Discerning “indistinguishable” quantum systems Adam Caulton∗ Abstract In a series of recent papers, Simon Saunders, Fred Muller and Michael Seevinck have collectively argued, against the folklore, that some non-trivial version of Leibniz’s principle of the identity of indiscernibles is upheld in quantum mechanics. They argue that all particles—fermions, paraparti- cles, anyons, even bosons—may be weakly discerned by some physical re- lation. Here I show that their arguments make illegitimate appeal to non- symmetric, i.e. permutation-non-invariant, quantities, and that therefore their conclusions do not go through. However, I show that alternative, sym- metric quantities may be found to do the required work. I conclude that the Saunders-Muller-Seevinck heterodoxy can be saved after all. Contents 1 Introduction 2 1.1 Getting clear on Leibniz’s Principle . . . . . . . . . . . . . . . . . . 2 1.2 The folklore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 A new folklore? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Muller and Saunders on discernment 6 2.1 The Muller-Saunders result . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Commentary on the Muller-Saunders proof . . . . . . . . . . . . . . 8 ∗c/o The Faculty of Philosophy, University of Cambridge, Sidgwick Avenue, Cambridge, UK, CB3 9DA. Email: aepw2@cam.ac.uk 1 3 Muller and Seevinck on discernment 10 3.1 The Muller-Seevinck result . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Commentary on the Muller-Seevinck result . . . . . . . . . . . . . . 12 4 A better way to discern particles 14 4.1 The basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 The variance operator . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Variance provides a discerning relation . . . . . . . . . . . . . . . . 16 4.4 Discernment for all two-particle states . . . . . . . . . . . . . . . . 18 4.5 Discernment for all many-particle states . . . . . . . . . . . . . . . 22 5 Conclusion 24 1 Introduction 1.1 Getting clear on Leibniz’s Principle What is the fate of Leibniz’s Principle of the Identity of Indiscernibles for quantum mechanics? It depends, of course, on how the Principle is translated into modern (enough) parlance for the evaluation to be made. Modern logic provides a frame- work in which some natural regimentations may be articulated, which, even if they would not have been of interest to Leibniz’s original project, are nevertheless worthy of investigation in their own right. One informal gloss of the Principle is that no two objects share all the same properties. Grant that we may regiment by taking ‘object’ in the Fregean-Quinean sense of an occupant of the first-order domain. Then what might count as a prop- erty? If, for each first-order model, we universally quantify over the interpretations afforded in that model to the distinguished predicates, then the Principle is con- tingent. That is, in some models the non-logical vocabulary is expressive enough to define identity; in others it is not. But this regimentation might be thought to make a metaphysical principle too much a hostage to the fortunes of language. Why only quantify over the properties for which we have the corresponding pred- icates? An alternative is to regiment the Principle so that, in each model, one gen- eralises over all subsets of the first-order domain. The result is a (second-order) logical truth, since for any object a there is its singleton {a}. But here I see at 2 least two objections. First, sets are not properties. But no worries: for any subset of the domain, there is at least one property to which it corresponds—namely, the property of belonging to that subset. Thus generalisation over subsets may be taken as covert generalisation over these properties at least, and discernibility by these properties entails discernibility simpliciter. (All this, of course, only so long as there are sets.) The second, more serious, objection is that the singleton sets one discerns by are precisely as discernible as the objects that are their unique members. In what sense, then, is it an achievement to have discerned those ob- jects with those sets? In other words: when the properties one quantifies over correspond to the subsets of the first-order domain, the Principle becomes trivial. That should come as no surprise—as I said, it is a logical truth—but it cannot be the regimentation of Leibniz’s Principle that we are looking for. The solution—or, at least, the solution I favour for the purposes of this paper— is to retreat to generalising over the interpretations assigned to the distinguished predicates, but suitably to relativise the Principle to the appropriate collection of properties and relations: namely, those for which the distinguished predicates stand. The Principle now reads: “No two objects share all the same properties expressible in the language.” One may then, if it is so desired, recover an absolute version of the Principle by ensuring that each property or relation taken to exist has its corresponding predicate in the non-logical vocabulary—assuming such a thing possible. With this regimentation, the Principle may be given an explicit first-order log- ical form. For each predicate, one forms a biconditional expressing co-satisfaction for x and y. If the predicate is 2-place or more, one universally generalises over the other argument places, and makes sure that there is a bi-conditional for each argument position of the predicate. The conjunction of all such bi-conditionals is then asserted to be co-extensional with ‘x = y’, and we have a (putative) ex- plicit definition of identity. The result is the Hilbert-Bernays (1934) axiom, made famous by Quine (e.g. 1960) and revived by Saunders (2003a, 2003b, 2006): ∀x∀y { x = y ≡ [ . . . ∧ (Fix ≡ Fiy) ∧ . . . . . . ∧ ∀z ((Gjxz ≡ Gjyz) ∧ (Gjzx ≡ Gjzy)) ∧ . . . . . . ∧ ∀z∀w ( (Hkxzw ≡ Hkyzw) ∧ (Hkzxw ≡ Hkzyw) ∧ (Hkzwx ≡ Hkzwy) ) ∧ . . . ]} (HB) Of course, one must assume that there are finitely many properties and relations, unless one cares to appeal to infinitary languages or some form of parameterization 3 (cf. Caulton and Butterfield 2011, §2.1). The question whether the Principle is true in any theory can now be made precise. If one takes a theory to be a set T of sentences, the question is: Does T logically entail (HB)? If one takes a theory to be a set M of models, the question is: Is (HB) true in every model in M? 1.2 The folklore Let T = quantum mechanics, or M = the models of quantum mechanics. Is (HB) a logical consequence of T , or true in all models in M—at least when the first- order variables are restricted to quantum particles? Until about eight years ago, the folklore has been that quantum particles cannot be discerned, so that Leibniz’s Principle fails. To explain this in more detail, it will be clearest to start with an even earlier folklore, inherited from the founders of quantum mechanics. This folklore has it that Pauli’s exclusion principle for fermions—or better: symmetrization for bosons and anti-symmetrization for fermions—means that: (a) bosons can be in the same state; but (b) fermions cannot be; so that (c) Leibniz’s Principle holds for fermions but not bosons. (For an expression of these three views, see e.g. Weyl 1928, 241.) In fact, these claims can and should be questioned. Under scrutiny, and certain interpretative assumptions, each of (a) to (c) fail, and it seems that Leibniz’s Principle is pan- demically false in quantum mechanics. For first: under the standard interpretation of the formalism,1 any two bosons or any two fermions of the same species are absolutely indiscernible, in the sense that no quantity (“observable”) exists which can discern them.2 For any assembly of fermions or bosons, and any state of that assembly (appropriately (anti-) sym- metrized), and any two particles in the assembly, the two particles’ probabilities for all single-particle quantities are equal; and so are appropriate corresponding two- particle conditional probabilities, including probabilities using conditions about a third constituent. In more technical language: according to the usual procedure of extracting the reduced density operator of a particle by tracing out the states 1I will not question this widespread interpretation of the formalism here, although, like Ear- man (ms.) and Dieks and Lubberdink (2011), I am greatly suspicious of it. 2For a discussion of absolute discernibility, see e.g. Saunders (2003b), Muller and Saunders (2008), and Caulton and Butterfield (2011). 4 for all the other particles in the assembly, we obtain the result that for all (anti-) symmetrized states of the assembly, one obtains equal reduced density operators for every particle. (Margenau 1944; French & Redhead 1988; Butterfield 1993; Huggett 1999; 2003; Massimi 2001; French & Krause 2006, 150-73.) Thus not only can fermions ‘be in the same state’, just as much as bosons can be—pace the informal slogan form of Pauli’s exclusion principle—also, a pair of indistinguishable particles of either species must be in the same state. This result appears to entail that Leibniz’s Principle is pandemically false in quantum theory.3 1.3 A new folklore? Such was the folklore until eight years ago. But this folklore has also recently been called into question by Simon Saunders, Fred Muller and Michael Seevinck. For (building on the Hilbert-Bernays account of identity) there are ways of distin- guishing particles that outstrip the notion of a quantum state for a particle (and its assocated probabilities, including conditional probabilities)—and yet which are supported by quantum theory. That is: the folklore has overlooked predicates on the right-hand side of (HB) which may, after all, sanction the right-to-left implication. For as the Hilbert-Bernays account teaches us, two objects can be discerned even if they share all their monadic properties and their relations to all other objects—and even if any relation that they hold to one another is held symmet- rically. That is: they can be discerned weakly.4 Thus if, for some relation R and two objects a and b, we have that Rab and Rba, then a and b must be distinct if either Raa or Rbb (or both) fails. It remains to provide such a relation that is legitimate within quantum me- chanics. This task was undertaken in its most general form for fermions by Muller and Saunders (2008), and for all particles by Muller and Seevinck (2009). (This work built upon an original suggestion by Saunders (2003b), which took inspira- tion from the fact that two particles in the spin singlet state may be said to have opposite spin (or to have vanishing combined total spin) without having to pick a spin direction.) In the following two sections (Sections 2 and 3), I will appraise the results in these two papers. I will conclude that Saunders, Muller and Seevinck were largely correct in their general conclusion that weakly discerning relations may be found, but that their proofs make incorrect assumptions—incorrect, that 3It may be of interest to note that these results can also be shown to hold for paraparticles, so long as one follows Messiah and Greenberg’s (1964) recommendation of working with ‘generalised rays’ (i.e. multi-dimensional subspaces) instead of one-dimensional rays; see Huggett (2003). 4The terminology originates with Quine (1960). 5 is, on their own terms—about which aspects of the quantum formalism represent genuine physical structure. I will propose a friendly amendment to the Saunders- Muller-Seevinck results in Section 4, and secure the fact that particles are always weakly discernible, whether they be bosons, fermions or paraparticles. 2 Muller and Saunders on discernment 2.1 The Muller-Saunders result Here I briefly present the main result contained in Muller and Saunders (2008). First I follow these authors in establishing three important distinctions in the way that particles may be discerned. 1. Absolute vs. relative vs. weak discernment. The first distinction relates to the logical form of the predicates used to discern the particles. As we have seen, all fermions and bosons are absolutely indiscernible; they are also relatively indiscernible. Thus our only hope is to discern them weakly. 2. Mathematical vs. physical discernment. Of course, it is crucial that the prop- erties and relations used to discern the particles be physical : we cannot appeal to elements of the theory’s mathematical formalism which have no representational function. Thus, for example, we cannot discern two particles in an assembly merely by appealing to the fact that the Hilbert space for that assembly is a tensor product of two copies of a factor Hilbert space. For all we know, this representative structure may be redundant; there may in fact only be one particle. So we must instead appeal to quantities in the formal- ism which genuinely represent physical quantities. Like Muller, Saunders and Seevinck, I call this sort of legitimate discernment ‘physical discernment’. I call instances of spurious discernment ‘mathematical discernment’—Muller and Saunders instead use the phrase ‘lexicon discernment’; but it is impor- tant to distinguish between mathematical objects (like Hilbert spaces) and mathematical language. Thus I restrict (HB) above to contain only physical predicates; mathematical predicates (such as set membership: ‘∈’) are not to be included. 3. Categorical vs. probabilistic discernment. The final distinction relates to the assumptions required to secure the discernment. Muller and Saunders call an instance of discernment ‘categorical’ just in case it requires no appeal to the Born rule, and ‘probabilistic’ otherwise. The main advantage of categorical, as opposed to probabilistic, discernment is that by by-passing probabilistic 6 notions its validity need not wait on any solution to the quantum measure- ment problem. However, this advantage is in my view only slight, since surely any solution to the measurement problem must anyway secure at least an ap- proximate vindication of the Born rule. Here the restriction is not on (HB) but the theory taken to entail it. Categorical discernment means entailment by quantum mechanics without the Born rule as a postulate. We are now in a position to state the main Muller-Saunders result: (SMS1) Fermions are categorically, weakly, physically discernible. Reconstruction of proof (cf. Muller and Saunders 2008, 536): We consider an assembly of only two fermions, so our Hilbert space is A(H⊗H); the result is easily extendible for more than two particles (cf. Muller and Saunders 2008, 534). Select some complete set of projection operators {Ei}, ∑ i Ei = 1, for the single- particle Hilbert space H and define Pij := Ei − Ej . Then define P (1) ij := Pij ⊗ 1 and P (2) ij := 1 ⊗ Pij , where the superscripts are labels for particles 1 and 2. We then define the following relation: Rt(x,y) iff ∑ i,j P (x) ij P (y) ij ρ = tρ, (1) where t ∈ R, ρ is the density operator representing the state of the assembly, and the indices i,j range over the projectors Ei. First we prove that 1 and 2 are categorically and weakly discerned by Rt for some value of t. To see that the discernment is categorical, it can be shown (cf. Muller and Saunders 2008, 533) that, with dim(H) > 2, for every state |Ψ〉 ∈ A(H⊗H), ∑ i,j P (1) ij P (2) ij |Ψ〉 = ∑ i,j P (2) ij P (1) ij |Ψ〉 = −2|Ψ〉 (2) and ∑ i,j ( P (1) ij )2 |Ψ〉 = ∑ i,j ( P (2) ij )2 |Ψ〉 = 2(d− 1)|Ψ〉 , (3) where d = dim(H). Thus every state of the assembly is an eigenstate of the operators used in the definition of Rt; and so we do not need to assume the Born rule. Rt therefore promises to provide categorical discernment. To see that Rt discerns the particles weakly for some t, note that Rt(1, 1) and Rt(2, 2) iff t = 2(d − 1), whereas Rt(1, 2) and Rt(2, 1) only if t = −2.5 So the relations R2(d−1) and R−2 both serve to weakly discern particles 1 and 2. 5Remember that ‘1’ and ‘2’ serve as particle labels in the expressions ‘Rt(1, 2)’, etc. 7 Finally, it remains to be shown that Rt is a physical relation. I turn to Muller and Saunder’s criteria (2008, pp. 527-8): (Req1) Physical meaning. All properties and relations should be trans- parently defined in terms of physical states and operators that corre- spond to physical magnitudes, as in [the weak projection postulate],6 in order for the properties and relations to be physically meaningful. (Req2) Permutation invariance. Any property of one particle is a prop- erty of any other; relations should be permutation-invariant, so binary relations are symmetric and either reflexive or irreflexive. (Req2) is clearly true of Rt. (Req1) is also true of Rt, provided that: (i) the projec- tors Ei are physically meaningful; and (ii) the physical meaningfulness of operators is preserved under mathematical operations; for our purposes these must include: arithmetical operations, i.e. addition and multiplication; and tensor multiplication with the identity. (Note: Muller and Saunders take (i) (along with (Req2)) to be sufficient to establish that Rt is a physical relation (2008, 534-5). However, it is clear that (ii) is also required.) � 2.2 Commentary on the Muller-Saunders proof I take no issue with Muller and Saunders’ claim that their relations Rt provide categorical and weak discernment. However, I question whether the relations Rt may properly be considered physical. I take no issue with the idea that projectors per se are physically meaningful (like Muller and Saunders, I agree that these can be considered to represent specific experimental questions with a yes/no answer); but Rt is defined in terms of non-symmetric projectors Ei ⊗ 1, etc. And it is compulsory—i.e., a necessary condition for representing a physical quantity—that the quantities obey the Indistinguishability Postulate (IP), which demands that all physical quantities be permutation invariant. (Cf. Messiah and Greenberg 1964.) This brings us to my criticism of (Req2), which has two components. First: it misapplies the correct idea that physical quantities must be symmetric. By requiring only that the relations defined from the quantum mechanical quantities be symmetric, (Req2) fails to rule out use of quantum mechanical quantities which may themselves be non-symmetric. To take a simple illustration of this point: ‘x 6The weak projection postulate is effectively Einstein, Podolsky and Rosen’s (1935) reality condition that the assembly’s being in an eigenstate of any self-adjoint operator Q with eigen- value q is a sufficient condition for the assembly’s possessing the property corresponding to the quantity’s Q having value q. This is an interpretative principle, which, like Muller and Saunders (2008) and Muller and Seevinck (2009), I take for granted. 8 is particle 1 and y is particle 2’ clearly fails to be a physical relation, both in the proper sense, and in terms of (Req2). But the relation ‘x is particle 1 and y is particle 2, or x is particle 2 and y is particle 1’ is equally unphysical, yet it does satisfy (Req2). It may be replied that this is where (Req1) comes in. But this brings us to the second component of my criticism of (Req2): it is redundant. For it is anyway necessary for a quantity to be symmetric to satisfy (Req1), since any non-symmetric quantity contravenes IP, and therefore cannot represent a ‘physical magnitude’. Indeed: since (Req1) already demands that the quantities be physical, why do we need any further requirement? It may be objected on behalf of Muller and Saunders that, while the quantities P (1) ij and P (2) ij indeed fail to be symmetric, the quantities defined in terms of them— namely, the ∑ i,j P (x) ij P (y) ij —are symmetric. This is indeed true: ∑ i,j ( P (1) ij )2 =∑ i,j ( P (2) ij )2 = 2(d− 1)1⊗1 and ∑ i,j P (1) ij P (2) ij = ∑ i,j P (2) ij P (1) ij = 2( ∑ i Ei ⊗Ei − 1⊗1), where 1 is the identity on H. (Note that the restrictions of both quantities to the anti-symmetric sector, A(H ⊗ H), are multiples of the identity on that sector.) But I see no force in the objection: the physical significance of these quantities was supposed to rest on their being constructions out of quantities like Ei ⊗1; yet it is precisely these quantities which run afoul of IP. Without any convincing account of the physical significance of the building blocks of the ∑ i,j P (x) ij P (y) ij , these quantities must be assessed for their physical significance on their own terms. But, since they are all multiples of the identity on the assembly’s state space, this significance is trivial: they all correspond to experimental questions which yield the same answer on every physical state.7 This triviality is a problem for Muller and Saunders, since it blocks the Rt from being physical relations. If we now attempt to redefine the Rt in a way that avoids misleading reference to the fraudulently physical quantities P (x) ij we obtain: Rt(x,y) iff (x = y and 2(d− 1)ρ = tρ) or (x 6= y and (−2)ρ = tρ) (4) This is equivalent to: Rt(x,y) iff (x = y and t = 2(d− 1)) or (x 6= y and t = −2). (5) So long as we have a definition of the Rt in terms of quantities that seems (i.e. from the point of view of the syntax) to treat the x = y and x 6= y cases equally, the fact that a different quantity (i.e. a different multiple of the identity) underlies 7I am very grateful to Nick Huggett for discussions about this point. 9 each of these two cases is tolerable. (In just the same way, Rxy and Rxx are strictly speaking different predicates—since one refers to a relation while the other refers to a monadic property—yet it is normal to treat any instance of Rxx as a special instance of Rxy. Indeed: weak discernment relies on this being legitimate.) But since the quantities ∑ i,j P (x) ij P (y) ij must be taken at face value—that is, as nothing but multiples of the identity—we must adopt definition (5) over definition (1), and definition (5) is hopelessly gerrymandered and unphysical. Thus Muller and Saunders’ proof that any two fermions are physically discernible does not go through. In Section 4, I propose an alternative relation which will discern fermions physi- cally and weakly, though not categorically. But first let me address the main results in Muller and Seevinck (2009). 3 Muller and Seevinck on discernment 3.1 The Muller-Seevinck result Muller and Seevinck use a similar framework to Muller and Saunders (2008): specifically, they carry over the three distinctions between kinds of discernment presented above, and the two requirements for physical significance, (Req1) and (Req2).8 There are two main results to discuss: the first concerns spinless parti- cles with infinite-dimensional Hilbert spaces; the second concerns spinning systems with finite-dimensional Hilbert spaces. I begin with their Theorem 1. (Note that I rephrase their Theorems; cf. Muller and Seevinck 2009, 189.) (SMS2) In an assembly with Hilbert space ⊗N L2(R3) and the associated algebra of quantities B( ⊗N L2(R3)), any two particles are categor- ically, weakly, physically discernible. Reconstruction of proof (cf. Muller and Seevinck 2009, 189): Again, for sim- plicity’s sake, I restrict attention to the case of two particles (N = 2). Let Q be the position operator for a single particle in some dimension (say x), and let be P 8Muller and Seevinck (2009, pp. 185-6) entertain adding a third requirement, to the effect that discernment by a relation is ‘authentic’ only if it is irreducible to monadic properties, and discernment by a monadic property is ‘authentic’ only if it is irreducible to relations. They reject this extra requirement, as do I; but my reason is different. My reason is that physical meaning (embodied in (Req1)) is all one could, and should, reasonably ask for—so long as that is taken to entail the requirement that the Indistinguishability Postulate is satisfied; cf. my commentary of Muller and Saunders’ proof in Section 2. 10 be the momentum operator in that same dimension. (So Q and P are (partially) defined on L2(R3); and I shall not go into detail about the partialness of the do- mains of definition, which are adequately discussed by Muller and Seevinck.) Now define Q(1) := Q ⊗ 1 and Q(2) := 1 ⊗ Q, and P (1) := P ⊗ 1 and P (2) := 1 ⊗ P , where 1 is the identity on L2(R3). We may now define a relation C as follows: C(x,y) iff [P (x),Q(y)]ρ = cρ, for some c 6= 0 , (6) where ρ is the density operator representing the state of the assembly. Now for every state we have C(1, 1) and C(2, 2), since [P (1),Q(1)] = [P (2),Q(2)] = −i~1⊗1. And we also have ¬C(1, 2) and ¬C(2, 1), since [P (1),Q(2)] = [P (2),Q(1)] = 0. Thus C weakly discerns particles 1 and 2. This discernment is categorical, since C holds or not categorically, i.e. without probabilistic assumptions. And the discernment is physical, since C meets (Req1) and (Req2). � Commentary. First of all I note that the restriction in (SMS2), that each particles’ state space be L2(R3) should count as no real restriction, since all real particles have spatial degrees of freedom. Second: since the discernment is cate- gorical, it is no restriction that the full (i.e. un-symmetrized) Hilbert space is used in the proof: the proof carries over for all restrictions to symmetry sectors. As in Section 2, again I take no issue with the claim that the discernment is weak and categorical, but I do deny that it is physical. The reason is the same as for Muller and Saunders (2008): namely, the proof uses unphysical quantities. (Thus I deny that (Req1) is satisfied.) Again we demand not just that the discerning relation be symmetric, but also that it be defined using only physical—a fortiori, only symmetric—quantities. And Q(x) and P (x), despite their tantalising intuitive interpretation, do not count as physical quantities. I now turn to Muller and Seevinck’s second main Theorem: (SMS3) In an assembly with a finite-dimensional Hilbert space ⊗N C2s+1, where s ∈ {1 2 , 1, 3 2 , . . .} and the associated algebra of quantities B( ⊗N C2s+1), any two particles are categorically, weakly, physically discernible using only their spin degrees of freedom. Reconstruction of proof (cf. Muller and Seevinck 2009, 193-7): Again I restrict attention to the case of two particles (N = 2). Let S = σxi + σyj + σkk be the quantity representing a single particle’s spin (so S acts on C2s+1). Then we define S1 := S ⊗1 and S2 := 1⊗ S, and the relation T as follows: T(x,y) iff for all ρ ∈D(C2s+1 ⊗ C2s+1), |(Sx + Sy)|2ρ = 4s(s + 1)~2ρ. (7) Recall that |S|2 = s(s+1)~21; this entails that |2S1|2 = |2S2|2 = 4s(s+1)~21⊗1; so T(1, 1) and T(2, 2) both hold. Meanwhile, |(S1+S2)|2 = |S|2⊗1+1⊗|S|2+2S⊗S = 11 2s(s+1)~21⊗1+2S⊗S. But the eigenvalue spectrum of |(S1 +S2)|2 never exceeds (2s)(2s+ 1)~2 < 4s(s+ 1)~2, so ¬T(1, 2) and ¬T(2, 1) both hold. This discernment is clearly weak. It is categorical, since it relies on no probabilistic assumptions, and it is physical, since T satisfies (Req1) and (Req2). � 3.2 Commentary on the Muller-Seevinck result I note that, in order to put the physical significance of T on firmer ground, Muller and Seevinck extend the EPR reality condition (cf. footnote 6) to a necessary and sufficient condition, which they call the ‘strong property postulate’. According to this postulate, the assembly possesses the property corresponding to the quantity’s Q having value q if and only if the assembly’s state is an eigenstate of the self- adjoint operator Q, with eigenvalue q. This strengthening is required to establish that the assembly does not possess combined total spin √ 4s(s + 1)~ when it not in an eigenstate of the total spin operator. Freedom from this stronger reality condition can be bought at the price of a concession to settle for probabilistic rather than categorical discernment. For we may define the new relation T ′: T ′(x,y) iff Tr ( ρ|(Sx + Sy)|2 ) = 4s(s + 1)~2. (8) It is clear that T ′ discerns iff the “de-modalized” version of T discerns. But the definition of T ′ involves a commitment to the Born rule, so T ′’s discernment is probabilistic. This trade-off between the strong reality condition and the Born rule will also be a feature of my proposals in the following section. The previous objection I levelled against (SMS1) and (SMS2) appears to be valid here too. For, even though |(S1 + S2)|2 and |2S1|2 = |2S2|2 are symmetric, once again their building blocks (S1 and S2) are not, and (it may be argued) it is only when defined in terms of these components that T is not a gerrymandered relation. However, my usual objection does not hold in this case. On the contrary, it seems reasonable to take T(x,y) as a natural physical relation, even though its explicit mathematical form depends on whether x = y or x 6= y. To see this, it should be enough that T can be parsed in English as the relation: ‘the combined total spin of x and y has the magnitude √ 4s(s + 1)~ in all states’. Combined total spin is a symmetric quantity, and it has obvious physical significance. Therefore I do not take issue with the discerning relation being physical. But I have two different objections in this case: one mild, the other more serious. The mild objection is that the relation T is different in a significant way 12 from the previous relations Rt and C. While Rt and C both apply to a given state of the assembly, the definition of T involves quantification over all states of the assembly. It is therefore a modal relation. But appeal to modal relations in this context is problematic, since it threatens to trivialise the search for a discerning relation for every state. It would turn out that Leibniz’s Principle is necessarily true if it is possibly true: a result that is at best controversial (though Saunders 2003b seems to endorse it, taking (HB) as an explicit definition of identity, as Quine 1960 also suggests). (Note, incidentally, that the use of modal relations cannot be criticised on the grounds that it assumes haecceitism. It is natural—at least in standard practice— to use Hilbert space labels to cross-identify systems between states, and this seems to have a whiff of haecceitism about it. However, this cross-identification strategy does not entail haecceitism, since the quantification over states may be restricted to the (anti-) symmetric sectors, in which all states are already permutation-invariant, so that the issue of haecceitism is moot.) This mild objection is easily met. We simply drop the quantification over states in the definition of T. If we do this, then the (unquantified) right-hand side of the definition (7) is still satisfied iff x = y, for all states ρ. We thereby drop the modal involvement. Thus we define a new relation, to be parsed as ‘the combined total spin of x and y has the magnitude √ 4s(s + 1)~’. The discernment remains categorical, since no probabilistic assumptions have been made. The serious problem is that (SMS3) is only applicable to assemblies whose con- stituent particles have non-zero spin. This might seem to be only a mild omission, since the only elementary spin-zero particle that actually exists is the Higgs boson, and for a treatment of that we turn to quantum field theory. However, it would be nice to establish the discernibility of quantum particles for all values of spin, not just for the sake of the Higgs boson, but for the sake of any hypothetical particle, actual or merely possible. To sum up: the same problem beleaguers the first two results (SMS1) and (SMS2), which aim to demonstrate the discernibility of (respectively) fermions and any particles with spatial degrees of freedom. The problem is that they both appeal to quantities which, in virtue of contravening IP, are non-physical. The third result, (SMS3), avoids this problem (modulo dropping some unnecessary modal involvement). However, it does not apply to particles with zero-spin. I now turn to my proposal for discerning any species of particle, for any value of spin. 13 4 A better way to discern particles Muller and Saunders’ Theorem 3 (539-40) contains the germ of a better way to secure discernment; i.e. a way free of the criticisms discussed in Sections 2 and 3. This section develops the germ. I proceed in stages. First I outline the basic idea, and propose a relation which weakly and physically discerns two particles in any two-particle assembly, using statistical variance. Then I investigate discernment for heterodox state spaces, in which particles may have definite position, and give a relation that will weakly in physically discern there too. Finally, I propose a relation that weakly and physically discerns any two particles in an assembly of any number of particles. 4.1 The basic idea My basic idea is that particles may be discerned by taking advantage of anti- correlations between single-particle states. In the case of fermions, this is ‘easy’ because of Pauli exclusion: in any basis the occupation number for any single- particle state never exceeds one. In the case of the other particles, it is more tricky, due to the fact that states for non-fermionic particles may have as terms product states with equal factors. In these states, two or more particles are fully correlated, so there does not seem to be any quantum property or relation which would discern them. The solution is to change the basis to one in which anti- correlations appear with non-zero amplitude; the quantity associated with this new basis can then form the basis of a discerning relation. Thus my strategy is discernment through anti-correlations, and the finding of anti-correlations through dispersion. For any state in which two particles are fully correlated, there will be dispersion in some other basis; in particular, the dispersion will involve branches with non-zero-amplitudes in which the particles are anti-correlated. 4.2 The variance operator For simplicity, I focus exclusively on the two particle case. We may take the assembly Hilbert space to be L2(R3) ⊗ L2(R3), but my results still carry over if we restrict to a symmetry sector, or add additional (e.g. spin) degrees of freedom. Anti-correlations between single-particle states in an eigenbasis for some single- particle quantity A may be indicated by means of the following ‘standard deviation’ 14 operator: ∆A := 1 2 (A⊗1−1⊗A) . (9) Actually, I will use its square ∆2A, the ‘variance’ operator, which, like ∆A, is self- adjoint (since A is). Unlike ∆A, ∆ 2 A is a symmetric operator, so it is in line with the Indistinguishability Postulate (IP), and is therefore eligible to represent a physical quantity. (∆A fails to be symmetric, since it is sent to minus itself under a permutation.) I also introduce the symmetric quantity A, which may be viewed as the statis- tical mean of A, taken over the two particles: A := 1 2 (A⊗1 + 1⊗A) . (10) Note that the over-line does not indicate an expectation value: A is an operator. By similarly defining A2 = 1 2 (A2 ⊗1 + 1⊗A2) we can express the variance operator more suggestively: ∆2A = 1 4 (A⊗1−1⊗A)2 = 1 4 ( A2 ⊗1 + 1⊗A2 − 2A⊗A ) = 1 2 ( A2 −A⊗A ) (11) and ∆2A = 1 4 ( A2 ⊗1 + 1⊗A2 − 2A⊗A ) = 1 2 ( A2 ⊗1 + 1⊗A2 ) − 1 4 ( A2 ⊗1 + 2A⊗A + 1⊗A2 ) = A2 −A 2 . (12) It is the latter equation (12) which justifies the term ‘variance’ for ∆2A and ‘standard deviation’ for ∆A. But note again that it is not the (c-numbered) sta- tistical variance of A over a given wavefunction; it is the variance of the operator A over the two particles: ∆2A is itself still an operator. The former equation (11) makes it most clear that ∆2A measures the anti-correlation between each of the two particles’ A-eigenstates. In particular, for any state all of whose terms are product states with equal factors in the A-basis: |Ψ〉 = ∑ k ck|φk〉⊗ |φk〉 , (13) 15 where A|φk〉 = ak|φk〉 , (14) it may be checked that the variance has eigenvalue zero. In general, however, a state with anti-correlations will not be an eigenstate of ∆2A. For a generic state-vector |Φ〉 = ∑ ij cij|φi〉⊗ |φj〉 (15) we have ∆2A|Φ〉 = 1 4 ∑ ij cij (ai −aj ) 2 |φi〉⊗ |φj〉 (16) so that 〈 ∆2A 〉 := 〈Φ|∆2A|Φ〉 = 1 4 ∑ ij |cij|2 (ai −aj ) 2 . (17) If we assume that A is non-degenerate (i.e., ai = aj implies i = j), then it is clear from (17) that there is a positive contribution to the value of 〈 ∆2A 〉 from every anti-correlation that has a non-zero amplitude. 4.3 Variance provides a discerning relation If a two-particle state has anti-correlations in a single-particle quantity A, we can build a symmetric, irreflexive relation which discerns them. The main idea is: if the expectation of the variance operator is non-zero, then this can be expressed as a relation between the two particles which neither particle bears to itself. Following Muller & Saunders (2008) and Muller & Seevinck (2009), we define the operators A(1) := A⊗1 ; A(2) := 1⊗A . (18) These quantities, being non-symmetric, are unphysical, but they can be used to define physical quantities: note, for example, that ∆A ≡ 12 ( A(1) −A(2) ) and A ≡ 1 2 ( A(1) + A(2) ) . We then define the relation R as follows: R(A,x,y) iff 1 4 ( A(x) −A(y) )2 ρ 6= 0 . (19) In English: R(A,x,y) holds for the state ρ if and only if ρ is not an eigenstate of the absolute difference between x’s and y’s operator A, with eigenvalue zero. 16 Here the variable A ranges over single-particle quantities and x and y range over particles. This definition implies that R(A, 1, 2) iff R(A, 2, 1), iff ∆2Aρ 6= 0, and ¬R(A, 1, 1) and ¬R(A, 2, 2). So R(A,x,y) is symmetric and irreflexive for each A. If ∆2A does not anihilate ρ, then we have R(A, 1, 2) and R(A, 2, 1); so in this case R(A,x,y) weakly discerns particles 1 and 2. Moreover, the discernment is categorical. The question remains whether this discernment is physical. I claim that it is, since the quantity 1 4 ( A(x) −A(y) )2 , which is symmetric, can be understood as a measure of anti-correlations between x and y for the single-particle quantity A— i.e., a measure of difference between x’s and y’s values for A. Thus it is no surprise that 1 4 ( A(x) −A(y) )2 = 0 for x = y; for no object can take a value for any quantity that is different from itself. I claim that, so long as the single-particle operator A has physical significance, so does 1 4 ( A(x) −A(y) )2 = 0. I emphasize that the physical meaning of 1 4 ( A(x) −A(y) )2 = 0 should not be thought of as depending on A(1) or A(2)’s having physical meaning. There is an important analogy here with relative distance. The relative distance between particle x and particle y need not be thought of as deriving its meaning from the absolute positions of x and y, even though the mathematical formalism of our theory may indeed allow us to define the relative distance in terms of these absolute positions. We need not take these mathematical definitions as representative of any physical fact, since we are not forced to admit that an element of the theory’s formalism which has a physical correlate also has physical correlates for all of its mathematical building blocks. This is because these mathematical building blocks may contain redundant structure which is not transmitted to all of their by-products. Such is the case of relative distance. And in fact, relative distance is more than an analogy: for (squared) relative distance is an instance of ∆2A, if we set A = Q, the single-particle position operator. Note that an additional assumption is required to transmit physical significance from 1 4 ( A(x) −A(y) )2 = 0 to R(A,x,y): we need to assume Muller and Seevinck’s ‘strong property postulate’. Recall that this states that any physical quantity of the assembly takes a certain value if and only if the assembly is in the appropriate eigenstate for that physical quantity’s corresponding operator. What is important here is the ‘only if’ component of the biconditional: this enables us to say that the difference in x’s and y’s values for A is non-zero just in case the assembly is not in the eigenstate with eigenvalue zero—including when the assembly is not in an eigenstate at all. I summarise the foregoing discussion in the following Lemmas: Lemma 1 For all two-particle assemblies, and all single-particle quantities A, the 17 relation R(A,x,y) has physical significance if A does, on the assumption of the strong property postulate. Lemma 2 For each state ρ of an assembly of two particles, and each single-particle quantity A, the relation R(A,x,y) discerns particles 1 and 2 weakly, cate- gorically and physically if and only if ∆2Aρ 6= 0, on the assumption of the strong property postulate. Proofs: See above. � As with (SMS3), in the previous Section, we can instead forego the strong property postulate and instead take advantage of the Born rule, to settle for prob- abilistic discernment. To do so we define the relation R′ as follows: R′(A,x,y) iff 1 4 Tr [ ρ ( A(x) −A(y) )2] 6= 0 . (20) Similar considerations to those above entail that R′(A, 1, 2) iff R′(A, 2, 1), iff 〈∆2A〉 6= 0. And ¬R(A, 1, 1) and ¬R(A, 2, 2). So R(A,x,y) weakly discerns particles 1 and 2 just in case 〈∆2A〉 6= 0. Thus: Lemma 3 For all single-particle quantities A, the relation R′(A,x,y) has physical significance if A does, on the assumption of the Born rule. Lemma 4 For each state ρ of the assembly, and each single-particle quantity A, the relation R′(A,x,y) discerns particles 1 and 2 weakly, probabilistically and physically, if and only if 〈∆2A〉 6= 0 for that state. Proofs: See above. � 4.4 Discernment for all two-particle states So far we have seen that two particles in a state with non-zero variance in some single-particle quantity A—i.e. two particles which are anti-correlated in A—may be discerned. To guarantee discernment in all two-particle states it remains to be shown that, for any such state, there will be some single-particle quantity whose eigenbasis has anti-correlations. In fact I prove a stronger result: namely that there is some single-particle quantity which discerns the two particles in all states of the assembly. Moreover, this quantity is familiar: it is position; and since all particles have spatial degrees of freedom, it will be a quantity that will always be available to discern. 18 Theorem 1 For each state ρ of an assembly of two particles, the relation R(Q,x,y) discerns particles 1 and 2 weakly, categorically and physically, where Q is the single-particle position operator, on the assumption of the strong property postulate. Proof: We assume the strong property postulate. From Lemma 2, we know that R(Q,x,y) discerns particles 1 and 2 weakly, categorically and physically, in the state ρ if and only if ∆2Qρ 6= 0. Let us first consider only pure states, and later generalise to all states. Pure states. Since we are working in the position representation, we use wave- functions rather than state-vectors or density operators. The most general form for the wavefunction of the assembly is Ψ(x, y) = ∑ ij cijφi(x)φj (y) , (21) where the φi are an orthonormal basis for L 2(R3). (We assume zero spin, but the proof is trivially extended for any non-zero value for spin.) Now (∆2QΨ)(x, y) = ∑ ij cij ( x2φi(x)φj (y) + φi(x)y 2φj (y) − 2xφi(x).yφj (y) ) = (∑ ij cijφi(x)φj (y) ) (x − y)2 = Ψ(x, y)(x − y)2 (22) (cf. Equation (16)). This is the zero function only if Ψ(x, y) = 0 whenever x 6= y. But then it cannot be represented in L2(R3) ⊗L2(R3), since it is not a function. Here I appeal to the fact that no wavefunction is “infinitely peaked” at the diagonal points of the configuration space. (The necessary Ψ can be written as a measure: Ψ(x, y) = f(x)δ(3)(x − y), for some function f ∈ L2(R3). I return to this point in Theorem 3, below.) We may conclude that (∆2QΨ)(x, y) 6= 0. It follows that ∆2Q|Ψ〉〈Ψ| 6= 0. Mixed states. We extend to density operators by taking convex combinations of (not necessarily othogonal) projectors. We have that ∆2Q (∑ i pi|Ψi〉〈Ψi| ) = ∑ i pi∆ 2 Q|Ψi〉〈Ψi| 6= 0 (23) since both the pi and the spectrum of ∆ 2 Q are positive. 19 From Lemma 2, we conclude that R(Q,x,y) discerns particles 1 and 2 weakly. The discernment is categorical since we made no probabilistic assumptions. Fi- nally, the discernment is physical, as follows from Lemma 1, the strong property postulate, and the fact that Q is physical. � We can now also prove Theorem 2 For each state ρ of an assembly of two particles, the relation R′(Q,x,y) discerns particles 1 and 2 weakly, probabilistically and physically; where Q is the single-particle position operator; on the assumption of the Born rule. Proof. We assume the Born rule. Then for any state ρ we have (cf. Equations 22, 23): 〈 ∆2Q 〉 = ∑ i pi ∫ d3x ∫ d3y |Ψi(x, y)|2(x − y)2 , (24) which is always positive (cf. Equation (17)). From Lemma 4, R′(Q,x,y) therefore discerns weakly. The discernment is probabilistic, since we assumed the Born rule. Finally, the discernment is physical, as follows from Lemma 3, the Born rule, and the fact that Q is physical. � It may be objected against the proofs of the foregoing two Theorems that I rely too heavily on a technical feature of the assembly’s Hilbert space, namely that it contains no states which exhibit no spread in (x − y)2. Effectively, unfavourable cases for discernment have been ruled out of the assembly’s Hilbert space a priori. But this objection is easily dealt with. Theorem 3 If we permit two-particle states to be represented by measures as well as by functions, then for all such states, either R(Q,x,y) or R(P,x,y) discerns particles 1 and 2 weakly, categorically and physically; where Q is the single-particle position operator, P is the single-particle momentum operator; on the assumption of the strong projection postulate. Proof. The guiding idea is that any state will exhibit spread in either relative position or relative momentum, so no state is annihilated by both ∆2Q and ∆ 2 P. We now allow measures, as well as functions, to represent states of the assembly. Recall from the proof of Theorem 1 that (∆2QΨ)(x, y) = 0 only if Ψ(x, y) = 0 whenever x 6= y. In this case Ψ(x, y) = f(x)δ(3)(x − y), for some measure f(x). Note at this point that the two particles cannot be fermions, since Ψ(x, y) = Ψ(y, x). We now move to the momentum basis by performing a Fourier transform 20 on Ψ: Ψ(k, l) = ∫ d3x ∫ d3y Ψ(x, y)e−ik.xe−il.y = ∫ d3x ∫ d3y f(x)δ(3)(x − y)e−i(k.x+l.y) = ∫ d3x f(x)e−i(k+l).x = f(k + l). (25) This yields (∆2PΨ)(k, l) = (k − l) 2f(k + l) , (26) which is the zero function only if f(k + l) = 0 whenever k 6= l. But we can only satisfy this requirement if f is the zero function. But in that case Ψ(x, y) is the zero function, and so does not represent a state. So if (∆2QΨ)(x, y) is the zero function, then (∆2PΨ)(k, l) can’t be. This result is easily extended to mixed states. With this result and Lemma 2 we conclude that either R(Q,x,y) or R(P,x,y) (or both) discerns particles 1 and 2 weakly. The discernment is categorical, since we made no probabilistic assumptions. Finally, the discernment is physical, given Lemma 1, the strong property postulate, and the fact that both Q and P are physical. � It only remains to state Theorem 4 If we permit two-particle states to be represented by measures as well as by functions, then for all such states, either R(Q,x,y) or R(P,x,y) discerns particles 1 and 2 weakly, probabilistically and physically; where Q is the single-particle position operator, P is the single-particle momentum operator; on the assumption of the Born rule. Proof: Left to the reader. � So we have established the weak discernibility of indistinguishable particles in any two-particle assembly. 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