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A formal framework for the study of the notion of undefined particle number in quantum mechanics

Abstract

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position—supported by E. Schrödinger—asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the line of solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions.

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Notes

  1. Although Manin has seemingly changed his position regarding this subject Manin (2010), the problem posed above still seems interesting to us and we will take it as a basis for our work.

  2. In the Concluding remarks of Caulton (2013), Caulton claims that the approaches of Muller, Saunders and Seevinck “...have been seen to fail, due to their surreptitious use of mathematical predicates that can be given no physical interpretation.”

  3. Similarly, in Dieks et al. (2010) it is claimed that (our emphasis): “All evidence points into the same direction: ‘identical quantum particles’ behave like money units in a bank account rather than like Blackean spheres. It does not matter what external standards we introduce, they will always possess the same relations to all (hypothetically present) entities. The irreflexive relations used by Saunders and others to argue that identical quantum particles are weakly discernible individuals lack the physical significance required to make them suitable for the job.”

  4. Related to this observation, see also Dieks et al. (2010) where a similar argument can be found for spins and the following observation is made regarding position measurements in QM: “To see how this complicates matters, think of a one-particle position measurement carried out on a many-particles system described by such a symmetrized state. The result found in such a measurement (for example, the click of a Geiger counter or a black spot on a photographic plate) is not linked to one of the ‘particle labels’; it is, in symmetrical fashion, linked to all of them. This already demonstrates how the classical limit of QM does not simply connect the classical particle concept to individual indices in the quantum formalism”.

  5. See also Teller (1989) and Morganti (2009) for a development of this notion and the problems posed by Teller.

  6. The Fock-space formulation is also discussed with great detail in French and Krause (2006), Chapter \(9\). See also Domenech et al. (2008b) and Domenech et al. (2009).

  7. We use “Cantorian” in analogy with the system NF of Quine (1953), Rosser (1953). But this should not lead to any confusion: the analogy is not too deep.

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da Costa, N.C.A., Holik, F. A formal framework for the study of the notion of undefined particle number in quantum mechanics. Synthese 192, 505–523 (2015). https://doi.org/10.1007/s11229-014-0583-2

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Keywords

  • Quantum mereology
  • Set theory
  • Undefined particle number
  • Quantum indistinguishability
  • Quantum superpositions