Abstract
Reconstructions of quantum theory are a novel research program in theoretical physics which aims to uncover the unique physical features of quantum theory via axiomatization. I focus on Hardy’s “Quantum Theory from Five Reasonable Axioms” (2001), arguing that reconstructions represent a modern usage of axiomatization with significant points of continuity to von Neumann’s axiomatizations in quantum mechanics. In particular, I show that Hardy and von Neumann share similar methodological ordering, have a common operational framing, and insist on the empirical basis of axioms. In the reconstruction programme, interesting points of discontinuity with historical axiomatizations include the stipulation of a generalized space of theories represented by a framework and the stipulation of analytic machinery at two levels of generality (first by establishing a generalized mathematical framework and then by positing specific formulations of axioms). In light of the reconstruction programme, I show that we should understand axiomatization attempts as being context–dependent, context which is contingent upon the goals of inquiry and the maturity of both mathematical formalism and theoretical underpinnings within the area of inquiry. Drawing on Mitsch (2022)’s account of axiomatization, I conclude that reconstructions should best be understood as provisional, practical, representations of quantum theory that are well suited for theory development and exploration. However, I propose my context–dependent re–framing of axiomatization as a means of enriching Mitsch’s account.
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Notes
Including no-go theorems, the Hilbert space formalism, and new concepts such as Bohr’s complementarity, to name a few (Goyal, 2022, p. 15)
In his discussion of mechanics, Hilbert discusses several differing approaches to its foundations, including Hertz’s perspective that force was explained by rigid connections between bodies and Boltzmann’s presentation that focused on the central forces between “any two mass points” (Corry, 1997, p. 142).
This is a result of the uncertainty principle, which rules out simultaneous measurements of incompatible properties or conjugated quantities (pairs of observables whose operators do not commute).
An expectation value is the average value predicted from a collection of measurements.
U here is some statistical operator that is both positive (real, non-negative valued) and linear (Rédei & Stöltzner, 2006, p. 5).
By unspecified I take Rédei and Stöltzner to be designating a general scope in terms of what those physical quantities could be, even if they can be specified at the time of measurement. This detail is not expanded on in Rédei and Stöltzner.
While this is true at this stage, it should be noted that von Neumann later abandons the Hilbert space formalism he was so integral in developing. He was aware of significant conceptual problems—including an infinite trace—in the derivation of the trace function (Rédei, 1996, p. 495). Von Neumann moved to develop his type II operator algebra (now called a von Neumann algebra) in an attempt to solve the conceptual problems in the Hilbert space formalism (Rédei, 1996, p. 495).
Although Hardy claims to obtain classical probability theory with the removal of axiom 5, the issue of how and when we are situated in a classical region in the space of theories is a subtle one. Hardy uses the continuity axiom to rule out various theories that do not correspond to quantum theory, including classical probability theory (Hardy, 2001, p. 15). It is sufficient to rule out alternatives to the Bloch sphere. However, omitting continuity could result in either classical or quantum theories in the absence of assumptions about the transformation properties of states. I am thankful to a reviewer for this insight.
Mitsch avoids the conflation of ‘physical interpretation’ by instead referring to ‘physical facts’ in Hilbert et al’s work. Mitsch maintains that ‘physical facts’ is a more apt terminology as it relates to Hilbert’s account (2022, p. 4).
It is possible to read von Neumann’s later shift from the Hilbert space formalism (HSF) to type II operator algebras as similar to the interest in different formulations in the reconstruction case. As R’edei notes, von Neumann abandons the idea that the HSF is the exclusive framework for quantum mechanics (1996, p. 495). Von Neumann looks to different mathematical formulations to solve conceptual problems in the HSF. It does appear that there is a potential similarity with the methodology of the reconstruction programme. However, the discussion in this paper is confined to von Neumann’s work within the HSF in the period up to 1932. Von Neumann began investigating the Type II framework after this period. It would be worth investigating this possible similarity with von Neumann’s later work in mind. I am thankful to a reviewer for this thought–provoking insight.
The Convex Operational Theories framework is a further specification of the Generalized Probability Theory framework (Chiribella & Spekkens, 2016, p. 8). Hence, Hardy’s ‘Five Axioms’ is within the GPT framework, stipulated specifically as a convex operational variety.
Stöltzner (2002) gives a much more detailed account of the different ways Hilbert deepens the foundations. There may be echoes of these different ways of deepening foundations in different implementations of the reconstruction project. For example, Stöltzner notes a type of deepening in the sense that deepening mathematical foundations might yield concepts that are physically more fundamental (2002, p. 257). This is similar in spirit to how Hardy uses a minimal Hilbert space formalism in order to highlight continuity as an important physical axiom of quantum theory. The potential connection between Stöltzner’s types of deepening and the reconstruction project will be explored in future work. I am grateful to a reviewer for this valuable insight.
To de-empiricise a theory was to disentangle empirical content from the analytic machinery. This was successful when a formalism was identified whose structure represented the important relations of the theory without empirical content.
This relates to how reconstructions are also distinct from the standard interpretational project in quantum mechanics: the standard interpretational approach accepts the formalism of quantum mechanics and instead aims to provide a physical interpretational of that formalism.
Mitsch is not referring to ‘practical’ in the sense of the ordering of analytic machinery and physical concepts that I have outlined prior from Hilbert et al.. Rather, ‘practical’ refers more to the notion that axiomatization contributes to scientific progress (Mitsch, 2022, p. 3).
Mitsch disagrees with Lacki that Hilbert was interested in axiomatization strictly in terms of logical clarification and rational reconstruction (2022, p. 3). Rather, Mitsch interprets Hilbert as having a more pragmatic and liberal idea of what axiomatization could achieve.
A process theory formalism is meant to capture the intuition that “the conceptual bare-bones of quantum theory concerns the manner in which systems and processes compose” (Selby et al., 2018, p. 1).
Which might even be mutually inconsistent, which is an unproblematic end in the interest of scraping out the mathematical core of a theory (Mitsch, 2022, p. 8).
For example, the physical implications of time–reversibility (Selby et al., 2018, p. 4) is difficult to comprehend.
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Acknowledgements
I would like to thank Doreen Fraser, Patricia Marino, and David DeVidi for helpful feedback on a draft of this paper, as well as attendees of the Foundations of Physics conference at the University of Bristol.
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Oddan, J. Reconstructions of quantum theory: methodology and the role of axiomatization. Euro Jnl Phil Sci 14, 20 (2024). https://doi.org/10.1007/s13194-024-00581-w
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DOI: https://doi.org/10.1007/s13194-024-00581-w