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Mind

Towards a Nominalization of Quantum Mechanics

Mark Balaguer

Published Apr 1, 1996• 58 citations
Publisher

Abstract

In this paper I will provide the beginnings of a nominalization of quantum mechanics, or QM. I will not completely nominalize QM, but I will pro-vide what I think is the most important part of such a nominalization, namely, a nominalistic recovery of the algebraic structure of Hilbert spaces. To nominalize QM is to reformulate it so that it no longer contains any reference to, or quantification over, abstract objects, i.e., objects which are aspatial, atemporal, acausal, mind-independent, etc. The reason for want-ing to do this is to show that we can be realists about QM without believ-ing in abstract objects. To argue that our physical theories can be nominalized is not to argue that nominalism—i.e., the view that there are no abstract objects—is true. It is merely to rebut a certain argument against nominalism, viz., the Quine-Putnam indispensability argument. I do not think there are any good arguments for nominalism, and I have elsewhere (1995) responded to what is widely regarded as the best such argument, viz., Benacerraf's (1973) epistemological argument against the existence of abstract objects. And I should also note here that even if it turns out that empirical science cannot be nominalized, it will not necessarily mean that nominalism is false, because it may be that nominalists can admit that empirical science contains indispensable abstract-object talk, and simply account for this fact; indeed, in another paper (forthcoming) I have tried to do just that. 1. How Field nominalizes The program of nominalization was initiated by Hartry Field (1980), who tried to nominalize Newtonian Gravitation Theory, a classical flat-space-time theory. David Malament has argued (1982) that Field's method can-not be extended to cover QM. I think Malament is wrong, and below, I will explain why. The first thing to do, however, is explain Field's method. I will concentrate on Field's nominalization of physical quantities, such as temperature and length (Field 1980, Ch. 7). For the sake of simplicity, my presentation will differ somewhat from Field's. For example, I concen-

DOI
10.1093/mind/105.418.209

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Hartry Field (1980) has developed an interesting nominalization strategy for Newtonian gravitation theory—a strategy that reformulates the theory without quantification over abstract entities. According to David Malament (1982), Field's strategy cannot be extended to quantum mechanics (QM), and so it only has a limited scope. In a recent work, Mark Balaguer has responded to Malament's challenge by indicating how QM can be nominalized, and by “doing much of the work needed to provide the nominalization” (Balaguer 1998, 114). In this paper, I critically assess Balaguer's proposal, and argue that it ultimately fails. Balaguer's strategy is incompatible with a number of interpretations of QM, in particular with Bas van Fraassen's version of the modal interpretation. And given that Balaguer's strategy invokes physically real propensities, it is unclear whether it is even compatible with nominalism. I conclude that the nominalization of QM remains a major problem for the nominalist.

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If you have a restless intellect, it is very likely that you have played at some point with the idea of investigating the meaning and conceptual foundations of quantum mechanics. It is also probable (albeit not certain) that your intentions have been stopped on their tracks by an encounter with some version of the "Shut up and calculate" command. You may have heard that everything is already understood. That understanding is not your job. Or, if it is, it is either impossible or very difficult. Maybe somebody explained to you that physics is concerned with "hows" and not with "whys"; that whys are the business of "philosophy" ---you know, that dirty word. That what you call "understanding" is just being Newtonian; which of course you cannot ask quantum mechanics to be. Perhaps they also added some norms: The important thing a theory must do is predict; a theory must only talk about measurable quantities. It may also be the case that you almost asked "OK, and why is that?", but you finally bit your tongue. If you persisted in your intentions and the debate got a little heated up, it is even possible that it was suggested that you suffered of some type of moral or epistemic weakness that tend to disappear as you grow up. Maybe you received some job advice such as "Don't work in that if you ever want to own a house". I have certainly met all these objections in my short career, and I think that they are all just wrong. In this somewhat personal document, I try to defend that making sense of quantum mechanics is an exciting, challenging, important and open scientific endeavor. I do this by compulsively quoting Feynman (and others), and I provide some arguments that you might want to use the next time you confront the mentioned "opinions". By analogy with the anti-rationalistic Copenhagen command, all the arguments are subsumed in a standard answer to it: "Shut up and let me think"

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In this paper I will provide the beginnings of a nominalization of quantum mechanics, or QM. I will not completely nominalize QM, but I will pro-vide what I think is the most important part of such a nominalization, namely, a nominalistic recovery of the algebraic structure of Hilbert spaces. To nominalize QM is to reformulate it so that it no longer contains any reference to, or quantification over, abstract objects, i.e., objects which are aspatial, atemporal, acausal, mind-independent, etc. The reason for want-ing to do this is to show that we can be realists about QM without believ-ing in abstract objects. To argue that our physical theories can be nominalized is not to argue that nominalism—i.e., the view that there are no abstract objects—is true. It is merely to rebut a certain argument against nominalism, viz., the Quine-Putnam indispensability argument. I do not think there are any good arguments for nominalism, and I have elsewhere (1995) responded to what is widely regarded as the best such argument, viz., Benacerraf's (1973) epistemological argument against the existence of abstract objects. And I should also note here that even if it turns out that empirical science cannot be nominalized, it will not necessarily mean that nominalism is false, because it may be that nominalists can admit that empirical science contains indispensable abstract-object talk, and simply account for this fact; indeed, in another paper (forthcoming) I have tried to do just that. 1. How Field nominalizes The program of nominalization was initiated by Hartry Field (1980), who tried to nominalize Newtonian Gravitation Theory, a classical flat-space-time theory. David Malament has argued (1982) that Field's method can-not be extended to cover QM. I think Malament is wrong, and below, I will explain why. The first thing to do, however, is explain Field's method. I will concentrate on Field's nominalization of physical quantities, such as temperature and length (Field 1980, Ch. 7). For the sake of simplicity, my presentation will differ somewhat from Field's. For example, I concen-
Hartry Field (1980) has developed an interesting nominalization strategy for Newtonian gravitation theory—a strategy that reformulates the theory without quantification over abstract entities. According to David Malament (1982), Field's strategy cannot be extended to quantum mechanics (QM), and so it only has a limited scope. In a recent work, Mark Balaguer has responded to Malament's challenge by indicating how QM can be nominalized, and by “doing much of the work needed to provide the nominalization” (Balaguer 1998, 114). In this paper, I critically assess Balaguer's proposal, and argue that it ultimately fails. Balaguer's strategy is incompatible with a number of interpretations of QM, in particular with Bas van Fraassen's version of the modal interpretation. And given that Balaguer's strategy invokes physically real propensities, it is unclear whether it is even compatible with nominalism. I conclude that the nominalization of QM remains a major problem for the nominalist.
In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude and phase degrees of freedom. Consequently, this account extends Hartry Field’s program outlined in Science Without Numbers (1980), responds to David Malament’s long-standing impossibility conjecture (1982), and establishes an important first step towards a genuinely intrinsic and nominalistic account of quantum mechanics. I will also compare the present account to Mark Balaguer’s (1996) nominalization of quantum mechanics and discuss how it might bear on the debate about “wave function realism.” In closing, I will suggest some possible ways to extend this account to accommodate spinorial degrees of freedom and a variable number of particles (e.g. for particle creation and annihilation). Along the way, I axiomatize the quantum phase structure as what I shall call a “periodic difference structure” and prove a representation theorem as well as a uniqueness theorem. These formal results could prove fruitful for further investigation into the metaphysics of phase and theoretical structure. (For a more recent version of this paper, please see "The Intrinsic Structure of Quantum Mechanics" available on PhilPapers.)
What is the proper metaphysics of quantum mechanics? In this dissertation, I approach the question from three different but related angles. First, I suggest that the quantum state can be understood intrinsically as relations holding among regions in ordinary space-time, from which we can recover the wave function uniquely up to an equivalence class (by representation and uniqueness theorems). The intrinsic account eliminates certain conventional elements (e.g. overall phase) in the representation of the quantum state. It also dispenses with first-order quantification over mathematical objects, which goes some way towards making the quantum world safe for a nominalistic metaphysics suggested in Field (1980, 2016). Second, I argue that the fundamental space of the quantum world is the low-dimensional physical space and not the high-dimensional space isomorphic to the ``configuration space.'' My arguments are based on considerations about dynamics, empirical adequacy, and symmetries of the quantum mechanics. Third, I show that, when we consider quantum mechanics in a time-asymmetric universe (with a large entropy gradient), we obtain new theoretical and conceptual possibilities. In such a model, we can use the low-entropy boundary condition known as the Past Hypothesis (Albert, 2000) to pin down a natural initial quantum state of the universe. However, the universal quantum state is not a pure state but a mixed state, represented by a density matrix that is the normalized projection onto the Past Hypothesis subspace. This particular choice has interesting consequences for Humean supervenience, statistical mechanical probabilities, and theoretical unity.
We still lack any consensus about what one is actually talking about as one uses quantum mechanics. There is a gap between the abstract terms in which the theory is couched and the phenomena the theory enables each of us to account for so well. Because it has no practical consequences for how we each use quantum mechanics to deal with physical problems, this cognitive dissonance has managed to coexist with the quantum theory from the very beginning. The absence of conceptual clarity for almost a century suggests that the problem might lie in some implicit misconceptions about the nature of scientific explanation that are deeply held by virtually all physicists, but are rarely explicitly acknowledged. I describe here such unvoiced but widely shared assumptions. Rejecting them clarifies and unifies a range of obscure remarks about quantum mechanics made almost from the beginning by some of the giants of physics, many of whom are held to be in deep disagreement. This new view of physics requires physicists to think about science in an unfamiliar way. My primary purpose is to explain the new perspective and urge that it be taken seriously. My secondary aims are to explain why this perspective differs significantly from what Bohr, Heisenberg, and Pauli had been saying from the very beginning, and why it is not solipsism, as some have maintained. To emphasize that this is a general view of science, and not just of quantum mechanics, I apply it to a long-standing puzzle in classical physics: the apparent inability of physics to give any meaning to 'Now'-the present moment.
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. Then in this framework, we reformulate the structure of quantum mechanics in the language of symplectic manifolds and avoid linear structure of Hilbert space in such a way that the results can be stated for an arbitrary symplectic manifold.
If you have a restless intellect, it is very likely that you have played at some point with the idea of investigating the meaning and conceptual foundations of quantum mechanics. It is also probable (albeit not certain) that your intentions have been stopped on their tracks by an encounter with some version of the "Shut up and calculate" command. You may have heard that everything is already understood. That understanding is not your job. Or, if it is, it is either impossible or very difficult. Maybe somebody explained to you that physics is concerned with "hows" and not with "whys"; that whys are the business of "philosophy" ---you know, that dirty word. That what you call "understanding" is just being Newtonian; which of course you cannot ask quantum mechanics to be. Perhaps they also added some norms: The important thing a theory must do is predict; a theory must only talk about measurable quantities. It may also be the case that you almost asked "OK, and why is that?", but you finally bit your tongue. If you persisted in your intentions and the debate got a little heated up, it is even possible that it was suggested that you suffered of some type of moral or epistemic weakness that tend to disappear as you grow up. Maybe you received some job advice such as "Don't work in that if you ever want to own a house". I have certainly met all these objections in my short career, and I think that they are all just wrong. In this somewhat personal document, I try to defend that making sense of quantum mechanics is an exciting, challenging, important and open scientific endeavor. I do this by compulsively quoting Feynman (and others), and I provide some arguments that you might want to use the next time you confront the mentioned "opinions". By analogy with the anti-rationalistic Copenhagen command, all the arguments are subsumed in a standard answer to it: "Shut up and let me think"