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Outline of a Generalization and a Reinterpretation of Quantum Mechanics Recovering Objectivity

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Abstract

The ESR model has been recently proposed in several papers to offer a possible solution to the problems raising from the nonobjectivity of physical properties in quantum mechanics (QM) (mainly the objectification problem of the quantum theory of measurement). This solution is obtained by embodying the mathematical formalism of QM into a broader mathematical framework and reinterpreting quantum probabilities as conditional on detection rather than absolute. We provide a new and more general formulation of the ESR model and discuss time evolution according to it, pointing out in particular that both linear and nonlinear evolution may occur, depending on the physical environment.

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Notes

  1. This position is called “realistic” by some authors [2]. It expresses, however, a very weak form of realism, which does not assume any a priori model for individual objects and their properties and does not imply ontological commitments about the theoretical entities of QM (one could indeed interpret individual objects as activations of preparation procedures [3]). Such a weak form of realism is obviously implied by stronger realistic interpretations and/or modifications of QM, as Bohm’s theory, many-worlds interpretation, GRW theory, etc.

  2. For the sake of simplicity, we consider the notions of physical system, physical property and state as primitive in this section. We note, however, that physical properties can be intuitively interpreted as dichotomic observables, which can be measured obtaining one of two possible outcomes (often labeled yes and no).

  3. Indeed, all mixtures are represented by density operators in QM, and every such operator admits infinitely many decompositions in terms of pure states. If a density operator represents a proper mixture, there exists a decomposition whose coefficients are interpreted as epistemic probabilities. If a density operator represents instead an improper mixture, all coefficients of its decomposition are to be interpreted as nonepistemic probabilities [11–13].

  4. If one puts \(\mathcal {N}=\emptyset \), the above scheme could refer to classical and statistical mechanics as well. Of course, for every \(S \in \mathcal {P}\) and \(E\in \mathcal {E}\), p(S, E)∈{0,1} in classical mechanics. Furthermore, p(S, E) admits an epistemic interpretation in these theories, at variance with QM (Section 1).

  5. According to a known epistemological perspective (received view [43, 44]), assigning an empirical interpretation of the theoretical entities implies establishing correspondence rules connecting the theoretical language of a physical theory with its observational language. We do not deepen this philosophical issue here, but stress that, generally, not all theoretical entities of a theory may have a direct empirical interpretation.

  6. It is well known that the attempt at describing the dichotomic registering devices (or, more generally, the apparatuses corresponding to observables) in QM, together with their interaction with the physical system Ω, raises the objectification problem mentioned in Section 1. More specifically, nonobjectivity transfers to the macroscopic level, as illustrated by famous paradoxes. We avoid such problem here by adopting the above straightforward empirical interpretation of the theoretical entities of QM on the macroscopic entities in π and \(\mathcal {R}\), as usual in elementary QM. Of course, in this presentation the question of whether QM can describe such macroscopic entities and their interaction with Ω (that is, ultimately, the question of the universality of QM [2]) remains unanswered. We come back on this issue in Section 6.

  7. We recall that this position is weakened by the modal interpretations of QM, which admit that, whenever 0≠p(S, E)≠1, E could be objective for some individual objects in the state S. Hence the modal interpretations of QM distinguish between dynamical states (that can be identified with the quantum states introduced above) and value states (the value state of an individual object α representing, in our present terms, the set of all quantum properties that are objective for α).

  8. Note that the family \(\{ \text {ext} S_{\mu } \cap \text {ext} S \}_{S_{\mu } \in \mathcal {S}_{\mu }, S \in \mathcal {S}}\) is a further partition of \(\mathcal {U}\), some elements of which may be void.

  9. It is then easy to show that \(\mathcal {S}_{\mu |S}=\{ S_{\mu } \in \mathcal {S}_{\mu } \ | \ \text {ext}S_{\mu } \cap \text {ext}S \ne 0 \}\).

  10. For the sake of simplicity, we consider here only the discrete case. Note that the sum can be extended to all microscopic states in \(\mathcal {S}_{\mu }\), because p(S μ |S) = 0 if \(S_{\mu } \notin \mathcal {S}_{\mu |S}\).

  11. We observe that the mapping \(\tau :\mathcal {S}_{n}^{\widehat {A}} \in \{\mathcal {S}_{1}^{\widehat {A}}\), \(\mathcal {S}_{2}^{\widehat {A}}, \ldots , \mathcal {S}_{W}^{\widehat {A}}\} \longrightarrow |a_{n}^{M}\rangle \in \{|a_{1}^{M}\rangle , |a_{2}^{M}\rangle ,\ldots ,|a_{W}^{M}\rangle \}\) canonically induces a homomorphism of \(\mathcal {H}\) onto the proper subspace of \(\mathcal {G}^{M}\) generated by the set \(\{ |{a_{1}^{M}}\rangle , |{a_{2}^{M}}\rangle , \ldots , |{a_{W}^{M}}\rangle \}\) of unit vectors of \(\mathcal {G}^{M}\).

  12. We do not use the term individual object in this section to avoid confusing an item of Ω with an item of the composite system (Ω,Ω M ).

  13. The evolution described by (59) coincides with the evolution postulated in [21]. Hence the latter is a special case of the general evolution described by (53).

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Acknowledgments

This work was supported by the Natural Science Foundations of China (11171301 and 10771191) and by the Doctoral Programs Foundation of Ministry of Education of China (J20130061).

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Authors and Affiliations

  1. Department of Mathematics and Physics, University of Salento, Via Arnesano, 73100, Lecce, Italy

    Claudio Garola

  2. School of Management and IQSCS, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Sandro Sozzo

  3. Department of Mathematics, Zhejiang University, Zhejiang, People’s Republic of China

    Junde Wu

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  1. Claudio Garola
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Correspondence to Sandro Sozzo.

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Garola, C., Sozzo, S. & Wu, J. Outline of a Generalization and a Reinterpretation of Quantum Mechanics Recovering Objectivity. Int J Theor Phys 55, 2500–2528 (2016). https://doi.org/10.1007/s10773-015-2887-5

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