Abstract
In this essay, I describe a novel interpretation of quantum theory, called the Platonic interpretation or Platonic quantum theory. This new interpretation is based on a sharp notion of ontology and on the Gelfand-Naimark-Segal (GNS) construction, in contrast to traditional approaches that start instead by assuming the textbook formalism of Hilbert spaces and wave functions. I show that the Platonic interpretation naturally emerges from the consideration of random variables that represent underlying physical properties but that lack a common sample space, in the sense that they do not share an overall probability distribution despite having individual probability distributions.
“One’s ideas must be as broad as Nature if they are to interpret Nature.” —Sherlock Holmes, in Arthur Conan Doyle’s A Study in Scarlet, 1887
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Notes
- 1.
If it turns out that a specific kind of system happens to be the only fundamental kind of system in the world, then one could potentially argue that only the ontic properties of that kind of system would fundamentally exist in reality.
- 2.
Just for purposes of mathematical simplicity, I will assume that our sample spaces are always discrete sets—perhaps after some suitable coarse-graining—so that one can assign their individual elements finite, nonzero probabilities.
- 3.
I will address the question of non-uniqueness of \(\mathcal {S}\) in this essay’s concluding section.
- 4.
I will likewise address the status of the complex numbers in quantum theory in the concluding section.
- 5.
A C∗-norm satisfies ∥cZ∥ = |c| ∥Z∥ (for c a complex number), ∥Z 1 + Z 2∥≤∥Z 1∥ + ∥Z 2∥, and ∥Z 1 Z 2∥≤∥Z 1∥ ∥Z 2∥, along with the C∗-condition ∥Z ∗ Z∥ = ∥Z∥2.
- 6.
Be careful to note the different orderings of Z 1 and Z 2 on the two sides. One can prove this identity using the previously stated desiderata by considering the inequality ω((Z 1 + cZ 2)∗(Z 1 + cZ 2)) ≥ 0 in the two special cases c = 1 and c = i.
- 7.
Technically speaking, the set of vectors obtained by acting with operators on the cyclic vector | Ψω〉 is a dense subset of the Hilbert space \(\mathcal {H}_{\omega }\).
- 8.
When applied to the commutative C∗-algebra appropriate to the case of a classical system, the GNS construction yields the Koopman-von Neumann formulation (Koopman 1931; von Neumann 1932a,b), a Hilbert-space picture for classical physics that has many applications in subjects like dynamical-systems theory. This classical Hilbert-space picture is not typically taken seriously as a physical statement about reality, and the Platonic interpretation essentially holds that one should apply the same skepticism toward the Hilbert-space picture in the quantum case as well.
- 9.
Technically, one must restrict to the case of GNS representations that are regular.
- 10.
More precisely, Fell’s theorem states that if ρ is any state map for a C∗-algebra \(\mathcal {A}\), and if A 1, …, A n are any finite collection of n > 0 elements in that C∗-algebra \(\mathcal {A}\), then given any faithful representation of \(\mathcal {A}\) with a Hilbert space \(\mathcal {H}\) and a corresponding set of operators \(\hat {A}_{1},\dots ,\hat {A}_{n}\) on \(\mathcal {H}\), and given any positive real number 𝜖 > 0, there exists a density operator \(\hat {\rho }\) on the Hilbert space \(\mathcal {H}\) such that \(\vert \rho (A_{i})-\mathrm {Tr}[\hat {\rho }\hat {A}_{i}]\vert <\epsilon \) for all i = 1, …, n. See Clifton and Halvorson (2001) for a review. Given that n can be any positive integer (say, the thousandth busy beaver number raised to the power of a googolplex), and 𝜖 can be any positive real number (say, 1∕n for that same choice of n), it is difficult to imagine how any faithful representation of \(\mathcal {A}\) could be empirically insufficient without \(\mathcal {A}\) itself being empirically insufficient. In particular, no finite amount of experimental data could ever distinguish between two faithful representations of \(\mathcal {A}\), even if one of those representations contained certain limiting operators, called parochial observables, that did not exactly exist in the other representation, or did not exactly exist even at the level of the original C∗-algebra \(\mathcal {A}\).
- 11.
One can show that the Schrödinger, Heisenberg, and interaction pictures for describing a quantum system’s time evolution correspond to specific choices of \(\hat {V}(t)\). These three pictures are particularly useful for solving problems and for making empirical predictions in practice. However, the set of possible choices of \(\hat {V}(t)\) vastly exceeds these three cases. For a review of non-abelian gauge transformations in the context of quantum field theory, see, for example, Weinberg (1996).
- 12.
The same sort of post-measurement conditionalization occurs explicitly in the de Broglie-Bohm interpretation, and also occurs implicitly in the Everett interpretation on a branch-by-branch basis.
- 13.
For example, according to the de Broglie-Bohm approach, the ontic properties of a three-dimensional particle with coordinate-space wave function Ψ(x, y, z) are the input arguments x, y, z, which are just the particle’s Cartesian position coordinates, whereas the ontic properties of a Klein-Gordon field with coordinate-space wave functional Ψ[ϕ] are the input arguments ϕ(x, y, z), which are just the Klein-Gordon field’s local values. Notice that the particle’s position operators \(\hat {x},\hat {y},\hat {z}\) mutually commute, as do the local Klein-Gordon field operators \(\hat {\phi }(x,y,z)\).
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Acknowledgements
I would like to thank David Albert, Jeremy Butterfield, Eddy Keming Chen, Juliusz Doboszewski, Johannes Fankhauser, Benjamin Feintzeig, Samuel Fletcher, Peter Galison, Ned Hall, Richard Healey, Mina Himwich, Carl Hoefer, David Kagan, Martin Lesourd, Logan McCarty, Tushar Menon, Ana Raclariu, Simon Saunders, Chip Sebens, Jeremy Steager, Chris Timpson, and David Wallace. I am especially grateful to Valia Allori for organizing the collection in which this essay appears, as well as to two anonymous referees for their insightful comments and questions.
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Barandes, J.A. (2022). Platonic Quantum Theory. In: Allori, V. (eds) Quantum Mechanics and Fundamentality . Synthese Library, vol 460. Springer, Cham. https://doi.org/10.1007/978-3-030-99642-0_16
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