## Abstract

When we speak about different interpretations of quantum mechanics it is suggested that there is one single quantum theory that can be interpreted in different ways. However, after an explicit characterization of what it is to interpret quantum mechanics, the right diagnosis is that we have a case of predictively equivalent rival theories. I extract some lessons regarding the resulting underdetermination of theory choice. Issues about theoretical identity, theoretical and methodological pluralism, and the prospects for a realist stance towards quantum theory can be properly addressed once we recognize that interpretations of quantum mechanics are rival theories.

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## Notes

- 1.
- 2.
Alberto Cordero, for example, defends a diagnosis of this type: “the underdetermination at hand is clearly one of

*limited scope*. The robust physical and structural commonalities between the competing theories are as numerous as they are widespread over the total explanatory narratives […]. So, although the case makes for an intense ontological debate, its corrosive power on belief seems confined to just some aspects of the full narrative” (Cordero 2001, pp. 307–308). - 3.
Ruetsche’s use of the term “kinematics” is very broad. It refers not only to spatiotemporal properties and relations, but to all observables, including those that are usually considered as dynamic.

- 4.
A more general formulation of the observables postulate is given in terms of positive operator valued measurements (POVMs) (see Busch et al. 1995, Chap. 1; Peres 1993, pp. 283–284). Interestingly, POVMs allow that some non-Hermitian operators represent observables as well (see Roberts 2018). However, for the purposes of this article it is simpler to retain the formulation in terms of Hermitian operators—as we will see, one of my arguments below directly refers to the representation of observables by this type of operators.

- 5.
Von Neumann’s view in his axiomatization of quantum theory was that the representation is bijective. However, in the 1950s came out discussions about superselection rules and superselection sectors (see Giulini 2016 for a historical and conceptual overview). Fortunately, this complex issue is not relevant for the arguments presented here.

- 6.
For brevity and simplicity, I have abridged Muller’s more technical formulation of the postulates. He does not include density operators as representatives of systems, nor the trace rule, I have added them.

- 7.
Notice that the added postulates may be of an epistemological or metaphysical nature. That is, it is possible to obtain a quantum theory out of

*OQM-QM*_{0}without adding any further physical–mathematical postulates, but adding epistemological-metaphysical ones—Bohr’s and Everett’s interpretations are of this sort, for example. Wallace (2008, p. 21) distinguishes between*pure*interpretations (no extra mathematical formalism apart from the template) and*modificatory*ones (which do add to or amend the mathematical machinery in the template). Now, just like modificatory ones, pure interpretations add to the core formalism in order to obtain a quantum theory—the difference is given by the*type*of postulates they add. - 8.
“Uninterpreted” here does not mean “devoid of physical content”. \( NQM \) is indeed connected, albeit in a loose way, to empirical reality through the Born rule. \( NQM \), though not a theory, is already physics, not just mathematics.

- 9.
QBists and quantum pragmatists are sometimes accused of being instrumentalists in disguise. The very reference to epistemic states of subjects or agents implies that this accusation is false. The QBist concept of

*participatory realism*(Fuchs 2017), and the fact that for the pragmatist the assignment of probabilities to propositions about quantum systems is objectively grounded (Healey 2012), are clear in that these proposals do not amount to an instrumentalist stance. Although they do not require a description of the quantum world from the theory, they nevertheless adopt a*sui generis*realist stance towards quantum mechanics. - 10.
- 11.
Spontaneous collapse proposals are modificatory interpretations of \( NQM \), and therefore quantum theories. However, given their predictive divergence with respect to their rivals, these theories are not subject to underdetermination.

- 12.
The theory is quasi-Newtonian in Bohm’s original presentation. In later interpretations of the theory, in which there is no pilot wave (the wavefunction is a nomological term) and the second-order equation of motion plays no role (see, for example, Dürr et al. 1996, 1997), the Newtonian flavor is absent. Now, issues about the interpretation of Bohmian mechanics are analogous to issues about interpretation in other physical theories, not analogous to the interpretive issues about

*NQM*. For a classification of the different ways to interpret Bohm’s theory (see my Acuña 2016). - 13.
For example, in Bohm’s theory ∇

*S*and \( {\hat p} \) are neither mathematically nor physically equivalent, but the expected value for the property represented by ∇*S*calculated in terms of the distribution postulate, and the expected value of a measurement outcome calculated in terms of the operator \( {\hat p} \) are numerically equal, as the equations above show. More generally, in Bohm’s theory operators are simply bookkeeping devices in measurement outcomes statistics—more about this below. Now, how different regions of the wave function of the measured system “branch out” and get correlated with the eigenvalues of the corresponding operator depends on the specific way in which the measured system and the apparatus interact [this actually all there is to contextuality in Bohm’s theory (see Bub 1997, pp. 165–169)]. Thus, in general, “measurements” are perturbative and do not reveal the possessed value of the property before the interaction, so they are better described as*experiments*in Bohm’s theory (see Bohm and Hiley 1993, Sect. 6.3). - 14.
Even in a minimalist reading of Bohm’s theory in which position is the

*only*beable of particles (see Esfeld et al. 2014), the position beable is still given by the configuration of the particles, not by the operator \( \hat{x} \) which is merely a bookkeeping device for statistics of experimental outcomes. - 15.
In this outline of the theorem, I use von Neumann’s own notation.

- 16.
See fn. 5 above.

- 17.
A quick argument (Bell 1966) to see that von Neumann’s assumptions forbid deterministic hidden-variables theories is the following. Let us assume that the values \( \lambda \ge n \) of a hidden-variable \( \lambda \) determine dispersion-free states for a fermion. We calculate the expectation value for the beables \( {\mathcal{S}}_{x} \) and \( {\mathcal{S}}_{y} \),

*represented by the non-commuting operators*\( \sigma_{x} \) and \( \sigma_{y} \), respectively, whose eigenvalues are \( \pm 1 \). We obtain, say, \( Exp\left( {{\mathcal{S}}_{x} ,\psi_{\lambda \ge n} } \right) = 1 \) and \( Exp\left( {{\mathcal{S}}_{y} ,\psi_{\lambda \ge n} } \right) = - 1 \). Now, from**B’***and***II**, for the dispersion-free case it must hold that \( Exp\left( {{\mathcal{S}}_{x} + {\mathcal{S}}_{y} ,\psi_{\lambda \ge n} } \right) = Exp\left( {{\mathcal{S}}_{x} ,\psi_{\lambda \ge n} } \right) + Exp\left( {{\mathcal{S}}_{y} ,\psi_{\lambda \ge n} } \right) \), but the eigenvalues of the operator \( \sigma_{x} + \sigma_{y} \) (*that represents the beable*\( {\mathcal{S}}_{x} + {\mathcal{S}}_{y} \)) are \( \pm \sqrt 2 \), so we get a contradiction. More generally, if in a hidden variables theory the beables are represented by Hermitian operators, for dispersion-free states the expected value of a beable is an eigenvalue of the corresponding operator. Hence, (\( {\mathbf{B}}^{\prime } \wedge {\mathbf{II}} \)implies that for dispersion-free states the expected value for quantities represented by an operator*)**O*which is a sum of two non-commuting operators*R*and*S*is equal to a sum of eigenvalues of*R*and*S*. However, in general, the eigenvalues of*O*are not equal to that sum. - 18.
Jammer (1974, p. 274, fn. 45) proposes an evaluation of the theorem that is close to Bub’s.

- 19.
From Bell’s quick argument (see fn. 17) we saw that in consistent deterministic hidden variables theories it must be the case that \( \neg \left( {{\mathbf{B}}' \wedge {\mathbf{II}}} \right) \), but \( \neg {\mathbf{B^{\prime}}} \) is incompatible with the trace rule, so avoidance of the no-go result requires that \( \neg {\mathbf{II}} \).

- 20.
Bub (2010) and Dieks (2017b), state that considering that passage, it seems that von Neumann understood that his theorem is not an impossibility proof of hidden variables theories, but a proof that deterministic hidden variables theories in Hilbert space

*which adopt the representation of properties by operators*(*i.e*., which adopt**I**and**II**) are not possible. - 21.
A complete treatment of how the trace rule follows from the Bohmian account of measurements and quantum equilibrium can be found in Holland (1993, Chap. 8). For the role that operators play in the statistics of measurement outcomes in the context of Bohmian mechanics—in which the account of observables in terms of POVMs is especially clarifying—see Daumer et al. (1997); Dürr and Teufel (2009, Chap. 12), and, especially Dürr et al. (2004).

- 22.
After our discussion of von Neumann’s theorem, we know that Hermitian operators cannot represent properties in Bohm’s theory, so the meaning of

*P3*_{M}and*P4*_{M}should not go beyond a constraint on measurement outcomes (more on this below). - 23.
Valentini and Westman (2005) have explored the idea of dropping the quantum equilibrium postulate in Bohmian mechanics and then derive it dynamically. By formulating the theory without a quantum equilibrium postulate it would be even clearer that Bohm’s theory is not an interpretation of

*NQM*. However, the issue of the foundations of the quantum distribution in Bohm’s theory is a contentious one—whether quantum equilibrium is a postulate or not is a controversial subject. - 24.
For a general overview of this controversy (see Weatherall 2018).

- 25.
For the canonical operators \( P \) and \( Q \) acting on a Hilbert space, the CCRs are \( \left[ {P_{i} ,P_{j} } \right] = \left[ {Q_{i} ,Q_{j} } \right] = 0, \left[ {P_{i} ,Q_{j} } \right] = - i\hbar \delta_{ij} I \), where \( I \) is the identity operator. In the case of spin quantum systems, the Jordan-Wigner theorem states that unitary equivalence holds if the canonical observables satisfy the canonical anticommutation relations (CARs). For a spin system and canonical operators \( \sigma_{x} ,\sigma_{y} ,\sigma_{z} \) acting on a Hilbert space, the CARs are \( \left[ {\sigma_{i} ,\sigma_{j} } \right] = i\sigma_{k} \), and \( \left( {\sigma_{x} } \right)^{2} = \left( {\sigma_{y} } \right)^{2} = \left( {\sigma_{z} } \right)^{2} = I \).

- 26.
One may wonder, though, if matrix and wave mechanics can really count as theories. They may turn out to be simply two unitarily equivalent quantizations unable to make it through the semantics phase. That is, they may be no more than templates for a quantum theory, and the equivalence between them holds only up to the structure-specifying stage.

- 27.
There are technical issues involved in the calculation of arrival times, but the conceptual distinction in Bohm’s theory is actually rather straightforward. See Muga and Leavens (2000) for a comprehensive treatment of this issue. See also Das and Dürr (2019) for a recent proposal of a feasible experimental test of the predictions of arrival times in Bohm’s theory.

- 28.
- 29.
Some may think this scheme is applicable in the case of relativistic versions of Bohm’s theory. The intuition would be that since Bohmian quantum field theory is not Lorentz-covariant it is incompatible with special relativity—so that we could discard the former. But this is too quick. First, it can be argued that the fact that Bohmian field theory is not Lorentz-covariant does not imply an incompatibility with special relativity, even if a preferred foliation that reflects the non-locality of the theory is added to Minkowski spacetime structure (see Maudlin 2008). Besides, Maudlin (2014) has convincingly argued that what Bell’s theorem proves is that any empirically viable quantum theory must be non-local, regardless of whether hidden variables are included or not, so that most quantum theories should ultimately add some extra structure to Minkowski spacetime to provide an account of non-locality after all. Secondly, there is not one official Bohmian relativistic field theory, but a variety of different approaches, where some of them are actually Lorentz-covariant at a fundamental level (see, e.g., Dürr et al. 1999, 2014; Lienert et al. 2017).

- 30.
The underdetermination Saatsi has in mind seems to be of the metaphysical type, not a result of empirical equivalence between rival theories.

- 31.
The Stone-von Neumann theorem and the Jordan-Wigner theorem guarantee that different quantizations with finite degrees of freedom are unitarily equivalent. Different quantizations with infinite degrees of freedom, which Ruetsche dubs

*QM*_{∞}—as in quantum field theory and quantum mechanics in the thermodynamical limit—are not necessarily unitarily equivalent. Thus, our minimal realism maneuver does not work for*QM*_{∞}. The alternatives that the realist can explore in this context are treated in Ruetsche (2011, 2015). - 32.
Muller (2015, p. 121) characterizes the meaning of Hermitian operators in

*NQM*in a similar way. If the eigenstate-eigenvalue link is added to the template, for a superposed state on the spectral decomposition basis of a Hermitian operator it cannot be said that the system has a definite value for the observable represented by the operator. This is why Muller states that Hermitian operators cannot be taken to represent properties in*NQM*. I think that this is not enough to support the conclusion. We could simply state that Hermitian operators do represent observables of systems (even in the sense of properties) in*NQM*, but that*if we assume the eigenstate-eigenvalue link*states which are superpositions of eigenstates do not have a definite value for the corresponding property. The eigenstate-eigenvalue link does not oblige us to deny that Hermitian operators represent properties in*NQM*. However, if we want to take*NQM*as the mathematical structure that is shared by all quantum theories, then in order to make sense of the meaning of Hermitian operators in Bohm’s theory we must indeed interpret them in a strictly phenomenological way. The QBist interpretation requires this restricted and phenomenological conception of Hermitian operators as well. - 33.
French and Ladyman (2003) defend ontic structural realism in quantum mechanics referring to issues about (non-)individuality in the context of indistinguishable particles. However, this argument does not work on the basis of the uninterpreted formalism.

*NQM*is not even enough to formulate issues about particle indistinguishability in a conceptually well-defined way. Furthermore, in Bohm’s theory particles are always distinguishable, so the arguments presented by French and Ladyman assume an interpretive stance, thus falling prey to underdetermination. For a critical assessment of French and Ladyman (2003) (see Morganti 2004). - 34.
For a discussion of this issue (see Bueno 2011).

- 35.

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## Acknowledgements

I thank four anonymous referees for their helpful comments on an earlier version of this work. I thank Angelo Bassi, Max Maaneli Derakshani and Francesca Vidotto for bibliographic references. The first version of this article was written during my stay as a Visiting Fellow in the Center for Philosophy of Science, University of Pittsburgh. I thank its Director Edouard Machery and everybody in the Center for their hospitality, and James Fraser and everyone in the reading group for their helpful comments on the first draft. This work was financially supported by FONDECYT Grant 11170608.

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Acuña, P. Charting the landscape of interpretation, theory rivalry, and underdetermination in quantum mechanics.
*Synthese* (2019). https://doi.org/10.1007/s11229-019-02159-z

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### Keywords

- Quantum mechanics
- Empirical equivalence
- Underdetermination
- Pluralism
- Theory identity