aa r X i v : . [ qu a n t - ph ] M a y To appear in: philosophy of science
Discerning Elementary Particles
F.A. Muller & M.P. Seevinck Utrecht, 30 April 2009Circa 7,000 words
Abstract.
We extend the quantum-mechanical results of Muller & Saunders (2008)establishing the weak discernibility of an arbitrary number of similar fermions infinite-dimensional Hilbert-spaces in two ways: (a) from fermions to bosons forall finite-dimensional Hilbert-spaces; and (b) from finite-dimensional to infinite-dimensional Hilbert-spaces for all elementary particles. In both cases this is per-formed using operators whose physical significance is beyond doubt. This confutesthe currently dominant view that (A) the quantum-mechanical description of simi-lar particles conflicts with Leibniz’s Principle of the Identity of Indiscernibles ( pii );and that (B) the only way to save pii is by adopting some pre-Kantian meta-physical notion such as Scotusian haecceittas or Adamsian primitive thisness. Wetake sides with Muller & Saunders (2008) against this currently dominant view,which has been expounded and defended by, among others, Schr¨odinger, Margenau,Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Mittelstaedt,Giuntini, Castellani, Krause and Huggett. Faculty of Philosophy, Erasmus University Rotterdam, Burg. Oudlaan 50, H5–16, 3062 PA Rotter-dam, E-mail: [email protected]; and: Institute for the History and Foundations of Science, UtrechtUniversity, Budapestlaan 6, IGG–3.08, 3584 CD Utrecht, The Netherlands E-mail: [email protected] Institute for the History and Foundations of Science, Utrecht University, Budapestlaan 6, IGG–3.08,3584 CD Utrecht, The Netherlands E-mail: [email protected] ontents
Introduction
According to the founding father of wave mechanics Erwin Schr¨odinger (1996: 121–122),one of the ontological lessons that quantum mechanics ( qm ) has taught us is, as he toldan audience in Dublin, February 1950, that the elementary building blocks of the physicalworld are entirely indiscernible : I beg to emphasize this and I beg you to believe it: it is not a question of our beingable 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 trulyhas no meaning.
Similar elementary particles have no ‘identity’, there is nothing that discerns one particlefrom another, neither properties nor relations can tell them apart, they are not individu-als. Thus Schr¨odinger famously compared the elementary particles to “the shillings andpennies in your bank account”, in contrast to the coins in your piggy-bank. In 1928, Her-mann Weyl (1950: 241) had preceded Schr¨odinger when he wrote that “even in principleone cannot demand an alibi from an electron”.Over the past decades, several philosophers have scrutinised this
Indiscernibility Thesis ( it ) by providing various rigorous arguments in favour of it: similar elementary particles(same mass, charge, spin, etc.), when forming a composite physical system, are indis-cernible by quantum-mechanical means. Leibniz’s metaphysical Principle of the Identityof Indiscernibles ( pii ) is thus refuted by physics ( qm ) — and perhaps is therefore not sometaphysical after all. This does not rule out conclusively that particles really are dis-cernible, but if they are, they have to be discerned by means that go above and beyondphysics ( qm ), such as by ascribing Scotusian haecceitas to the particles, or ascribing sib-ling attributes to them from scholastic and neo-scholastic metaphysics. Nevertheless fewphilosophers have considered this move to save the discernibility of the elementary parti-cles to be attractive — if this move is mentioned, then usually as a possibility and rarelyas a plausibility. Mild naturalistic inclinations seem sufficient to accommodate it in ourgeneral metaphysical view of the world. We ought to let well-established scientific knowl-edge inform our metaphysical view of the world whenever possible and appropriate, andthis is exactly what Schr¨odinger begged us to do. The only respectable metaphysics is nat-uralised metaphysics; see further Ladyman & Ross (2007: 1–38). Prominent defenders of it include: Margenau (1944); Cortes (1976), who brandished pii “a false principle”; Bar-nette (1978), French & Redhead (1988); Giuntini & Mittelstaedt (1989), who argued thatalthough demonstrably valid in classical logic, in quantum logic the validity of pii cannotbe established; French (1989a), who assured us in the title that pii “is not contingently1rue either”; French (1989a; 1989b; 1998; 2006), Redhead & Teller (1992), Butterfield(1993), Castellani & Mittelstaedt (2000), Massimi (2001), Teller (1998), French & Rickles(2003), Huggett (2003), French & Krause (2006: Ch. 4).There have however been dissenters. B.C. van Fraassen (1991) is one of them; seeMuller & Saunders (2006: 517–518) for an analysis of his arguments. We follow theother dissenters: S.W. Saunders and one of us (Saunders (2006), Muller & Saunders(2008)). They neither claim that fermions are individuals nor do they rely on a partic-ular interpretation of qm . On the basis of standard mathematics (standard set-theoryand classical predicate logic) and only uncontroversial postulates of qm (notably leavingout the projection postulate, the strong property postulate and the quantum-mechanicalprobabilities), they demonstrate that similar fermions are weakly discernible , i.e. theyare discerned by relations that are irreflexive and symmetric, in every admissible stateof the composite system. So according to Muller & Saunders, the elementary buildingblocks 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 ratherserious restriction because most applications of qm to physical systems employ complexwave-functions and these live in the infinite-dimensional Hilbert-space L ( R N ); nonethe-less they confidently conjecture that their result will hold good for infinite-dimensionalHilbert-spaces as well (their Conjecture 1). Furthermore, Muller & Saunders (2008: 534–535) need to assume for their proof there is a maximal self-adjoint operator acting onfinite-dimensional Hilbert-spaces that is physically significant. In the case of dimension2 of a single-fermion Hilbert-space, Pauli’s spin-1 / probabilistic kind(the discerning relation involves quantum-mechanical probabilities and therefore theirproof needs the Probability Postulate of qm ), whereas the discerning relation of thefermions is of a categorical kind (no probabilities involved). More precisely, the categori-cal discernibility of bosons turns out to be a contingent matter: in some states they arecategorically discernible, in others, e.g. direct-product states, they are not; this preventsone to conclude that bosons are categorical discernibles simpliciter . But the boson’s prob-abilistic discernibility is a quantum-mechanical necessity; Theorem 3 ( ibid. ) establishes itfor two bosons, with no restrictions on the dimensionality of Hilbert-space but conditionalon whether a particular sort of operator can be found (again, a maximal self-adjoint oneof physical significance). The fermions are also probabilistic discernibles; their Theorem 2states it for finite-dimensional Hilbert-spaces only and is therefore equally restrictive as2heir Theorem 1.The central aim of the current paper is the completion of the project initiated anddeveloped in Muller & Saunders (2008), by demonstrating that all restrictions in their discernibility theorems can be removed by proving more general theorems and provingthem differently than they have done, employing only quantum mechanical operatorsthat have obvious physical significance. We shall then be in a position to conclude inutter generality that all kinds of similar particles in all their physical states, pure andmixed, 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 thedeath-knell for it , and, by implication, 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 weintroduce some terminology, state explicitly what we need of qm , and address the issueof what has the license to discern elementary particles (Section 2). For the motivation and further elaboration of the terminology we are about to introduce,we refer to Muller & Saunders (2008: 503–505) because we follow them closely (readersof that paper can jump to the next Section of the current paper). Here we only mentionwhat is necessary in order to keep the current paper comparatively self-contained.We call physical objects in a set absolutely discernible , or individuals , iff for everyobject there is some physical property that it has but all others lack; and relationallydiscernible iff for every object there is some physical relation that discerns it from allothers (see below). An object is indiscernible iff it is both absolutely and relationallyindiscernible, and hence discernible iff it is discernible either way or both ways. Objectsthat 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 (1981: 129–133) was the first to inquire into different kinds of discernibility; hediscovered there are only two independent logical categories of relational discernibility (bymeans of a binary relation): either the relation is irreflexive and asymmetric, in whichcase we speak of relative discernibility ; or the relation is irreflexive and symmetric, inwhich case we speak of weak discernibility . We call attention to the logical fact that ifrelation R discerns particles and relatively , then its complement relation , defined as ¬ R , is also asymmetric but reflexive; and if R discerns particles and weakly , then itscomplement relation ¬ R is reflexive and symmetric but does not hold for a = b whenever3 holds for a = b .Leibniz’s Principle of the Identity of Indiscernibles ( pii ) for physical objects statesthat 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 andthen inquire into the logical relations between these and pii ; see Muller & Saunders (2008:504–505). Similarly one can distinguish three indiscernibility theses as the correspondingnegations of the Leibnizian principles. We restrict ourselves to the
Indiscernibility Thesis ( it ): there are composite systems of similar physical objects that consist of absolutely andrelationally indiscernible physical objects. Then either it is a theorem of logic that pii holds and it fails, or conversely: ⊢ pii ←→ ¬ it . (1)Next we rehearse the postulates of qm that we shall use in our Discernibility Theo-rems.The State Postulate (StateP) associates some super-selected sector Hilbert-space H to every given physical system S and represents every physical state of S by a statisticaloperator W ∈ S ( H ); the pure states lie on the boundary of this convex set S ( H ) of allstatistical operators and the mixed states lie inside. If S consists of N similar elementaryparticles, then the associated Hilbert-space is a direct-product Hilbert-space H N = H ⊗· · · ⊗ H of N identical single-particles Hilbert-spaces.The Weak Magnitude Postulate (WkMP) says that every physical magnitude is rep-resented by an operator that acts on 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 Symmetrisation Postulate, we need to define first the orthogonalprojectors Π ± N of the lattice P ( H N ) of all projectors, defined asΠ + N ≡ N ! N ! X π ∈ P N U π and Π − N ≡ N ! N ! X π ∈ P N sign( π ) U π , (2)where sign( π ) ∈ {± } is the sign of the permutation π ∈ P N on { , , . . . , N } (+1 if itis even, − U π is a unitary operator acting on H N corresponding topermutation π (these U π form a unitary representation on H N of the permutation group P N ). The projectors (2) lead to the following permutation-invariant orthogonal subspaces: H N + ≡ Π + N (cid:2) H N (cid:3) and H N − ≡ Π − N (cid:2) H N (cid:3) , (3)which are called the be - symmetric (Bose-Einstein) and the fd - symmetric (Fermi-Dirac)subspaces of H N , respectively. These subspaces can, alternatively, be seen as generated4y the symmetrised and anti-symmetrised versions of the products of basis-vectors in H N .Only for N = 2, we have that H N − ⊕ H N + = H N .The Symmetrisation Postulate (SymP) states for a composite system of N > N -fold direct-product Hilbert-space H N the following: (i) the projectors Π ± N (2) are super-selection operators; (ii) integer and half-integer spin particles are confined tothe be - and the fd -symmetric subspaces (3), respectively; and (iii) all composite systemsof 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 A mathematically by ordered pair h A, a i , where A is the operator representing A and a ∈ 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 h A, a i ; the StrongProperty Postulate (StrPP) adds the converse conditional to WkPP. (We mention thateigenstates can be mixed , so that physical systems in mixed states can posses propertiestoo (by WkPP). see Muller & Saunders (2008: 513) for details.) WkPP implies that everyphysical 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 thesuper-selected operators. We call these possessed quantitative physical properties super-selected and we call physical systems, e.g. particles, that have the same super-selectedquantitative physical properties similar (this is the precise definition of ‘similar’, a wordthat we have been using loosely until now, following Dirac).We also adopt the following Semantic Condition (SemC). When talking of a physicalsystem at a given time, we ascribe to it at most one quantitative physical propertyassociated with physical magnitude A :(SemC) If physical system S possesses h A, a i and h A, a ′ i , then a = a ′ . (4)For example, particles cannot possess two different masses at the same time and (4)is the generalisation of this in the language of qm . In other words: if S possessesquantitative physical property h A, a i , then S does not posses property h A, a ′ i for every a ′ = a . Statement (4) is neither a tautology nor a theorem of logic, but we agree withMuller & Saunders (2008: 515) in that “it seems absurd to deny it all the same”.Notice there is neither mention of measurements nor of probabilities in the postulatesmentioned above, let alone interpretational glosses such as dispositions.For an outline of the elementary language of qm , we refer again to Muller & Saun-ders (2008: 520–521). In this language, the proper formulation of pii is that physicallyindiscernible physical systems are identical (Muller & Saunders 2008: 521–523): PhysInd ( a , b ) −→ a = b , (5)5here ‘ a ’ and ‘ b ’ are physical-system variables, ranging over all physical systems, andwhere PhysInd ( a , b ) comprises everything that is in principle permitted to discern the par-ticles: roughly, all physical relations and all physical properties. The properties and therelations may involve, in their definition, probabilities, in which case we call them proba-bilistic ; otherwise, in the absence of probabilities, we call them categorical . So the threelogical kinds of discernibility — (a) absolute and (r) relational, which further branchesin (r.w) weak and (r.r) relative discernibility — come in a probabilistic and a categoricalvariety. In their analysis of the traditional arguments in favour of it , Muller & Saunders(2008: 524–526) make the case that, setting conditional probabilities aside, (r) relationaldiscernibility has been largely overlooked by the tradition. (Parenthetically, Leibniz alsoincluded relations in his pii because he held that all relations reduce to properties andthus 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 dowith his formulation; see Muller & Saunders (2008: 504–505).) What in particular hasbeen overlooked, and is employed by Muller & Saunders, are properties of wholes that arerelations between their constitutive parts: the distance between the Sun and the Earthis a property of the solar system; the Coulomb-interaction between the electron and theproton is a property of the Hydrogen atom; etc.Muller & Saunders (2008: 524–528) argue at length that only those properties andrelations are permitted to occur in PhysInd (5) that meet the following two requirements.(Req1)
Physical meaning . All properties and relations, as they occur WkPP, shouldbe transparently defined in terms of physical states and operators that correspondto physical magnitudes in order for the properties and relations to be physicallymeaningful.(Req2)
Permutation invariance . Any property of one particle is a property ofany other; relations should be permutation-invariant, so binary relations should besymmetric and either reflexive or irreflexive.All proponents of the Indiscernibility Thesis ( it ) have considered quantum-mechanicalmeans of discerning similar particles that obey these two Requirements (see the referenceslisted 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 flourishingliterature on indiscernibility and inquire briefly whether this motivates a third require-ment, call it Req3. One easily shows that absolute discernibles are relational discerniblesby defining a relation (expressed by dyadic predicate R M ) in terms of the discerning6roperties (expressed by monadic predicate M ); see Muller & Saunders (2008: 529). Onecould submit that this is not a case of ‘genuine’ but of ‘fake’ relational discernibility, be-cause there is nothing inherently relational about the way this relational discernibility isachieved: R M is completely reducible to property M , which already discerns the particlesabsolutely. Similarly, one may also object that a case of absolute discernibility impliedby relational discernibility by means of a monadic predicate M R that is defined in termsof 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 (2007: 36), who calls ‘fake’ and‘genuine’ more neutrally “contextual” and “intrinsic”, respectively). Definitions: physicalsystems 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 predicatesof which some discern a and b absolutely ; a and b are genuine individuals, or genuine ab-solute discernibles, iff they are discerned by some monadic predicate that is not reducibleto dyadic predicates of which some discern a and b relationally ; discernibles are fake iffthey are not genuine. Hence there is a prima facie case for adding a third Requirementthat excludes fake discernibility:(Req3) Authenticity.
Predicates expressing discerning relations and properties mustbe genuine.In turn, a fake property or relation can be defined rigorously as its undefinabilityin terms of the predicates in the language of qm that meet Req1 and
Req2. In orderto inquire logically into genuineness and fakeness, thus defined, at an appreciable levelof rigour, the entire formal language must be spelled out and all axioms of qm must bespelled out in that formal language. Such a logical inquiry is however far beyond the scopeof the current paper. Nonetheless we shall see that our discerning relations plausibly are genuine.But besides formalise-fobia, there is a respectable reason for not adding Req3 to ourlist. 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 byall 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 presumablyprove by induction over the complexity of the defined predicates. So indiscernibles remainindiscernibles, whether we require the candidate properties and relations to be genuine ornot.( b ) Suppose next that the particles turn out to be discernibles. ( b.i ) If they arediscerned by a relation that turns out to be definable in terms of genuine properties oneof which discerns the particles absolutely, then the relationals become individuals — good7ews for admirers of pii . But the important point to notice is that discernibles remaindiscernibles. ( b.ii ) If the particles are discerned by a property that turns turns out to bedefinable in terms of genuine relations one of which discerns the particles relationally, thenthe individuals loose their individuality and become relationals. They had a fake-identityand are now exposed as metaphysical imposters. But again, the important point to noticeis that discernibles remain discernibles.To conclude, adding Req3 will not have any consequences for crossing the borderbetween discernibles and indiscernibles. This seems a respectable reason not to add Req3to our list of two Requirements. We first prove a Lemma, from which our categorical discernibility theorems then imme-diately follow.
Lemma 1 (StateP, WkMP, WkPP, SemC)
Given a composite physical system of N > similar particles and its associated direct-product Hilbert-space H N . If there aretwo single-particle operators, A and B , acting in single-particle Hilbert-space H , and theycorrespond to physical magnitudes A and B , respectively, and there is a non-zero number c ∈ C such that in every pure state | φ i ∈ H in the domain of their commutator thefollowing holds: [ A, B ] | φ i = c | φ i , (6) then all particles are categorically weakly discernible. Proof.
Let a , b , j be particle-variables, ranging over the set { , , . . . , N } of N particles.We proceed Step-wise, as follows.[S1] Case for N = 2, pure states.[S2] Case for N = 2, mixed states.[S3] Case for N >
2, pure states.[S4] Case for
N >
2, mixed states.[S1].
Case for N = 2 , pure states. Assume the antecedent. Define the following operatorson H = H ⊗ H : A ≡ A ⊗ and A ≡ ⊗ A , (7)8here the operator is the identity-operator on H ; and mutatis mutandis for B . Definenext the following commutator-relation : C ( a , b ) iff ∀ | Ψ i ∈ D : (cid:2) A a , B b (cid:3) | Ψ i = c | Ψ i , (8)where D ⊆ H ⊗ H is the domain of the commutator. An arbitrary vector | Ψ i can beexpanded: | Ψ i = d X j,k =1 γ jk | φ j i ⊗ | φ k i , d X j,k =1 | γ jk | = 1 , (9)where d is a positive integer or ∞ , and {| φ , | φ i , . . . } is a basis for H that lies in thedomain of the commutator [ A, B ]. Then using expansion (9) and eq. (6) one quickly showsthat ( a = b ): (cid:2) A a , B a (cid:3) | Ψ i = c | Ψ i , (10)and that for a = b : (cid:2) A a , B b (cid:3) | Ψ i = 0 | Ψ i 6 = c | Ψ i , (11)because by assumption c = 0. By WkPP, the composite system then possesses thefollowing four quantitative physical properties (when substituting or for a in thefirst, and for a and for b , or conversely, in the second): (cid:10)(cid:2) A a , B a (cid:3) , c (cid:11) and (cid:10)(cid:2) A a , B b (cid:3) , (cid:11) ( a = b ) . (12)In virtue of the Semantic Conditional (4), the composite system then does not possessthe following four quantitative physical properties (recall that c = 0): (cid:10)(cid:2) A a , B a (cid:3) , (cid:11) and (cid:10)(cid:2) A a , B b (cid:3) , c (cid:11) ( a = b ) . (13)The composite system possesses the property (12) that is a relation between its con-stituent parts, namely C (8), which is reflexive: C ( a , a ) for every a due to (10). Similarly,but now using SemC (4), the composite system does not possess the property (13) that is a relation between its constituent parts, namely C . Therefore is not related to ,and is not related to either, because ¬ C ( a , b ) and ¬ C ( b , a ) ( a = b ); and then, dueto the following theorem of logic: ⊢ (cid:0) ¬ C ( a , b ) ∧ ¬ C ( b , a ) (cid:1) −→ (cid:0) C ( a , b ) ←→ C ( b , a ) (cid:1) , (14)we conclude that C is symmetric (Req2). Since by assumption A and B correspond tophysical magnitudes, relation C (8) is physically meaningful (Req1) and hence is admis-sible, because it meets Req1 and Req2. 9urther, it was just shown that the relation C (8) is reflexive and symmetric but failsfor a = b due to (11), which means that C discerns the two particles weakly in every purestate | Ψ i of the composite system. Since probabilities do not occur in C (8), the particlesare discerned categorically .[S2]. Case for N = 2 , mixed states. The equations in (8) can also be written as anequation for 1-dimensional projectors that project onto the ray that contains | Ψ i : (cid:2) A a , B b (cid:3) | Ψ ih Ψ | = c | Ψ ih Ψ | . (15)Due to the linearity of the operators, this equation remains valid for arbitrary linearcombinations of projectors. This includes all convex combinations of projectors, whichexhausts the set S ( H ⊗ H ) of all mixed states. The commutator-relation C (8) is easilyextended to mixed states W ∈ S ( H ⊗ H ) and the ensuing relation also discerns theparticles categorically and weakly.[S3], [S4].
Case for
N > , pure and mixed states. Cases [S1] and [S2] are immediatelyextended to the N -particle cases, by considering the following N -factor operators: A j ≡ ⊗ · · · ⊗ ⊗ A ⊗ ⊗ · · · ⊗ , (16)where A is the j -th 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]. Q.e.d.Theorem 1 (StateP, WkMP, WkPP, SemC)
In a composite physical system of afinite number of similar particles, all particles are categorically weakly discernible in everyphysical state, pure and mixed, for every infinite-dimensional Hilbert-space.
Proof.
In Lemma 1, choose for A the linear momentum operator b P , for B the Cartesianposition-operator b Q , and for c the value − i ℏ . The physical significance of these operatorsand their commutator, which is the celebrated canonical commutator (cid:2) b P , b Q (cid:3) = − i ℏ , (17)is beyond doubt and so is the ensuing commutator relation C (8), which we baptise theHeisenberg-relation . The operators b P and b Q act on the infinite-dimensional Hilbert-spaceof the complex wave-functions L ( R ), which is isomorphic to every infinite-dimensional Hilbert-space.
Q.e.d.
But is Theorem 1 not only applicable to particles having spin-0 and have we forgottento mention this? Yes and No. Yes, we have deliberately forgotten to mention this. No, itis a corollary of Theorem 1 that it holds for all spin-magnitudes, which is the content ofthe next theorem. 10 orollary 1 (StateP, WkMP, WkPP, SemC)
In a composite physical system of N > similar particles of arbitrary spin, all particles are categorically weakly discerniblein every admissible physical state, pure and mixed, for every infinite-dimensional Hilbert-space. Proof.
To deal with spin, we need SymP. The actual proof of the categorical weakdiscernibility for all particles having non-zero spin-magnitude is at bottom a notationalvariant of the proof of Theorem 1. Let us sketch how this works for N = 2. We beginwith the following Hilbert-space for a single particle: H s ≡ (cid:0) L ( R ) (cid:1) s +1 , (18)which is the space of spinorial wave-functions Ψ, i.e. column-vectors of 2 s + 1-entries,each entry being a complex wave-function of L ( R ). Notice that H s is an (2 s + 1)-fold Cartesian-product set, which becomes a Hilbert-space by carrying the Hilbert-spaceproperties of L ( R ) over to H s . For instance, the inner-product on H s is just the sum ofthe inner-products of the components of the spinors: h Ψ | Φ i = s +1 X k =1 h Ψ k | Φ k i , (19)where Ψ k is the k -th entry of Ψ (and similarly for Φ k ), which provides the norm of H s ,which in turn generates the norm-topology of H s , etc. The degenerate case of the spinorhaving only one entry is the case of s = 0, which we treated in Theorem 1. So we proceedhere with s >
0. In particular, the number of entries 2 s + 1 is even iff the particles havehalf-integer spin, and odd iff the particles have integer spin, in units of ℏ .Let e k ∈ C s +1 be such that its k -th entry is 1 and all others 0. They form the standardbasis for C s +1 and are the eigenvectors of the z -component of the spin-operator b S z ,whose eigenvalues are traditionally denoted by m (in the terminology of atomic physics:‘magnetic quantum number’): b S z e k = m k e k , (20)where m = − s , m = − s + 1, . . . , m s − = s − m s = + s . Let φ , φ , . . . be a basis for L ( R ); then this is a basis for single-particle spinor space H s (18): (cid:8) e k φ m ∈ H s | k ∈ { , , . . . , s + 1 } , m ∈ N + (cid:9) . (21)Recall that the linear momentum-operator b P and the Cartesian position operator b Q onan arbitrary complex wave-function φ ∈ L ( R ) are the differential-operator times − i ℏ and the multiplication-operator, respectively: b P : D P → L ( R ) , φ b P φ, where (cid:0) b
P φ (cid:1) ( q ) ≡ − i ℏ ∂φ ( q ) ∂ q (22)11nd b Q : D Q → L ( R ) , φ b Qφ, where (cid:0) b Qφ (cid:1) ( q ) ≡ ( q x + q y + q z ) φ ( q ) , (23)where domain D P = C ( R ) ∩ L ( R ) and domain D Q ⊂ L ( R ) consist of all wave-functions ψ such that | q | ψ ( q ) → | q | → ∞ . The action of b P and b Q is straight-forwardly extended to arbitrary spinorial wave-functions by letting the operators actcomponent-wise on the 2 s + 1 components. The canonical commutator of b P and b Q (17)then carries over to spinor space H (18). We can now appeal to the general Lemma 1and conclude that the two arbitrary spin-particles are categorically and weakly discernible. Q.e.d.
Another possibility to finish the proof is more-or-less to repeat the proof of Lemma 1but now with spinorial wave-functions. For step [S1], the case N = 2, the state space ofthe composite system becomes: H s ≡ (cid:0) L ( R ) (cid:1) s +1 ⊗ (cid:0) L ( R ) (cid:1) s +1 , (24)where the spinorial wave-functions now have (2 s + 1) entries — (2 s + 1) N for N spin- s particles. A basis for H s (24) is (cid:8) e k φ m ⊗ e j φ l ∈ H s | k, j ∈ { , , . . . , s + 1 } , m, l ∈ N + (cid:9) . (25)An arbitrary spinorial wave-function Ψ ∈ H s of the composite system can then be ex-panded as follows:Ψ( q , q ) = s +1 X k,j =1 ∞ X m,l =1 γ ml (2 s + 1) e k φ m ( q ) ⊗ e j φ l ( q ) , (26)where the γ ml form a squarely-summable sequence, i.e. a Hilbert-vector in ℓ ( N ), of norm 1.With the usual definitions, b P ≡ b P ⊗ , b P ≡ ⊗ b P , b Q ≡ b Q ⊗ , b Q ≡ ⊗ b Q , (27)one obtains all the relevant commutators on H s by using expansion (26). The discerningrelation (8) on the direct-product spinor-space H s then becomes C ( a , b ) iff ∀ Ψ ∈ D : (cid:2) b P a , b Q b (cid:3) Ψ = − i ℏ Ψ , (28)where D ⊂ H s is the domain of the commutator. Etc.We close this Section with a number of systematic remarks.12 emark 1 . Notice that in contrast to the proof of Theorem 1, the proof Corollary 1relies, besides on StateP, WkMP, WkPP and SemC, on the Symmetrisation Postulate(SymP) only in so far as that without this postulate the distinction between integer andhalf-integer spin particles makes little sense and, more importantly, the tacit claim thatthis distinction exhausts all possible composite systems of similar particles is unfounded.Besides this, SymP does not perform any deductive labour in the proof. Specifically, thedistinction between Bose-Einstein and Fermi-Dirac states never enters the proof, whichmeans 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 non-commutativityof the physical magnitudes, which is one of the algebraic hall-marks of quantum physics.Good thing. The physical meaning of relation C (28) can be understood as follows:momentum and position pertain to two particles differently from how they pertain toa single particle. Admittedly this is something we already knew for a long time, sincethe advent of qm . What we didn’t know, but do know now, is that this old knowledgeprovides the ground for discerning similar particles weakly and categorically. Remark 3 . The spinorial wave-function Ψ must lie in the domain D of the commutatorof the unbounded operators b P and b Q , which domain is a proper subspace of H s so thatthe members of H s \D fall outside the scope of relation C (8). The domain D does howeverlies dense in H s , even the domain of all polynomials of b P and b Q does so (the non-Abelianring on D they generate) — D is the Schwarz-space of all complex wave-functions thatare continuously differentiable and fall off exponentially. This means that every wave-function that does not lie in Schwarz-space can be approximated with arbitrary accuracyby means of wave-functions that do lie in Schwarz-space. This is apparently good enoughfor physics. Then it is good enough for us too. Remark 4 . A special case of Theorem 1 is that two bosons in symmetric direct-productstates, sayΨ( q , q ) = φ ( q ) φ ( q ) , (29)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 muchmetaphysical 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 oneaccepts those postulates, one should accept every consequence of those postulates. Thisis part of what it means to accept deductive logic , which we do accept. We admit that13he discernibility of two bosons in state (29) is an unexpected if not bizarre consequence.But in comparison to other bizarre consequences of qm , such as inexplicable correlationsat a distance ( epr ), animate beings that are neither dead nor alive (Schr¨odinger’s im-mortal cat), kettles of water on a seething fire that will never boil (quantum Zeno), ananthropocentric and intentional concept taken as primitive (measurement), states of mat-ter defying familiar states of aggregation ( be -condensate), in comparison to all that, theweak discernibility of bosons in direct-product states is not such a hard nut to swallow.Get real, it’s peanuts. Remark 5 . Every ‘realistic’ quantum-mechanical model of a physical system, whetherin atomic physics, nuclear physics or solid-state physics, employs wave-functions. Thismeans that now , and only now , we can conclude that the similar elementary particles ofthe real world are categorically and weakly discernible. Conjecture 1 of Muller & Saunders(2008: 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 informationtheory. There one chooses to pay attention to spin-degrees of freedom only and ignoresall others — position, linear momentum, energy. This is not to deny there are physicalmagnitudes such as position, momentum or energy, or that these physical magnitudes donot apply in the quantum-information-theoretic models. Of course not. Ignoring thesephysical magnitudes is a matter of expediency if one is not interested in them. Idealisationand approximation are part and parcel of science. No one would deny that quantum-mechanical models using spinorial wave-functions in infinite-dimensional Hilbert-spacematch 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 forfinite-dimensional Hilbert-spaces. In the case of finite-dimensional Hilbert-spaces, considering C d suffices, because every d -dimensional Hilbert-space is isomorphic to C d ( d ∈ N + ). The proof is a vast generalisationof the total-spin relation T of Muller & Saunders (2008: 535). Theorem 2 (StateP, WkMP, StrPP, SymP)
In a composite physical system of N > similar particles, all particles are categorically weakly discernible in every physicalstate, pure and mixed, for every finite-dimensional Hilbert-space by only using their spindegrees of freedom. roof. Let a , b , j be particle-variables, ranging over the set { , , . . . , N } of N particles.We proceed again Step-wise, as follows.[S1] Case for N = 2, pure states.[S2] Case for N = 2, mixed states.[S3] Case for N >
2, all states.[S1].
Case for N = 2 , pure states. We begin by considering two similar particles, labeled , , of spin-magnitude s ℏ , which is a positive integer or a half-integer; a and b are againvariables over this set. The single particle Hilbert-space is C s +1 , which is isomorphic toevery ( s N -particles the associated Hilbert-space isthe N -fold ⊗ -product of C s +1 . According to SymP, when we have considered integer andhalf-integer spin particles, we have considered all particles.We begin by considering the spin-operator of a single particle acting in C s +1 : b S = b S x + b S y + b S z , (30)where b S x , b S y and b S z are the three spin-operators along the three perpendicular spatialdirections ( x, y, z ). The operators b S and S z are self-adjoint and commute and thereforehave a common set of orthonormal eigenvectors | s, m i ; their eigenvector-equations are: b S | s, m i = s ( s + 1) ℏ | s, m i and b S z | s, m i = m ℏ | s, m i , (31)where eigenvalue m ∈ {− s, − s + 1 , . . . , s − , + s } (see e.g. Cohen-Tannoudji et al. (1977:Ch. X) or Sakurai (1995: Ch. 3). Next we consider two particles.The total spin operator of the composite system is b S ≡ b S + b S , where b S ≡ b S ⊗ , b S ≡ ⊗ b S , (32)and its z -component is b S z = b S z ⊗ + ⊗ b S z , (33)which all act in C s +1 ⊗ C s +1 . The set (cid:8)b S , b S , b S , b S z (cid:9) (34)is a set of commuting self-adjoint operators. These operators therefore have a commonset of orthonormal eigenvectors | s ; S, M i . Their eigenvector equations are: b S | s ; S, M i = s ( s + 1) ℏ | s ; S, M i , b S | s ; S, M i = s ( s + 1) ℏ | s ; S, M i , b S | s ; S, M i = S ( S + 1) ℏ | s ; S, M i , b S z | s ; S, M i = M ℏ | s ; S, M i . (35)15ne easily shows that S ∈ { , , . . . , s } and M ∈ {− S, − S + 1 , . . . , S − , S } .We note that every vector | φ i ∈ C s +1 ⊗ C s +1 has a unique expansion in terms ofthese orthonormal eigenvectors | s ; S, M i , because they span this space: | φ i = s X S =0 + S X M = − S γ ( M, S ) | s ; S, M i , (36)where γ ( M, S ) ∈ C [0 ,
1] and their moduli sum to 1. Since the vectors | s ; m, m ′ i ≡| s, m i ⊗ | s, m ′ i also form a basis of C s +1 ⊗ C s +1 , so that | φ i = + s X m = − s + s X m ′ = − s α ( m, m ′ ; s ) | s ; m, m ′ i , (37)where α ( m, m ′ ; s ) ∈ C [0 ,
1] and their moduli sum to 1, these two bases can be expandedin each other. The expansion-coefficients α ( m, m ′ ; s ) of the basis-vector | s ; S, M i are thewell-known ‘Clebsch-Gordon coefficients’. See for instance Cohen-Tannoudji (1977: 1023).Let us now proceed to prove Theorem 2. Consider the following categorical ‘Total-spinrelation’: T ( a , b ) iff ∀ | φ i ∈ C s +1 ⊗ C s +1 : (cid:0)b S a + b S b (cid:1) | φ i = 4 s ( s + 1) ℏ | φ i . (38)One easily verifies that relation T (38) meets Req1 and Req2.We now prove that relation T (38) discerns the two fermions weakly. Case 1 : a = b . We then obtain the spin-magnitude operator of a single particle, say a : (cid:0)b S a + b S a (cid:1) | s ; S, M i = 4 b S a | s ; S, M i = 4 s ( s + 1) ℏ | s ; S, M i , (39)which extends to arbitrary | φ i by expansion (36): (cid:0)b S a + b S a (cid:1) | φ i = 4 s ( s + 1) ℏ | φ i . (40)By WkPP, the composite system then possesses the following quantitative physicalproperty (when substituting or for a ): (cid:10) b S a , s ( s + 1) ℏ (cid:11) . (41)This property (41) is a relation between the constituent parts of the system, namely T (38), and this relation is reflexive: T ( a , a ) for every a due to (40). Case 2 : a = b . The basis states | s ; S, M i are eigenstates (35) of the total spin-operator b S (32): (cid:0)b S a + b S b (cid:1) | s ; S, M i = S ( S + 1) ℏ | s ; S, M i , (42)16hich does not extend to arbitrary vectors | φ i but only to superpositions of basis-vectorshaving the same value for S , that is, to vectors of the form: | s ; S i = + S X M = − S γ ( M, S ) | s ; S, M i . (43)Since S is maximally equal to 2 s , the eigenvalue S ( S + 1) belonging to vector | s ; S i (43)is always smaller than 4 s ( s + 1) = 2 s ( s + 1) + 2 s , because s >
0. Therefore relation T (38) fails for a = b for all S : (cid:0)b S a + b S b (cid:1) | s ; S, M i 6 = s ( s + 1) ℏ | s ; S, M i . (44)The composite system does indeed not possess, by SemC (4), the following two quan-titative physical properties of the composite system (substitute for a and for b orconversely): (cid:10)(cid:0)b S a + b S b (cid:1) , s ( s + 1) ℏ (cid:11) , (45)which is expressed by predicate T as a relation between its constituent parts, and ,because the system does possess this property according to WkPP: (cid:10)(cid:0)b S a + b S b (cid:1) , S ( S + 1) ℏ (cid:11) . (46)However, superpositions of basis-vectors having a different value for S , such as √ (cid:0) | s ; 0 , i + | s ; 1 , M i (cid:1) , (47)where M is −
1, 0 or +1, are not eigenstates of the total spin-operator (32). Preciselyfor these states we need to appeal to StrPP, because according to the converse of WkPPthis is sufficient to conclude that the composite system does not possess physical property(45), so that also for these states relation T ( a , b ) fails for a = b . From this fact andthe theorem of predicate logic (14), we then conclude that T is symmetric (Req2). Sincethe operators involved correspond to physical magnitudes, e.g. spin, relation T (38) isphysically meaningful (Req1) and hence is admissible, because it also meets Req2 ( T isreflexive and symmetric).Therefore total-spin-relation T (38) discerns the two spin- s particles weakly. Since noprobability measures occur in the definiens of 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 totake care of. 17 ase 1 : a = b . Rewriting relation T (38) for 1-dimensional projectors is easy. Sincethe spin s of the constituent particles is fixed, the 1-dimensional projector that projectson the ray that contains | s ; S, M i is an eigenoperator (eigenstate) of ( b S a + b S a ) havingthe same eigenvalue 4 s ( s + 1) (39). Consequently, every (convex) sum of 1-dimensionalprojectors that project on vectors with the same value of S has this same eigenvalue andwe proceed as before in [S2] of Lemma 1, by an appeal to WkPP and a generalisation of T (38) to mixed states: T ( a , b ) iff ∀ W ∈ S ( C s +1 ⊗ C s +1 ) : (cid:0)b S a + b S b (cid:1) W = 4 s ( s + 1) ℏ W . (48)For (convex) sums of projectors that project on vectors of different value of S , we needStrPP again, as in step [S1] above. Relation T ( a , a ) (48) holds also for mixed states. Case 2 : a = b . The 1-dimensional projector on | s ; S, M i now is an eigenoperator(eigenstate) of (cid:0)b S a + b S b (cid:1) having eigenvalue S ( S + 1) ℏ (35). Since S s for every S ,this eigenvalue is necessarily smaller than 4 s ( s + 1) for all S . Then either every convexmixture of the 1-dimensional projectors has an eigenvalue smaller than 4 s ( s + 1) too, orit is not an eigenstate of (cid:0)b S a + b S b (cid:1) at all (when the mixture consists of projectors ondifferent states | s ; S, M i and | s ; S ′ , M ′ i , S = S ′ ). In virtue of StrPP, relation T (38) thendoes not hold for its parts (for a = b ), for all states, mixed and pure, because the systemdoes not possess the required physical property.So T (48) is reflexive and symmetric (Req2) and certainly physically meaningful(Req1). In conclusion two similar particles in in every finite-dimensional are categori-cally weakly discernible in all admissible states, both pure and mixed.[S3]. Case for
N > , 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 andthen repeat the proof we have just given, in [S1] and [S2], to show they are weakly andcategorically discernible. When we can discern an arbitrary particle, say a , from everyother particle, we have discerned all particles. Q.e.d.
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 provethey are discernible because to assume they are not identical (for if they were, we wouldhave single particle, and not N > 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, e.g. 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-identicaland we prove they are qualitatively not-identical. Or still in other words, we assume18umerical diversity and prove weak qualitative diversity. See further Muller & Saunders(2008: 541–543) for an elaborate discussion of precisely this issue. Remark 2 . Of course Theorem 1 implies probabilistic versions. The
Probability Pos-tulate (ProbP) of qm gives the Born probability measure over measurement outcomesfor pure states and gives Von Neumann’s extension to mixed states, which is the trace-formula. By following the strategy of Muller & Saunders (2008: 536–537) to carry overcategorical proofs to probabilistic proofs, one easily proves the probabilistic weak discerni-bility 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 anempirically superfluous postulate. StrPP also leads almost unavoidably to nothing lessthan the Projection Postulate (see Muller & Saunders 2008: 514). Foes of the ProjectionPostulate are not committed to Theorem 2. They will find themselves metaphysically inthe following situation (provided they accept the whiff of interpretation WkPP): similarelementary particles in infinite-dimensional Hilbert-spaces are weakly discernible, in cer-tain 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 stateswhen there always is a maximal operator of physical significance (see Introduction), butfor 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 ofHilbert-spaces in all kinds of states are weakly discernible. This may be seen as anargument in favour of StrPP: it leads to a uniform nature of elementary particles whendescribed quantum-mechanically and the proofs make no distinction between fermionsand bosons.
Remark 4 . The so-called
Second Underdetermination Thesis says roughly that thephysics underdetermines the metaphysics — the
First Underdetermination Thesis thenis the familiar Duhem-Quine thesis of the underdetermination of theory by all actualor by all possible data; see Muller (2009).
Naturalistic metaphysics , as recently hasbeen vigorously defended by Ladyman & Ross (2007: 1–65), surely follows scientifictheory wherever scientific theory leads us, without prejudice, without clinging to so-calledcommon 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 theyare metaphysical) that similar elementary particles are categorical (and by implicationprobabilistic) relationals, more specifically weak discernibles. Those who have held that qm underdetermines the metaphysics in this regard (see references in the Introduction),in this case the nature of the elementary particle, are guilty of engaging in unnaturalmetaphysics (for elaboration, see Muller (2009: Section 4)).19 Conclusion: Leibniz Reigns
We have demonstrated that for every set S N of N similar particles, in infinite-dimensionaland finite-dimension Hilbert-spaces, in all their physical states, pure and mixed, similarparticles can be discerned by physically meaningful and permutation-invariant means,and therefore are not physically indiscernible:QM − ⊢ ∀ N ∈ { , , . . . } , ∀ a , b ∈ S N : a = b −→ ¬ PhysInd ( a , b ) , (49)where QM − now stands for StP, WkMP, StrPP and SymP, which is logically the same ashaving proved pii (5):QM − ⊢ ∀ N ∈ { , , . . . } , ∀ a , b ∈ S N : PhysInd ( a , b ) −→ a = b , (50)and by theorem of logic (1) as having disproved it . HenceQM − ⊢ pii ∧ ¬ it . (51)Therefore all claims to the contrary, that qm refutes pii , or is inconsistent with pii , orthat pii cannot be established (see the references in Section 1 for propounders of theseclaims) find themselves in heavy weather. Quantum-mechanical particles are categoricalweak 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 aline are categorical relationals, categorical weak discernibles to be precise. Elementaryparticles are exactly like points in this regard.Leibniz is back from exile and reigns over all quantum-mechanically possible worlds, salva veritate . 20 eferences Barnette, R.L. (1978), ‘Does Quantum Mechanics Disprove the Principle of the Identity of In-discernibles?’,
Philosophy of Science (1978) 466–470.Brading, K., Castellani, E. (2003), Symmetries in Physics: New Reflections , Cambridge: Cam-bridge University Press, 2003.Butterfield, J.N. (1993), ‘Interpretation and Identity in Quantum Theory’,
Studies in Historyand Philosophy of Science (1993) 443–476.Castellani, E. (1998), Interpreting Bodies. Classical and Quantum Objects in Modern Physics ,Princeton, New Jersey: Princeton University Press, 1998.Castellani, E., Mittelstaedt, P. (1998), ‘Leibniz’s Principle, Physics and the Language of Physics’,
Foundations of Physics , (2000) 1587–1604.Cohen-Tannoudji, C. et al. (1977), Quantum Mechanics , Volume II, New York: John Wiley &Sons, 1977.Cortes, A. (1976), ‘Leibniz’s Principle of the Identity of Indiscernibles: A False Principle’,
Phi-losophy of Science (1976) 491–505.Dalla Chiara, M.L., Toraldo di Francia, G. (1993), ‘Individuals, Kinds and Names in Physics’,in Corsi, G., et al. (eds.), Bridging the Gap: Philosophy, Mathematics, Physics , Dordrecht:Kluwer Academic Publishers, pp. 261–283.Dalla Chiara, M.L., Giuntini, R., Krause, D. (1998), ‘Quasiset Theories for Micro-objects: AComparision’, in Castellani (1998, 142–152).Fraassen, B.C. van. (1991),
Quantum Mechanics. An Empiricist View , Oxford: Clarendon Press,1991.French, S., Redhead, M.L.G. (1988), ‘Quantum Physics and the Identity of Indiscernibles,
BritishJournal for the Philosophy of Science (1988) 233–246.French, S. (1989a), ‘Why the Principle of the Identity of Indiscernibles is not contingently TrueEither’, Synthese (1989) 141–166.French, S. (1989b), ‘Identity and Indiscernibility in Classical and Quantum Physics’, Aus-tralasian Journal of Philosophy (1989) 432–446.French, S. (1998), ‘On the Withering Away of Physical Objects’, in Castellani (1998, 93–113).French, S., Rickles, D. (2003), ‘Understanding Permutation Symmetry’, in Brading & Castellani(2003, 212–238). rench, S. (2006), ‘Identity and Individuality in Quantum Theory’, Stanford Encyclopedia ofPhilosophy , E.N. Zalta (ed.), url = h http://plato.stanford.edu/archives/spr2006/entries/qt-idind/ i French, S., Krause, D. (2006),
Identity in Physics: A Historical, Philosophical and Formal Anal-ysis , Oxford: Clarendon Press, 2006.Huggett, N. (2003), ‘Quarticles and the Identity of Indiscernibles’, in Brading & Castellani(2003, pp. 239–249).Krause, D. (1992), ‘On a Quasi-set Theory’,
Notre Dame Journal of Formal Logic (1992)402–411.Ladyman, J. (2007), ‘On the Identity and Diversity of Objects in a Structure’, Proceedings ofthe Aristotelian Society , Supplementary Volume (2007) 23–43.Ladyman, J., Ross, D. (2007), Every Thing Must Go. Metaphysics Naturalized , Oxford: OxfordUniversity Press, 2007.Margenau, H. (1944), ‘The Exclusion Principle and its Philosophical Importance’,
Philosophyof Science (1944) 187–208.Massimi, M. (2001), ‘The Exclusion Principle and the Identity of Indiscernibles: a Response toMargenau’s Argument’, British Journal for the Philosophy of Science (2001) 303–330.Muller, F.A. (2009), ‘Withering Away, Weakly’, to appear in Synthese .Muller, F.A., Saunders, S.W. (2008), ‘Discerning Fermions’,
British Journal for the Philosophyof Science (2008) 499–548.Redhead, M.L.G., Teller, P. (1992), ‘Quantum Physics and the Identity of Indiscernibles’, BritishJournal for the Philosophy of Science (1992) 201–218.Sakurai, J.J. (1995). Modern Quantum Mechanics , Revised Edition, New York: Addison WesleyPublishing Company.Saunders, S. (2006), ‘Are quantum particles objects?’,
Analysis (2006) 52–63.Teller, P. (1998), ‘Quantum Mechanics and Haecceities’, in Castellani (1991, 114–141).Weyl, H. (1931),
The Theory of Groups and Quantum Mechanics , London: Methuen & Com-pany, 1931., London: Methuen & Com-pany, 1931.