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The British Journal for the Philosophy of Science

Discerning Fermions

Frederik Müller, Simon W. Saunders

Published Aug 24, 2008• 155 citations
Publisher

Abstract

We demonstrate that the quantum-mechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finite-dimensional Hilbert spaces, is not in conflict with Leibniz's Principle of the Identity of Indiscernibles (PII). We discern the fermions by means of physically meaningful, permutation-invariant categorical relations, i.e. relations independent of the quantum-mechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all their admissible states; but their categorical discernibility turns out to be a state-dependent matter. In all demonstrated cases of discernibility, the fermions and the bosons are discerned (i) with only minimal assumptions on the interpretation of quantum mechanics; (ii) without appealing to metaphysical notions, such as Scotusian haecceitas, Lockean substrata, Postian transcendental individuality or Adamsian primitive thisness; and (iii) without revising the general framework of classical elementary predicate logic and standard set theory, thus without revising standard mathematics. This confutes: (a) the currently dominant view that, provided (i) and (ii), the quantum-mechanical description of such composite physical systems always conflicts with PII; and (b) that if PII can be saved at all, the only way to do it is by adopting one or other of the thick metaphysical notions mentioned above. Among the most general and influential arguments for the currently dominant view are those due to Schrödinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Giuntini, Mittelstaedt, Castellani, Krause and Huggett. We review them succinctly and critically as well as related arguments by van Fraassen and Massimi. 1. Introduction: The Currently Dominant View1.1. Weyl on Leibniz's principle1.2. Intermezzo: Terminology and Leibnizian principles1.3. The rise of the currently dominant view1.4. Overview2. Elements of Quantum Mechanics2.1. Physical states and physical magnitudes2.2. Composite physical systems of similar particles2.3. Fermions and bosons2.4. Physical properties2.5. Varieties of quantum mechanics3. Analysis of Arguments3.1. Analysis of the Standard Argument3.2. Van Fraassen's analysis3.3. Massimi's analysis4. The Logic of Identity and Discernibility4.1. The language of quantum mechanics4.2. Identity of physical systems4.3. Indiscernibility of physical systems4.4. Some kinds of discernibility5. Discerning Elementary Particles5.1. Preamble5.2. Fermions5.3. Bosons6. Concluding Discussion

DOI
10.1093/bjps/axn027

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Journal for General Philosophy of Science•Jun 15, 2021•0 citations
We demonstrate that the quantum-mechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finite-dimensional Hilbert spaces, is not in conflict with Leibniz's Principle of the Identity of Indiscernibles (PII). We discern the fermions by means of physically meaningful, permutation-invariant categorical relations, i.e. relations independent of the quantum-mechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all their admissible states; but their categorical discernibility turns out to be a state-dependent matter. In all demonstrated cases of discernibility, the fermions and the bosons are discerned (i) with only minimal assumptions on the interpretation of quantum mechanics; (ii) without appealing to metaphysical notions, such as Scotusian haecceitas, Lockean substrata, Postian transcendental individuality or Adamsian primitive thisness; and (iii) without revising the general framework of classical elementary predicate logic and standard set theory, thus without revising standard mathematics. This confutes: (a) the currently dominant view that, provided (i) and (ii), the quantum-mechanical description of such composite physical systems always conflicts with PII; and (b) that if PII can be saved at all, the only way to do it is by adopting one or other of the thick metaphysical notions mentioned above. Among the most general and influential arguments for the currently dominant view are those due to Schrödinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Giuntini, Mittelstaedt, Castellani, Krause and Huggett. We review them succinctly and critically as well as related arguments by van Fraassen and Massimi. 1. Introduction: The Currently Dominant View1.1. Weyl on Leibniz's principle1.2. Intermezzo: Terminology and Leibnizian principles1.3. The rise of the currently dominant view1.4. Overview2. Elements of Quantum Mechanics2.1. Physical states and physical magnitudes2.2. Composite physical systems of similar particles2.3. Fermions and bosons2.4. Physical properties2.5. Varieties of quantum mechanics3. Analysis of Arguments3.1. Analysis of the Standard Argument3.2. Van Fraassen's analysis3.3. Massimi's analysis4. The Logic of Identity and Discernibility4.1. The language of quantum mechanics4.2. Identity of physical systems4.3. Indiscernibility of physical systems4.4. Some kinds of discernibility5. Discerning Elementary Particles5.1. Preamble5.2. Fermions5.3. Bosons6. Concluding Discussion
We maximally extend the quantum-mechanical results of Muller and Saunders (2008) establishing the ‘weak discernibility’ of an arbitrary number of similar fermions in finite-dimensional Hilbert spaces. This confutes the currently dominant view that (A) the quantum-mechanical description of similar 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 heavy metaphysical notion such as Scotusian haecceitas or Adamsian primitive thisness. We take sides with Muller and Saunders (2008) against this currently dominant view, which has been expounded and defended by many.
In a series of recent papers, Simon Saunders, Fred Muller, and Michael Seevinck have collectively argued, against the folklore, that some nontrivial version of Leibniz’s principle of the identity of indiscernibles is upheld in quantum mechanics. They argue that all particles—fermions, paraparticles, anyons, even bosons—may be weakly discerned by some physical relation. Here I show that their arguments make illegitimate appeal to nonsymmetric, that is, permutation-noninvariant, quantities and that therefore their conclusions do not go through. However, I show that alternative, symmetric quantities may be found to do the required work. I conclude that the Saunders-Muller-Seevinck heterodoxy can be saved.
Saunders has recently claimed that “identical quantum particles” with an anti-symmetric state (fermions) are weakly discernible objects , just like irreflexively related ordinary objects in situations with perfect symmetry (Black’s spheres, for example). Weakly discernible objects have all their qualitative properties in common but nevertheless differ from each other by virtue of (a generalized version of) Leibniz’s principle, since they stand in relations an entity cannot have to itself. This notion of weak discernibility has been criticized as question begging, but we defend and accept it for classical cases likes Black’s spheres. We argue, however, that the quantum mechanical case is different. Here the application of the notion of weak discernibility indeed is question begging and in conflict with standard interpretational ideas. We conclude that the introduction of the conceptual resource of weak discernibility does not change the interpretational status quo in quantum mechanics.
Saunders has recently claimed that ``identical quantum particles'' with an anti-symmetric state (fermions) are weakly discernible objects, just like irreflexively related ordinary objects in situations with perfect symmetry (Black's spheres, for example). Weakly discernible objects have all their qualitative properties in common but nevertheless differ from each other by virtue of (a generalized version of) Leibniz's principle, since they stand in relations an entity cannot have to itself. This notion of weak discernibility has been criticized as question begging, but we defend and accept it for classical cases likes Black's spheres. We argue, however, that the quantum mechanical case is different. Here the application of the notion of weak discernibility indeed is question begging and in conflict with standard interpretational ideas. We conclude that the introduction of the conceptual resource of weak discernibility does not change the interpretational status quo in quantum mechanics.
Muller and Saunders ([2008]) purport to demonstrate that, surprisingly, bosons and fermions are discernible; this article disputes their arguments, then derives a similar conclusion in a more satisfactory fashion. After briefly explicating their proof and indicating how it escapes earlier indiscernibility results, we note that the observables which Muller and Saunders argue discern particles are (i) non-symmetric in the case of bosons and (ii) trivial multiples of the identity in the case of fermions. Both problems undermine the claim that they have shown particles to be physically discernible. We then prove two results concerning observables that are truly physical: one showing when particles are discernible and one showing when they are not (categorically) discernible. Along the way we clarify some frequently misunderstood issues concerning the interpretation of quantum observables. 1 Background2 Criticisms 2.1 Bosons 2.2 Fermions3 Reformulating the Insight 3.1 What weakly discerns? 3.2 General results4 Conclusion
Elementary particles appear in different ways.Particles of the same kind (similar particles) share all state-independent properties like mass, charge, and spin quantum number; there are many similar particles, but they are indistinguishable with respect to a certain class of properties.However, the case might even be stronger: similar particles apparently share all state-dependent properties, like spatial location and spin projection.This threatens the validity of Leibniz's famous Principle of the Identity of Indiscernibles (PII; see Rodriguez-Pereyra (2014) for a historical reconstruction).As it is usually understood, quantum mechanics requires for similar particles to be in permutation invariant states.Take the following state, for example: