Abstract
Itamar Pitowsky contends that quantum states are derived entities, bookkeeping devices for quantum probabilities, which he understands to reflect the odds rational agents would accept on the outcomes of quantum gambles. On his view, quantum probability is subjective, and so are quantum states. We disagree. We take quantum states, and the probabilities they encode, to be objective matters of physics. Our disagreement has both technical and conceptual aspects. We advocate an interpretation of Gleason’s theorem and its generalizations more nuanced—and less directly supportive of subjectivism—than Itamar’s. And we contend that taking quantum states to be physical makes available explanatory resources unavailable to subjectivists, explanatory resources that help make sense of quantum state preparation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
For details of Pitowsky’s account of quantum gambles, see his (2003, Sec. 1).
- 3.
A further technical quibble, readily addressed, is that the version Itamar discusses applies only to the special setting of non-relativistic QM; §4.2 extends his discussion.
- 4.
- 5.
- 6.
This is not to say that T1∗ is uncontroversial among philosophers of physics. Bohmians would deny that T1∗ captures the fundamental event structure.
- 7.
The sum of probabilities for a transfinite collection is understood as the sup of sums of finite subcollections. The sup exists because the sequence of sums over ever larger finite subcollections gives a non-decreasing bounded sequence of real numbers, and every such sequence has a least upper bound.
- 8.
- 9.
- 10.
An awkwardness for Itamar’s account is the existence, in the case where \(\dim (\mathcal {H})=2\), of probability measures on \(\mathcal {P}(\mathfrak {B}(\mathcal {H}))\) that don’t extend to quantum states on \(\mathfrak {B}(\mathcal {H})\). In order to maintain that quantum states are just bookkeeping devices for the credences of rational agents contemplating bets on quantum event structures, it seems that Itamar must either deny that two dimensional Hilbert spaces afford quantum event structures, or articulate and motivate rationality constraints (in addition to the constraint of conformity to the axioms of the probability calculus!) that prevent agents from adopting “stateless” credences.
- 11.
For a contrary viewpoint, see Feintzeig and Weatherall (2019).
- 12.
Bayesian statisticians are divided on the issue of additivity requirements. Some versions of decision theory, such as that of Savage (1972), operate with mere finite additivity. Kadane et al. (1986) argue that various anomalies and paradoxes of Bayesian statistics can be resolved by the use of so-called improper priors and that such priors can be interpreted in terms of finitely additive probabilities. For more considerations in favor of mere finite additivity see Seidenfeld (2001).
- 13.
- 14.
This is a slight generalization of the result of Cassinelli and Zanghi (1983).
- 15.
One of us (LR) has reservations about the interpretational morals being drawn from the BDL theorem; see Appendix 6.
- 16.
Let \(\mathfrak {N}\) be a von Neumann algebra acting on \(\mathcal {H}\). A vector \(|\varOmega \rangle \in \mathcal {H}\) is cyclic iff \(\mathfrak {N}|\varOmega \rangle \) is dense in \(\mathcal {H}\). It is separating vector iff A|Ω〉 = 0 implies A = 0 for any \(A\in \mathfrak {N}\) such that \([A,\mathfrak {N}]=0\).
- 17.
Or rather that is all there is supposed to be to what Pitowsky (2003) and Bub and Pitowsky (2010) call the ‘big’ measurement problem. They think that after the big problem is dissolved there still remains the ‘small’ measurement problem, viz. to explain the emergence of the classical world we observe.
- 18.
This is hardly to deny that Bohmians are realists. They’re just realists about something else—the Bohm theory. Standard QM stands to the Bohm theory as an effective theory stands to a more fundamental theory whose coarse-grained implications it mimics. Thus realism about the Bohm theory induces something like “effective realism” about QM. The point is that this represents less realism about the theory than Itamar’s position. For more on effective realism, see Williams (2017).
References
Araki, H. (2009). Mathematical theory of quantum fields. Oxford: Oxford University Press.
Bratelli, O., & Robinson, D. W. (1987). Operator algebras and quantum statistical mechanics I (2nd ed.). Berlin: Springer.
Bub, J., & Pitowsky, I. (2010). Two dogmas of quantum mechanics. In S. Saunders, J. Barrett, A. Kent, & D. Wallace (Eds.), Many worlds? Everett, quantum theory, & reality (pp. 433–459). Oxford: Oxford University Press.
Buchholz, D., & Wichmann, E. H. (1986). Causal independence and the energy-level density of states in local quantum field theory. Communications in Mathematical Physics, 106, 321–344.
Buchholz, D., Doplicher, S., & Longo, R. (1986). On Noether’s theorem in quantum field theory. Annals of Physics, 170, 1–17.
Cassinelli, G., & Zanghi, N. (1983). Conditional probabilities in quantum mechanics. Nuovo Cimento B, 73, 237–245.
de Finetti, B. (1972). Probability, induction and statistics. London: Wiley.
de Finetti, B. (1974). Theory of probability (2 Vols). New York: Wiley.
Drish, T. (1979). Generalizations of Gleason’s theorem. International Journal of Theoretical Physics, 18, 239–243.
Eilers, M., & Horst, E. (1975). Theorem of Gleason for nonseparable Hilbert spaces. International Journal of Theoretical Physics, 13, 419–424.
Feintzeig, B., & Weatherall, J. (2019). Why be regular? Part II. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 65, 133–144.
Haag, R. (1992). Local quantum physics. Berlin: Springer.
Halvorson, H. (2001). On the nature of continuous physical quantities in classical and quantum mechanics. Journal of Philosophical Logic, 30, 27–50.
Halvorson, H. (2007). Algebraic quantum field theory. In J. Earman & J. Butterfield (Eds.), Handbook of the philosophy of science, philosophy of physics part A (pp. 731–864). Amsterdam: Elsevier.
Hamhalter, J. (2003). Quantum measure theory. Dordrecht: Kluwer Academic.
Hegerfeldt, G. C., & Mayato, R. (2012). Discriminating between von Neumann and Lüders reduction rule. Physical Review A, 85, 032116.
Kadane, J. B, Schervish, M. J., & Seidenfeld, T. (1986). Statistical implications of finitely additive probability. In P. K. Goel & A. Zellner (Eds.), Bayesian inference and decision techniques (pp. 59–76). Amsterdam: Elsevier.
Kadison, R. V., & Ringrose, J. R. (1987). Fundamentals of the theory of operator algebras (2 Vols). Providence: Mathematical Society.
Kumar, C. S., Shukla, A., & Mahesh, T. S. (2016). Discriminating between Lüders and von Neumann measuring devices: An NMR investigation. Physics Letters A, 380, 3612–2616.
Loewer, B. (2012). Two accounts of laws and time. Philosophical Studies, 160, 115–137.
Maeda, S. (1990). Probability measures on projections in von Neumann algebras. Reviews in Mathematical Physics, 1, 235–290.
Petz, D., & Rédei, M. (1995). John von Neumann and the theory of operator algebras. The Neumann Compendium, 163–185.
Pitowsky, I. (2003). Betting on the outcomes of measurements: A Bayesian theory of quantum probability. Studies in the History and Philosophy of Modern Physics, 34, 395–414.
Pitowsky, I. (2006). Quantum mechanics as a theory of probability. In: W. Demopoulos & I. Pitowsky (Eds.), Physical theory and its implications; essays in honor of Jeffrey Bub (pp. 213–240). Berlin: Springer.
Raggio, G. A. (1988). A remark on Bell’s inequality and decomposable normal states. Letters in Mathematical Physics, 15, 27–29.
Ruetsche, L. (2011). Why be normal? Studies in History and Philosophy of Modern Physics, 42, 107–115.
Savage, L. J. (1972). Foundations of statistics (2nd ed.). New York: Wiley.
Seidenfeld, T. (2001). Remarks on the theory of conditional probability. In V. F. Hendricks, et al. (Eds.), Probability theory (pp. 167–178). Dordrecht: Kluwer Academic.
Skyrms, B. (1992). Coherence, probability and induction. Philosophical Issues, 2, 215–226.
Srinvas, M. D. (1980). Collapse postulate for observables with continuous spectra. Communications in Mathematical Physics, 71, 131–158.
Takesaki, M. (1972). Conditional expectations in von Neumann algebras. Journal of Functional Analysis, 9, 306–321.
Williams, P. (2017). Scientific realism made effective. The British Journal for the Philosophy of Science, 70, 209–237.
Acknowledgements
This work reflects a collaboration with the late Aristides Arageorgis extending over many years. We are indebted to him. And we thank Gordon Belot, Jeff Bub, and the editors of this volume for valuable guidance and feedback on earlier drafts.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Earman, J., Ruetsche, L. (2020). Quantum Mechanics As a Theory of Observables and States (And, Thereby, As a Theory of Probability). In: Hemmo, M., Shenker, O. (eds) Quantum, Probability, Logic. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-030-34316-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-34316-3_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-34315-6
Online ISBN: 978-3-030-34316-3
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)