Discerning Fermions

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 principle 1.2. Intermezzo: Terminology and Leibnizian principles 1.3. The rise of the currently dominant view 1.4. Overview 2. Elements of Quantum Mechanics2.1. Physical states and physical magnitudes 2.2. Composite physical systems of similar particles 2.3. Fermions and bosons 2.4. Physical properties 2.5. Varieties of quantum mechanics 3. Analysis of Arguments3.1. Analysis of the Standard Argument 3.2. Van Fraassen's analysis 3.3. Massimi's analysis 4. The Logic of Identity and Discernibility4.1. The language of quantum mechanics 4.2. Identity of physical systems 4.3. Indiscernibility of physical systems 4.4. Some kinds of discernibility 5. Discerning Elementary Particles5.1. Preamble 5.2. Fermions 5.3. Bosons 6. Concluding Discussion