Skip to main content
SpringerLink
Log in

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

Synthese volume 192, pages 505–523 (2015)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

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.

References

  • Aerts, D., & Daubechies, I. (1979). A characterization of subsystems in physics. Letters in Mathematical Physics, 3, 11–17.

  • Aerts, D., & Daubechies, I. (1979). A mathematical condition for a sublattice of a propositional system to represent a physical subsystem, with a physical interpretation. Letters in Mathematical Physics, 3, 19–27.

    Article  Google Scholar 

  • Arenhart, J. (2013a). Wither away individuals. Synthese, 190(16), 3475–3494.

    Article  Google Scholar 

  • Arenhart, J. (2013b). Weak discernibility in quantum mechanics: Does it save PII? Axiomathes, 23(3), 461–484.

    Article  Google Scholar 

  • Arenhart, J., & Krause, D. (2014). From primitive identity to the non-individuality of quantum objects. Studies in History and Philosophy of Modern Physics, 46(Part B), 273–282.

    Article  Google Scholar 

  • Ballentine, L. (1998). Quantum mechanics: A modern development. Hackensack: World Scientific Publishing Co., Pte. Ltd.

    Book  Google Scholar 

  • Beltrametti, E. G., & Cassinelli, G. (1981). The logic of quantum mechanics. Reading: Addison-Wesley.

    Google Scholar 

  • Bigaj, T. (2013). On discernibility and symmetries. Erkenntnis, 1–19. doi:10.1007/s10670-014-9616-y.

  • Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843.

    Article  Google Scholar 

  • Borghini, A., & Lando, G. (2011). Natural properties, supervenience, and mereology. Humana. Mente Journal of Philosophical Studies, 19, 79–104.

    Google Scholar 

  • Bratteli, O., & Robinson, D. W. (1997). Operator algebras and quantum statistical mechanics (Vol. 2). Berlin: Springer.

    Book  Google Scholar 

  • Brignole, D., & da Costa, N. C. A. (1971). On supernormal Ehresmann-Dedecker universes. Mathematische Zeitschrift, 122(4), 342–350.

    Article  Google Scholar 

  • Butterfield, J. (1993). Interpretation and identity in quantum theory. Studies in History and Philosophy of Science, 24, 443–476.

    Article  Google Scholar 

  • Caulton, A., & Butterfield, J. (2012a). On Kinds of Indiscernibility in logic and metaphysics. British Journal for the Philosophy of Science, 63(1), 27–84.

    Article  Google Scholar 

  • Caulton, A., & Butterfield, J. (2012b). Symmetries and paraparticles as a motivation for structuralism. British Journal for the Philosophy of Science, 63(2), 233–285.

    Article  Google Scholar 

  • Caulton, A. (2013). Discerning ‘Indistinguishable’ quantum systems. Philosophy of Science, 80, 49–72.

    Article  Google Scholar 

  • Clifton, R., & Halvorson, H. (2001). Are Rindler quanta real? Inequivalent particle concepts in quantum field theory. British Journal for the Philosophy of Science, 52, 417–470.

    Article  Google Scholar 

  • da Costa, N. C. A. (1980). Ensaio sobre os Fundamentos da Lógica. São Paulo: HUCITEC.

    Google Scholar 

  • da Costa, N. C. A., & Bueno, Y. O. (2009). Non reflexive logics. Revista Brasileira de Filosofia, 58, 181–208.

    Google Scholar 

  • Dalla Chiara, M. L., & Toraldo di Francia, G. (1995). Identity questions from quantum theory. In K. Gavroglu, et al. (Eds.), Physics, philosophy and the scientific community (pp. 39–46). Dordrecht: Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • Dalla Chiara, M. L., Giuntini, R., & Krause, D. (1998). Quasiset theories for microobjects: A comparision. In E. Castellani (Ed.), Interpreting bodies: Classical and quantum objects in modern physics (pp. 142–152). Princeton: Princeton University Press.

    Google Scholar 

  • Dalla Chiara, M. L., Giuntini, R., & Greechie, R. (2004). Reasoning in quantum theory. Dordrecht: Kluwer Acad. Pub.

    Book  Google Scholar 

  • Darby, G., & Watson, D. (2010). Lewis’s principle of recombination: Reply to Efird and Stoneham. Dialectica, 64(3), 435–445.

    Article  Google Scholar 

  • de la Harpe, P., Jones, V. (1995). An introduction to C\(^{\ast }\)-algebras.

  • Dieks, D. (2010). Are ‘Identical Quantum Particles’ weakly discernible objects? In M. Suárez, M. Dorato, & M. Rédei (Eds.), EPSA philosophical issues in the sciences (pp. 21–30). Berlin: Springer.

    Chapter  Google Scholar 

  • Dirac, P. A. M. (1927). The quantum theory of the emission and absorption of radiation. Proceedings of the Royal Society of London Series A, 114, 243–265.

    Article  Google Scholar 

  • Domenech, G., Holik, F., de Ronde, C. (2008). Entities, Identity and the formal structure of quantum mechanics. arXiv:1203.3007v1.

  • Domenech, G., & Holik, F. (2007). A discussion on particle number and quantum indistinguishability. Foundations of Physics, 37, 855–878.

    Article  Google Scholar 

  • Domenech, G., Holik, F., & Krause, D. (2008). Q-spaces and the foundations of quantum mechanics. Foundations of Physics, 38, 969–994.

    Article  Google Scholar 

  • Domenech, G., Holik, F., Kniznik, L., & Krause, D. (2009). No labeling quantum mechanics of indiscernible particles. International Journal of Theoretical Physics, 49, 3085–3091.

    Article  Google Scholar 

  • Domenech, G., Holik, F., & Massri, C. (2010). A quantum logical and geometrical approach to the study of improper mixtures. Journal of Mathematical Physics, 51, 052108.

    Article  Google Scholar 

  • Duncan, A., & Janssen, M. (2008). Pascual Jordan’s resolution of the conundrum of the wave-particle duality of light. Studies in History and Philosophy of Science Part B, 39, 3.

    Article  Google Scholar 

  • Dvurečenskij, A., & Pulmannová, S. (2000). New trends in quantum structures. Dordrecht: Kluwer Acad. Pub.

    Book  Google Scholar 

  • Engesser, K., Gabbay, D. M., & Lehmann, D. (Eds.). (2009). Handbook Of quantum logic and quantum structures (quantum logic). North-Holland: Elsevier.

    Google Scholar 

  • French, S., & Redhead, M. (1988). Quantum physics and the identity of indiscernibles. British Journal for the Philosophy of Science, 39, 233–246.

    Article  Google Scholar 

  • French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Giuntini, R. (1991). Quantum logic and hidden variables. Mannheim: BI Wissenschaftsverlag.

    Google Scholar 

  • Greechie, J. R. (1981). Current issues in quantum logic. In E. Beltrameti & B. van Fraassen (Eds.), A non-standard quantum logic with a strong set of states (pp. 375–380). New York: Plenum.

    Google Scholar 

  • Gudder, S. P. (1978). In A. R. Marlow (Ed.) Mathematical foundations of quantum theory. Academic Press, New York.

  • Halmos, P. (1963). Naive set theory. New York: D. Van Nostrand Company.

    Google Scholar 

  • Hawley, K. (2006). Weak discernibility. Analysis, 66(4), 300–303.

    Article  Google Scholar 

  • Hawley, K. (2009). Identity and Indiscernibility. Mind, 118(469), 101–119.

    Article  Google Scholar 

  • Holik, F. (2006). Aportes hacia una incorporación de la teoría de cuasiconjuntos en el formalismo de la mecánica cuántica. Master Thesis at the University of Buenos Aires.

  • Holik, F. (2010). Compound quantum systems: An algebraic approach. PhD. Thesis at the University of Buenos Aires.

  • Holik, F. (2011). Neither name, nor number. In Probing the meaning of quantum mechanics: Physical, philosophical, and logical perspectives. World Scientific. arXiv:1112.4622v1.

  • Holik, F., Massri, C., & Ciancaglini, N. (2012). Convex quantum logic. International Journal of Theoretical Physics, 51, 1600–1620.

    Article  Google Scholar 

  • Huggett, N. (2000). Philosophical foundations of quantum field theory. The British Journal for the Philosophy of Science, 51, 617–637.

    Article  Google Scholar 

  • Jauch, J. M. (1968). Foundations of quantum mechanics. Cambridge: Addison-Wesley.

    Google Scholar 

  • Kalmbach, G. (1983). Orthomodular lattices. San Diego: Academic Press.

    Google Scholar 

  • Kalmbach, G. (1986). Measures and Hilbert lattices. Singapore: World Scientific.

    Book  Google Scholar 

  • Krause, D. (2003). Why quasi-sets? Boletim da Sociedade Paranaense de Matematica, 20, 73–92.

    Google Scholar 

  • Kunen, K. (1980). Set theory, an introduction to indpendence proofs. Amsterdam: North-Holland.

    Google Scholar 

  • Ladyman, J., & Bigaj, T. (2010). The principle of the identity of indiscernibles and quantum mechanics. Philosophy of Science, 77, 117–136.

    Article  Google Scholar 

  • Ladyman, J., Linnebo, Ø., & Pettigrew, R. (2012). Identity and discernibility in philosophy and logic. The Review Of Symbolic Logic, 5(1), 162–186.

    Article  Google Scholar 

  • Mackey, G. W. (1957). Quantum mechanics and Hilbert space. American Mathematical Monthly, Supplement 64, 45–57.

  • Manin Y. I. (1976). Problems of present day mathematics I: Foundations. In F. E. Browder (Ed.), Mathematical Problems Arising From Hilbert Problems, Proceedings of Symposia in Pure Mathematics XXVIII (p. 36), Providence: American Mathematical Society.

  • Manin, Y. I. (1977). A course in mathematical logic. Berlin: Springer.

    Book  Google Scholar 

  • Manin, Y. (2010). A course in mathematical logic for mathematicians. New York: Springer.

    Book  Google Scholar 

  • Mittelstaedt, P. (1998). The interpretation of quantum mechanics and the measurement process. Cambridge: Cambridge University Press.

    Google Scholar 

  • Morganti, M. (2009). A new look at relational holism in quantum mechanics. Philosophy of Science, 76, 1027–1038.

    Google Scholar 

  • Muller, F.A. (2014). “The Rise of Relationals”, to appear. In: Mind.

  • Muller, F. A., & Saunders, S. (2008). Discerning Fermions. British Journal for the Philosophy of Science, 59, 499–548.

    Article  Google Scholar 

  • Muller, F. A., & Seevinck, M. P. (2009). Discerning elementary particles. Philosophy of Science, 76, 179–200.

    Article  Google Scholar 

  • Pavičić, M., Megill, D. (2008). Is quantum logic a logic?. In K. Engesser, D. Gabbay, and D. Lehmann (Eds.) Handbook of quantum logic and quantum structures, Vol. Quantum logic (pp. 23–47). Amsterdam: Elsevier.

  • Piron, C. (1976). Foundations of quantum physics. Cambridge: Addison-Wesley.

    Google Scholar 

  • Post, H. (1963) Individuality and physics. The listener, 70, 534–537; reprinted in Vedanta for East and West 32, (1963), 14–22, cited in [2].

  • Pták, P., & Pulmannova, S. (1991). Orthomodular structures as quantum logics. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Putnam, H. (1968). Is Logic Empirical? Boston studies in the philosophy of science, vol. 5. In Robert S. Cohen, Marx W. Wartofsky (Eds.) (Dordrecht: D. Reidel, 1968) (pp. 216–241). .

  • Quine, W. V. O. (1953). From a logical point of view, chapter V. Cambridge: Harvard University Press.

    Google Scholar 

  • Randall, C. H., & Foulis, D. J. (1981). Interpretation and foundations of quantum theory. In H. Neumann (Ed.), (pp. 21–28). Bibliographisches Institut, Mannheim.

  • Redhead, M., & Teller, P. (1991). Particles, particle labels, and quanta: the toll of unacknowledged metaphysics. Foundations of Physics, 21, 43–62.

    Article  Google Scholar 

  • Redhead, M., & Teller, P. (1992). Particle labels and the theory of indistinguishable particles in quantum mechanics. British Journal for the Philosophy of Science, 43, 201–218.

    Article  Google Scholar 

  • Robertson, B. (1973). Introduction to field operators in quantum mechanics. American Journal of Physics, 41, 678.

    Article  Google Scholar 

  • Rosser, J. B. (1953). Logic for mathematicians. New York: McGraw-Hill.

    Google Scholar 

  • Santorelli, A., Krause, D., & Sant’Anna, A. (2005). A critical study on the concept of identity in Zermelo–Fraenkel like axioms and its relationship with quantum statistics. Logique & Analyse, 189–192, 231–260.

    Google Scholar 

  • Saunders, S. (2003). Physics and Leibniz’s principles. In K. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections (pp. 289–307). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Saunders, S. (2006). Are quantum particles objects? Analysis, 66, 52–63.

    Article  Google Scholar 

  • Schrödinger, E. (1998) What is an elementary particle?. In E. Castellani (Ed.), Interpreting bodies: classical and quantum objects in modern physics (pp. 197–210). Princeton: Princeton Un. Press.

  • Schroer, B. (2003). Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics (CBPF-NF–018/03). Brazil

  • Simons, P. (1987). Parts: A study in ontology. Clarendon Press-Oxford: Oxford University Press.

    Google Scholar 

  • Takeuti, G. (1981). Quantum set theory. In E. Beltrametti, B. C. van Frassen (Eds.), Current issues in quantum logic (pp. 302–322). Plenum, New York.

  • Teller, P. (1986). Relational holism and quantum mechanics. British Journal for the Philosophy of Science, 37, 71–81.

    Google Scholar 

  • Teller, P. (1989). Relativity, relational holism and the bell inequalities. In J. Cushing & E. McMullin (Eds.), Philosophical consequences of quantum theory (pp. 208–223). Notre Dame: University of Notre Dame Press.

    Google Scholar 

  • Titani, S., & Kozawa, H. (2003). Quantum Set Theory. International Journal of Theoretical Physics, 42, 2575–2602.

    Article  Google Scholar 

  • van Fraassen, B. C., & Peschard, I. (2008). Identity over time: Objectively and subjectively. Philosophical Quarterly, 58, 15–35.

    Google Scholar 

  • Varadarajan, V. (1968). Geometry of quantum theory I. Princeton: van Nostrand.

    Book  Google Scholar 

  • Varadarajan, V. (1970). Geometry of quantum theory II. Princeton: van Nostrand.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, Federal University of Santa Catarina, Florianópolis, SC, 88040-900, Brazil

    Newton C. A. da Costa

  2. Instituto de Física La Plata (IFLP), CONICET, 115 y 49, 1900, La Plata, Argentina

    Federico Holik

  3. Center Leo Apostel for Interdisciplinary Studies and Department of Mathematics, Brussels Free University, Krijgskundestraat 33, 1160, Brussels, Belgium

    Federico Holik

Authors
  1. Newton C. A. da Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federico Holik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Federico Holik.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-014-0583-2

Keywords

search

Navigation