.

We first prove a lemma, from which our categorical discernibility theorems then immediately follow.

Lemma 1 (StateP, WkMP, WkPP, SemC). Given a composite physical system of $N\geq 2$ similar particles and its associated direct-product Hilbert space $\mathstrut{\cal H} ^{N}$. If there are two single-particle operators, $A$ and $B$, acting in single-particle Hilbert space $\mathstrut{\cal H} $, and they correspond to physical magnitudes $\mathstrut{\cal A} $ and $\mathstrut{\cal B} $, respectively, and there is a nonzero number $c\in \mathstrut{\Bbb C} $ such that in every pure state $\vert \phi \rangle \in \mathstrut{\cal H} $ in the domain of their commutator the following holds:

then all particles are categorically weakly discernible.

Proof. Let a , b , j be particle variables, ranging over the set {1, 2, … , N } of $N$ particles. We proceed stepwise, as follows.

1. [S1] Case for $N=2$, pure states.

2. [S2] Case for $N=2$, mixed states.

3. [S3] Case for $N> 2$, pure states.

4. [S4] Case for $N> 2$, mixed states.

[S1]. Case for $N=2$, pure states. Assume the antecedent. Define the following operators on $\mathstrut{\cal H} ^{2}=\mathstrut{\cal H} \otimes \mathstrut{\cal H} $:

where the operator 1 is the identity-operator on $\mathstrut{\cal H} $; and mutatis mutandis for B. Define next the following commutator relation:

where $\mathstrut{\cal D} \subseteq \mathstrut{\cal H} \otimes \mathstrut{\cal H} $ is the domain of the commutator. An arbitrary vector $\vert \Psi \rangle $ can be expanded:

where $d$ is a positive integer or $\infty $, and $\{ \vert \phi _{1},\,\vert \phi _{2}\rangle ,\,\ldots \} $ is a basis for $\mathstrut{\cal H} $ that lies in the domain of the commutator $[ A,\, B] $. Then, using expansion (9) and equation (6), one quickly shows that ($\boldsymbol{a}=\boldsymbol{b}$):

and that for $\boldsymbol{a}\neq \boldsymbol{b}$:

because by assumption $c\neq 0$. By WkPP, the composite system then possesses the following four quantitative physical properties (when substituting 1 or 2 for a in the first, and 1 for a and 2 for b , or conversely, in the second):

In virtue of SemC (4), the composite system then does not possess the following four quantitative physical properties (recall that $c\neq 0$):

The composite system possesses the property (12) that is a relation between its constituent parts, namely, $\mathsf{C}$ (8), which is reflexive: $\mathsf{C}( \boldsymbol{a},\,\boldsymbol{a}) $ for every a due to (10). Similarly, but now using SemC (4), the composite system does not possess the property that is a relation between its constituent parts, namely, $\mathsf{C}$. Therefore 1 is not related to 2 , and 2 is not related to 1 either, because $\neg \mathsf{C}( \boldsymbol{a},\,\boldsymbol{b}) $ and $\neg \mathsf{C}( \boldsymbol{b},\,\boldsymbol{a}) $ ($\boldsymbol{a}\neq \boldsymbol{b}$); and then, due to the following theorem of logic,

we conclude that $\mathsf{C}$ is symmetric (Req2). Since by assumption $A$ and $B$ correspond to physical magnitudes, relation $\mathsf{C}$ (8) is physically meaningful (Req1) and hence is admissible, because it meets Req1 and Req2.

Further, it was just shown that the relation $\mathsf{C}$ (8) is reflexive and symmetric but fails for $\boldsymbol{a}\neq \boldsymbol{b}$ due to (11), which means that $\mathsf{C}$ discerns the two particles weakly in every pure state $\vert \Psi \rangle $ of the composite system. Since probabilities do not occur in $\mathsf{C}$ (8), the particles are discerned categorically.

[S2]. Case for $N=2$, mixed states. The equations in (8) can also be written as an equation for one-dimensional projectors that project onto the ray that contains $\vert \Psi \rangle $:

Due to the linearity of the operators, this equation remains valid for arbitrary linear combinations of projectors. This includes all convex combinations of projectors, which exhausts the set $\mathstrut{\cal S} ( \mathstrut{\cal H} \otimes \mathstrut{\cal H} ) $ of all mixed states. The commutator relation $\mathsf{C}$ (8) is easily extended to mixed states $W\in \mathstrut{\cal S} ( \mathstrut{\cal H} \otimes \mathstrut{\cal H} ) $ and the ensuing relation also discerns the particles categorically and weakly.

[S3], [S4]. Case for $N> 2$, pure and mixed states. Cases [S1] and [S2] are immediately extended to the $N$-particle cases, by considering the following $N$-factor operators:

where $A$ is the jth factor and $j$ a particle variable running over the $N$ labeled particles, and similarly for $B^{( j) }$. The extension to the mixed states then proceeds as in [S2]. QED.

Proof. In Lemma 1, choose for $A$ the linear momentum operator $\hat{P}$, for $B$ the Cartesian position operator $\hat{Q}$, and for $c$ the value $-\mathrm{i}\,\hslash $. The physical significance of these operators and their commutator, which is the celebrated canonical commutator

is beyond doubt and so is the ensuing commutator relation $\mathsf{C}$ (8), which we baptize the ‘Heisenberg relation’. The operators $\hat{P}$ and $\hat{Q}$ act on the infinite-dimensional Hilbert space of the complex wave functions $L^{2}( \mathstrut{\Bbb R} ^{3}) $, which is isomorphic to every infinite-dimensional Hilbert space. QED.

But is Theorem 1 not only applicable to particles having spin-0 and have we forgotten to mention this? Yes and no. Yes, we have deliberately forgotten to mention this. No, it is a corollary of Theorem 1 that it holds for all spin magnitudes, which is the content of the next theorem.

Proof. To deal with spin, we need SymP. The actual proof of the categorical weak discernibility for all particles having nonzero spin magnitude is at bottom a notational variant of the proof of Theorem 1. Let us sketch how this works for $N=2$. We begin with the following Hilbert space for a single particle:

which is the space of spinorial wave functions $\Psi $, that is, column vectors of $2s+1$-entries, each entry being a complex wave function of $L^{2}( \mathstrut{\Bbb R} ^{3}) $. Notice that $\mathstrut{\cal H} _{s}$ is a $( 2s+1) $-fold Cartesian product set, which becomes a Hilbert space by carrying the Hilbert space properties of $L^{2}( \mathstrut{\Bbb R} ^{3}) $ over to $\mathstrut{\cal H} _{s}$. For instance, the inner product on $\mathstrut{\cal H} _{s}$ is just the sum of the inner products of the components of the spinors:

where $\Psi _{k}$ is the kth entry of $\Psi $ (and similarly for $\Phi _{k}$), which provides the norm of $\mathstrut{\cal H} _{s}$, which in turn generates the norm topology of $\mathstrut{\cal H} _{s}$, and so on. The degenerate case of the spinor having only one entry is the case of $s=0$, which we treated in Theorem 1. So we proceed here with $s> 0$. In particular, the number of entries $2s+1$ is even iff the particles have half-integer spin, and odd iff the particles have integer spin, in units of $\hslash $.

Let $\mathbf{e}_{k}\in \mathstrut{\Bbb C} ^{2s+1}$ be such that its kth entry is 1 and all others 0. They form the standard basis for $\mathstrut{\Bbb C} ^{2s+1}$ and are the eigenvectors of the $z$-component of the spin operator $\hat{S}_{z}$, whose eigenvalues are traditionally denoted by $m$ (in the terminology of atomic physics: ‘magnetic quantum number’):

where $m_{1}=-s$, $m_{2}=-s+1$, $\ldots $, $m_{s-1}=s-1$, $m_{s}=+s$. Let $\phi _{1},\,\phi _{2},\,\ldots $ be a basis for $L^{2}( \mathstrut{\Bbb R} ^{3}) $; then this is a basis for single-particle spinor space $\mathstrut{\cal H} _{s}$ (18):

Recall that the linear momentum-operator $\hat{P}$ and the Cartesian position operator $\hat{Q}$ on an arbitrary complex wave function $\phi \in L^{2}( \mathstrut{\Bbb R} ^{3}) $ are the differential operator times $-\mathrm{i}\,\hslash $ and the multiplication operator, respectively:

and

where domain $\mathstrut{\cal D} _{P}=C^{1}( \mathstrut{\Bbb R} ^{3}) \cap L^{2}( \mathstrut{\Bbb R} ^{3}) $ and domain $\mathstrut{\cal D} _{Q}\subset L^{2}( \mathstrut{\Bbb R} ^{3}) $ consist of all wave functions $\psi $ such that $\vert \boldsymbol{q}\vert ^{2}\psi ( \boldsymbol{q}) \rightarrow 0$ when $\vert \boldsymbol{q}\vert \rightarrow \infty $. The action of $\hat{P}$ and $\hat{Q}$ is straightforwardly extended to arbitrary spinorial wave functions by letting the operators act componentwise on the $2s+1$ components. The canonical commutator of $\hat{P}$ and $\hat{Q}$ then carries over to spinor space $H_{2}$ (18). We can now appeal to the general Lemma 1 and conclude that the two arbitrary spin particles are categorically and weakly discernible. QED.

Another possibility to finish the proof is more or less to repeat the proof of Lemma 1 but now with spinorial wave functions. For step [S1], the case $N=2$, the state space of the composite system becomes:

where the spinorial wave functions now have $( 2s+1) ^{2}$ entries—$( 2s+1) ^{N}$ for $N$ spin-$s$ particles. A basis for $\mathstrut{\cal H} ^{2}_{s}$ (24) is

An arbitrary spinorial wave function $\Psi \in H^{2}_{s}$ of the composite system can then be expanded as follows:

where the $\gamma _{ml}$ form a squarely summable sequence, that is, a Hilbert vector in $l^{2}( \mathstrut{\Bbb N} ) $, of norm 1.

With the usual definitions,

one obtains all the relevant commutators on $\mathstrut{\cal H} ^{2}_{s}$ by using expansion (26). The discerning relation on the direct-product spinor space $\mathstrut{\cal H} ^{2}_{s}$ then becomes

where $\mathstrut{\cal D} \subset \mathstrut{\cal H} ^{2}_{s}$ is the domain of the commutator, and so on. We close this section with a number of systematic remarks.

Remark 1. Notice that in contrast to the proof of Theorem 1, the proof Corollary 1 relies—besides on StateP, WkMP, WkPP, and SemC—on SymP only in so far as that without this postulate the distinction between integer and half-integer spin particles makes little sense and, more importantly, the tacit claim that this distinction exhausts all possible composite systems of similar particles is unfounded. Besides this, SymP does not perform any deductive labor in the proof. Specifically, the distinction between Bose-Einstein and Fermi-Dirac states never enters the proof, which means that any restriction on Hilbert rays and on statistical operators, as SymP demands, leaves the proof valid: the theorem holds for all particles in all sorts of states … fermions, bosons, quons, parons, quarticles, anyons and what have you.

Remark 2. The proofs of Theorem 1 and Corollary 1 exploit the noncommutativity of the physical magnitudes, which is one of the algebraic hallmarks of QM. Good thing. The physical meaning of relation $\mathsf{C}$ (28) can be understood as follows: momentum and position pertain to two particles differently from how they pertain to a single particle. Admittedly this is something we already knew for a long time—since the advent of QM. What we didn’t know, but do know now, is that this old knowledge provides the ground for discerning similar particles weakly and categorically.

Remark 3. The spinorial wave function $\Psi $ must lie in the domain $\mathstrut{\cal D} $ of the commutator of the unbounded operators $\hat{P}$ and $\hat{Q}$, which domain is a proper subspace of $\mathstrut{\cal H} ^{2}_{s}$ so that the members of $\mathstrut{\cal H} ^{2}_{s}\setminus \mathstrut{\cal D} $ fall outside the scope of relation $\mathsf{C}$ (8). The domain $\mathstrut{\cal D} $ does however lie dense in $\mathstrut{\cal H} ^{2}_{s}$, even the domain of all polynomials of $\hat{P}$ and $\hat{Q}$ does so (the non-Abelian ring on $\mathstrut{\cal D} $ they generate)—$\mathstrut{\cal D} $ is the Schwarz space of all complex wave functions that are continuously differentiable and fall off exponentially. This means that every wave function that does not lie in Schwarz space can be approximated with arbitrary accuracy by means of wave functions that do lie in Schwarz space. This is apparently good enough for physics. Then it is good enough for us too.

Remark 4. A special case of Theorem 1 is that two bosons in symmetric direct-product states, say

are also weakly discernible. This seems a hard nut to swallow. If two bosons in state (29) are discernible, then something must have gone wrong. Perhaps we attach too much metaphysical significance to a mathematical result?

Our position is the following. The weak discernibility of the two bosons in state (29) is a deductive consequence of a few postulates of QM. Rationality dictates that if one accepts those postulates, one should accept every consequence of those postulates. This is part of what it means to accept deductive logic, which we do accept. We admit that the discernibility of two bosons in state (29) is an unexpected, if not bizarre, consequence. But in comparison to other bizarre consequences of QM—for example, inexplicable correlations at a distance (EPR), animate beings that are neither dead nor alive (Schrödinger's immortal cat), kettles of water on a seething fire that will never boil (quantum Zeno), an anthropocentric and intentional concept taken as primitive (measurement), states of matter defying familiar states of aggregation (BE-condensate)—the weak discernibility of bosons in direct-product states is not such a hard nut to swallow. Get real: it's peanuts.

Remark 5. Every ‘realistic’ QM model of a physical system, whether in atomic physics, nuclear physics, or solid state physics, employs wave functions. This means that now, and only now, we can conclude that the similar elementary particles of the real world are categorically and weakly discernible. Conjecture 1 of Muller and Saunders (Reference Muller and Saunders2008, 537) has been proved.

Parenthetically, do finite-dimensional Hilbert spaces actually have applications at all? Yes they have, in quantum optics and even more prominently in quantum information theory. There one chooses to pay attention to spin degrees of freedom only and ignores all others—position, linear momentum, energy. This is not to deny there are physical magnitudes such as position, momentum, or energy, or that these physical magnitudes do not apply in the quantum information theoretic models. Of course not. Ignoring these physical magnitudes is a matter of expediency if one is not interested in them. Idealization and approximation are part and parcel of science. No one would deny that QM models using spinorial wave functions in infinite-dimensional Hilbert space match physical reality better—if at all—than finite-dimensional models do that only consider spin, and it is for those better models that we have proved our case. Nevertheless, we next proceed to prove the discernibility of elementary particles for finite-dimensional Hilbert spaces.

4. To Discern in Finite-Dimensional Hilbert Spaces.

In the case of finite-dimensional Hilbert spaces, considering $\mathstrut{\cal C} ^{d}$ suffices, because every $d$-dimensional Hilbert space is isomorphic to $\mathstrut{\cal C} ^{d}$ ($d\in \mathstrut{\Bbb N} ^{+}$). The proof is a vast generalization of the total-spin relation $\mathsf{T}$ of Muller and Saunders (Reference Muller and Saunders2008, 5x).

Proof. Let $\boldsymbol{a},\,\boldsymbol{b},\,\boldsymbol{j}$ be particle variables, ranging over the set {1, 2, … , N } of $N$ particles. We proceed again stepwise, as follows:

1. [S1] Case for $N=2$, pure states.

2. [S2] Case for $N=2$, mixed states.

3. [S3] Case for $N> 2$, all states.

[S1]. Case for $N=2$, pure states. We begin by considering two similar particles, labeled 1 , 2 , of spin magnitude $s\hslash $, which is a positive integer or a half-integer; $\boldsymbol{a}$ and $\boldsymbol{b}$ are again variables over this set. The single particle Hilbert space is $\mathstrut{\Bbb C} ^{2s+1}$, which is isomorphic to every $( s2+1) $-dimensional Hilbert space; for $N$-particles the associated Hilbert space is the $N$-fold $\otimes $-product of $\mathstrut{\Bbb C} ^{2s+1}$. According to SymP, when we have considered integer and half-integer spin particles, we have considered all particles.

We begin by considering the spin operator of a single particle acting in $\mathstrut{\Bbb C} ^{2s+1}$:

where $\hat{S}_{x}$, $\hat{S}_{y}$, and $\hat{S}_{z}$ are the three spin operators along the three perpendicular spatial directions ($x,y,z$). The operators $\hat{S}^{2}$ and $S_{z}$ are self-adjoint and commute, and therefore have a common set of orthonormal eigenvectors $\vert s,\, m\rangle $; their eigenvector equations are

where eigenvalue $m\in \{ {\mbox{$-$}} s,\,{\mbox{$-$}} s+1,\,\ldots ,\, s-1,\, +s\} $ (see, e.g., Cohen-Tannoudji et al. Reference Cohen-Tannoudji1977, Chapter 10; or Sakurai Reference Sakurai1995, Chapter 3). Next we consider two particles.

The total spin operator of the composite system is

and its $z$-component is

which all act in $\mathstrut{\Bbb C} ^{2s+1}\otimes \mathstrut{\Bbb C} ^{2s+1}$. The set

is a set of commuting self-adjoint operators. These operators therefore have a common set of orthonormal eigenvectors $\vert s\mathrm{;}\,\, S,\, M\rangle $. Their eigenvector equations are:

One easily shows that S ∊ {0, 1, …, 2s} and M ∊ {−S, −S + 1, …, $S-1$, S}.

We note that every vector $\vert \phi \rangle \in \mathstrut{\Bbb C} ^{2s+1}\otimes \mathstrut{\Bbb C} ^{2s+1}$ has a unique expansion in terms of these orthonormal eigenvectors $\vert s\mathrm{;}\,\, S,\, M\rangle $, because they span this space:

where $\gamma ( M,S) \in \mathstrut{\Bbb C} [ 0,\, 1] $ and their moduli sum to $1$. Since the vectors $\vert s\mathrm{;}\,\, m,\, m^{\prime }\rangle \equiv \vert s,\, m\rangle \otimes \vert s,\, m^{\prime }\rangle $ also form a basis of $\mathstrut{\Bbb C} ^{2s+1}\otimes \mathstrut{\Bbb C} ^{2s+1}$, so that

where $\alpha ( m,\, m^{\prime }\mathrm{;}\,\, s) \in \mathstrut{\Bbb C} [ 0,\, 1] $ and their moduli sum to 1, these two bases can be expanded in each other. The expansion coefficients $\alpha ( m,\, m^{\prime }\mathrm{;}\,\, s) $ of the basis vector $\vert s\mathrm{;}\,\, S,\, M\rangle $ are the well-known ‘Clebsch-Gordon coefficients’. See, for instance, Cohen-Tannoudji Reference Cohen-Tannoudji1977, 1023.

Let us now proceed to prove Theorem 2. Consider the following categorical ‘total spin relation’:

One easily verifies that relation $\mathsf{T}$ (38) meets Req1 and Req2. We now prove that relation $\mathsf{T}$ (38) discerns the two fermions weakly.

Case 1: $\boldsymbol{a}=\boldsymbol{b}$. We then obtain the spin magnitude operator of a single particle, say a :

which extends to arbitrary $\vert \phi \rangle $ by expansion (36):

By WkPP, the composite system then possesses the following quantitative physical property (when substituting 1 or 2 for a ):

This property (41) is a relation between the constituent parts of the system, namely, $\textsf{T}$ (38), and this relation is reflexive: $\mathsf{T}( \boldsymbol{a},\,\boldsymbol{a}) $ for every a due to (40).

Case 2: $\boldsymbol{a}\neq \boldsymbol{b}$. The basis states $\vert s\mathrm{;}\,\, S,\, M\rangle $ are eigenstates (35) of the total spin operator $\hat{S}$ (32):

which does not extend to arbitrary vectors $\vert \phi \rangle $ but only to superpositions of basis vectors having the same value for S, that is, to vectors of the form

Since S is maximally equal to $2s$, the eigenvalue $S( S+1) $ belonging to vector $\vert s\mathrm{;}\,\, S\rangle $ (43) is always smaller than $4s( s+1) =2s( s+1) +2s$, because $s> 0$. Therefore relation $\textsf{T}$ fails for $\boldsymbol{a}\neq \boldsymbol{b}$ for all S:

The composite system does indeed not possess, by SemC (4), the following two quantitative physical properties of the composite system (substitute 1 for a and 2 for b or conversely):

which is expressed by predicate $\textsf{T}$ as a relation between its constituent parts, 1 and 2 , because the system does possess this property according to WkPP:

However, superpositions of basis vectors having a different value for S, such as

where M is −1, 0, or +1, are not eigenstates of the total spin operator (32). Precisely for these states we need to appeal to StrPP, because according to the converse of WkPP this is sufficient to conclude that the composite system does not possess physical property (45), so that also for these states relation $\mathsf{T}( \boldsymbol{a},\,\boldsymbol{b}) $ fails for $\boldsymbol{a}\neq \boldsymbol{b}$. From this fact and the theorem of predicate logic (14), we then conclude that $\textsf{T}$ is symmetric (Req2). Since the operators involved correspond to physical magnitudes, for example, spin, relation $\textsf{T}$ (38) is physically meaningful (Req1) and hence is admissible, because it also meets Req2 ($\textsf{T}$ is reflexive and symmetric).

Therefore total spin relation $\textsf{T}$ (38) discerns the two spin-s particles weakly. Since no probability measures occur in the definiens of $\textsf{T}$; it discerns them also categorically.

[S2]. Case for $N=2$, mixed. The extension from pure to mixed states runs as before, as in step [S2] of the proof of Lemma 1. There is however one subtle point we need to take care of.

Case 1: $\boldsymbol{a}=\boldsymbol{b}$. Rewriting relation $\textsf{T}$ (38) for one-dimensional projectors is easy. Since the spin s of the constituent particles is fixed, the one-dimensional projector that projects on the ray that contains $\vert s\mathrm{;}\,\, S,\, M\rangle $ is an eigenoperator (eigenstate) of $( \hat{S}_{\boldsymbol{a}}+\hat{S}_{\boldsymbol{a}}) ^{2}$ having the same eigenvalue $4s( s+1) $ (39). Consequently, every (convex) sum of one-dimensional projectors that project on vectors with the same value of S has this same eigenvalue and we proceed as before in [S2] of Lemma 1, by an appeal to WkPP and a generalization of $\textsf{T}$ (38) to mixed states:

For (convex) sums of projectors that project on vectors of different value of S, we need StrPP again, as in step [S1] above. Relation $\mathsf{T}( \boldsymbol{a},\,\boldsymbol{a}) $ (48) holds also for mixed states.

Case 2: $\boldsymbol{a}\neq \boldsymbol{b}$. The one-dimensional projector on $\vert s\mathrm{;}\,\, S,\, M\rangle $ now is an eigenoperator (eigenstate) of $( \hat{S}_{a}+\hat{S}_{b}) ^{2}$ having eigenvalue $S( S+1) \hslash ^{2}$ (35). Since $S\leq 2s$ for every S , this eigenvalue is necessarily smaller than $4s( s+1) $ for all S . Then either every convex mixture of the one-dimensional projectors has an eigenvalue smaller than $4s( s+1) $ too, or it is not an eigenstate of $( \hat{S}_{a}+\hat{S}_{b}) ^{2}$ at all (when the mixture consists of projectors on different states $\vert s\mathrm{;}\,\, S,\, M\rangle $ and $\vert s\mathrm{;}\,\, S^{\prime },M^{\prime }\rangle $, $S\neq S^{\prime }$). In virtue of StrPP, relation $\textsf{T}$ (38) then does not hold for its parts (for $\boldsymbol{a}\neq \boldsymbol{b}$), for all states, mixed and pure, because the system does not possess the required physical property.

So $\textsf{T}$ (48) is reflexive and symmetric (Req2) and certainly physically meaningful (Req1). In conclusion, two similar particles in every finite-dimensional are categorically weakly discernible in all admissible states, both pure and mixed.

[S3]. Case for $N> 2$, all states. Consider a subsystem of two particles, say a and b , of the N-particle system. We can consider these two to form a composite system and then repeat the proof we have just given, in [S1] and [S2], to show they are weakly and categorically discernible. When we can discern an arbitrary particle, say a , from every other particle, we have discerned all particles. QED.

We end this section again with a few more systematic remarks.

Remark 1. In our proofs we started with N particles. Is it not circular, then, to prove they are discernible because to assume they are not identical (for if they were, we would have single particle, and not $N> 1$ particles), implies we are somehow tacitly assuming they are discernible? Have we committed the fallacy of propounding a petitio principii?

No we have not. We assume the particles are formally discernible, for example, by their labels, but then demonstrate on the basis of a few postulates of QM that they are physically discernible. Or in other words, we assume the particles are quantitatively not identical and we prove they are qualitatively not identical. Or still in other words, we assume numerical diversity and prove weak qualitative diversity. (See Muller and Saunders Reference Muller and Saunders2008, 541–543, for an elaborate discussion of precisely this issue.)

Remark 2. Of course Theorem 1 implies probabilistic versions. The probability postulate (ProbP) of QM gives the Born probability measure over measurement outcomes for pure states and gives Von Neumann's extension to mixed states, which is the trace formula. By following the strategy of Muller and Saunders (Reference Muller and Saunders2008, 536–537) to carry over categorical proofs to probabilistic proofs, one easily proves the probabilistic weak discernibility of similar particles, notably then without using WkPP and SemC (4).

Remark 3. In contrast to Theorem 1, Theorem 2 relies on StrPP, which arguably is an empirically superfluous postulate. StrPP also leads almost unavoidably to nothing less than the projection postulate (see Muller and Saunders Reference Muller and Saunders2008, 514). Foes of the projection postulate are not committed to Theorem 2. They will find themselves metaphysically in the following situation (provided they accept the whiff of interpretation WkPP): similar elementary particles in infinite-dimensional Hilbert spaces are weakly discernible, in certain classes of states in finite-dimensional Hilbert spaces they are also weakly discernible, fermions in finite-dimensional Hilbert spaces are weakly discernible in all admissible states when there always is a maximal operator of physical significance (see Section 1), but for other classes of states in finite-dimensional Hilbert spaces the jury is still out.

For those who have no objections against StrPP, all similar particles in all kinds of Hilbert spaces in all kinds of states are weakly discernible. This may be seen as an argument in favor of StrPP: it leads to a uniform nature of elementary particles when described quantum mechanically and the proofs make no distinction between fermions and bosons.

Remark 4. The so-called Second Underdetermination Thesis says roughly that the physics underdetermines the metaphysics—the ‘First Underdetermination Thesis’ then is the familiar Duhem-Quine thesis of the underdetermination of theory by all actual or by all possible data; see Muller Reference Muller2009. ‘Naturalistic metaphysics’, as has been recently and vigorously defended by Ladyman and Ross (Reference Ladyman and Ross2007, 1–65), surely follows scientific theory wherever scientific theory leads us, without prejudice, without clinging to so-called common sense, and without tacit adherence to what they call ‘domesticated metaphysics’. Well, QM leads us by means of mathematical proof to the metaphysical statements (if they are metaphysical) that similar elementary particles are categorical (and by implication probabilistic) relationals, more specifically weak discernibles. Those who have held that QM underdetermines the metaphysics in this regard (see references in Section 1), in this case the nature of the elementary particle, are guilty of engaging in unnatural metaphysics (for elaboration, see Muller Reference Muller2009, Section 4).

5. Conclusion: Leibniz Reigns.

We have demonstrated that for every set $\mathstrut{\cal S} _{N}$ of N similar particles, in infinite-dimensional and finite-dimension Hilbert spaces, in all their physical states, pure and mixed, similar particles can be discerned by physically meaningful and permutation-invariant means, and therefore are not physically indiscernible:

where $\mathrm{QM}\,^{-}$ now stands for StateP, WkMP, StrPP and SymP, which is logically the same as having proved PII (5):

and by theorem of logic (1) as having disproved IT. Hence

Therefore all claims to the contrary, that QM refutes PII, or is inconsistent with PII, or that PII cannot be established (see the references in Section 1 for propounders of these claims) find themselves in heavy weather. Quantum mechanical particles are categorical weak discernibles, and therefore not indiscernibles as propounders of IT have claimed. Similar elementary particles are like points on a line, in a plane, or in Euclidean space: absolutely indiscernible yet not identical (there is more than one of them!). Points on a line are categorical relationals, categorical weak discernibles to be precise. Elementary particles are exactly like points in this regard. Leibniz is back from exile and reigns over all quantum mechanically possible worlds, salva veritate.