Hidden Variables in Quantum Mechanics from the Perspective of Boltzmannian Statistical Mechanics
Abstract
:1. Introduction: No-Hidden-Variables Theorems
In ordinary quantum mechanics there just is nothing but the wavefunction for calculating probabilities. There is then no question of making the result on one side redundant on the other by more fully specifying events in some [past] space-time region. We have a violation of local causality. ([4], p. 241)
However, it has turned out that quantum mechanics cannot be ‘completed’ into a locally causal theory, at least as long as one allows, as Einstein, Podolsky and Rosen did, freely operating experimenters. ([4], p. 242)
Efforts to construct such deeper levels of description, in which properties of individual systems do have preexisting values revealed by the act of measurement, are known as hidden-variables programs. A frequently offered analogy is that a successful hidden-variables theory would be to quantum mechanics as classical mechanics is to classical statistical mechanics (see, for example, A. Einstein, in Schilpp [13], 1949, p. 672): quantum mechanics would survive intact, but would be understood in terms of a deeper and more detailed picture of the world. (p. 803)
- Section 2 will provide a general analysis of measurements in the framework of Boltzmannian statistical mechanics and show that the identification of measurement outcomes with preexisting values of the measured system is neither an a priori assumption nor generally justified. As soon as one goes beyond “ideal” thermodynamic measurements, statistical mechanics reproduces several features of quantum mechanical probabilities that are frequently (and misleadingly) advertised as “non-classical”.
- Section 3 will apply this framework to (versions of) the Kochen–Specker and Bell theorem to shed further light on their implications. Regarding the former, it will corroborate Bell’s (1966) analysis [8] by showing how, once the measuring device and the active role of the measurement process are taken into account, “contextuality” arises as a natural feature of random variables. This makes KS-type theorems relatively unremarkable. Translating the GHZ-Bell theorem into the Boltzmannian framework will illustrate again why Bell-type theorems are much more significant by pointing to nonlocality (due to entanglement) as the key innovation of quantum physics.
- To get to the point more quickly, the main discussion adopts some established but potentially misleading terminology (including the term “hidden variables”). Section 4 will try to rectify that and provide a more accurate framing of the hidden-variables debate as one about quantum ontology.
2. Measurements in Statistical Mechanics
2.1. Boltzmannian Measurement Scheme
- The measured system (phase space ) couples to some measurement device (phase space ). Let be the total phase space equipped with a probability measure . In special cases, when the measurement is performed on a macroscopic system and the relevant outcomes can be observed directly, we may omit the measurement devices and identify with .
- We consider a partition of into macrostates, of which at least a subset corresponds to the possible measurement outcomes as indicated by our instruments. will denote the initial macrostate (“ready state”) of the experimental setup. (Other macrostates, which we ignore for simplicity, might correspond to null results, i.e., failed measurements). If the outcome indicated by the macrostate is associated with the numerical value , we can define a value function
- The total system evolves according to a flow on that depends on the dynamics of the experiment, in particular on the interactions between the measured system and the measurement device. We assume that the measurement experiment starts at (with initial conditions in ) and that the result is obtained at some later time . In general, T can be chosen from a long time interval during which we have a stable record of the measurement outcome.
- The random variable that describes the measurement experiment is thenThis is to say that the initial microstate X (of the entire setup) evolves with the dynamical flow , and at , we read out the measured value . The distribution of Z under the phase space measure , conditionalized on the initial macrostate , yields the probabilistic predictions for the measurement experiment. In particular:
- (a)
- The right random variable may be on , if we consider two independent measurements on identically prepared systems.
- (b)
- The right random variables may be of the form and where is the dynamics of a new experiment, suitable for measuring A and B simultaneously. In many cases, this new experiment may require an entirely different setup, so that are not even defined on the same phase space as .
- (c)
- If we want to perform the original measurements consecutively, the relevant random variables would take the form
- (d)
- We may also consider the random variables
2.2. (In)Compatible Measurements
- (i)
- The microscopic dynamics are time-translation invariant, so that and .
- (ii)
- The dynamical flows and commute, so that .
- (iii)
- The measurement outcome for A is stable under the dynamics and vice versa. Formally: and . The idea is that once a stable record of the A-result has formed, it will be unaffected by the B-measurement (and vice versa).
- A:
- The die is rotated with equal probability by or about the x-axis and we observe in which direction () the arrow on top is pointing.
- B:
- The die is rotated with equal probability by or about the y-axis and we observe in which direction the arrow on top is pointing.
2.3. Two Examples
3. Boltzmannian Analysis of No-Go Theorems
- (a)
- All observables have the eigenvalues and .
- (b)
- The observables along each column and each row commute.
- (c)
- The product along each row is 1.
- (d)
- The product along the first column and along the second column is 1, while the product along the third column is .
3.1. Who’s Afraid of Contextuality?
3.2. Why Bell’s Theorem Is Not Silly
- (a)
- is an eigenstate of with eigenvalue .
- (b)
- is an eigenstate of with eigenvalue .
- As indicated above, nonlocality can be regarded as a special case of contextuality, as it entails that the value of one observable cannot be predetermined independently of which other (compatible) observables are measured alongside it. It is therefore sometimes suggested that the key improvement of Bell-type theorems over KS-type theorems is that they provide a better justification for expecting non-contextuality, namely the locality assumption (together with the spacelike separation of the pertinent measurements), which implies that the measurements cannot disturb each other [7]. This framing overlooks the crucial point that the definition of non-contextuality relies on the algebraic structure of quantum observables, whereas Bell-locality is not about observable operators at all. The former assumption, underlying the KS theorem, proves gratuitous in a hidden-variables framework in which observable operators are no longer fundamental. A violation of locality, in contrast, has profound dynamical implications for any theory that seeks to reproduce the predictions of quantum mechanics.
- In the Boltzmannian framework, it is most natural to think of the failure of (24) as a violation of “parameter independence”, meaning that, e.g., depends on what measurements (x-spin or y-spin) are performed on the other two particles. These correspond, after all, to different settings of the Stern–Gerlach magnet (or polarizer, in the case of photons) and thus to different interactions with the particles. However, in principle, we could also reproduce the quantum predictions for the GHZ experiment by allowing to depend on the outcomes (but not the parameter choices) for the other two other measurements. This should not be confused with the innocuous observation that the measurement results are correlated. Random variables that violate “outcome independence” would still mean that the dynamical evolution of the state variables describing a particle and its measurement is nonlocally influenced by distant events.
- Finally, one should emphasize that Bell’s theorem (especially in its 1990 version [2]) is more general than the argument discussed here and does not presuppose the Boltzmannian framework. Our analysis merely corroborates (and perhaps further elucidates) Bell’s finding that the nonlocality of quantum mechanics cannot be cured by additional variables.
4. Misleading Terminology
This word very strongly suggests the ascertaining of some preexisting property of some thing, any instrument involved playing a purely passive role. Quantum experiments are simply not like that, as we learned especially from Bohr. The results have to be regarded as the joint product of ’system’ and ’apparatus’, the complete experimental set-up. But the misuse of the word ’measurement’ makes it easy to forget this and then to expect that the ’results of measurements’ should obey some simple logic in which the apparatus is not mentioned. The resulting difficulties soon show that any such logic is not ordinary logic. It is my impression that the whole vast subject of ’Quantum Logic’ has arisen in this way from the misuse of a word. I am convinced that the word ’measurement’ has now been so abused that the field would be significantly advanced by banning its use altogether, in favour for example of the word ’experiment’. ([4], p. 166)
It seems to me that the term “probability” is often abused nowadays. […] A probability statement presupposes the full reality of its subject. No reasonable person will make a guess as to whether Caesar’s die at the Rubicon had a five on top. The quantum mechanists sometimes pretend that probability statements are to be applied especially to events with a blurred reality. (Schrödinger to Einstein. November 18, 1950. Letter 238 in ([43], pp. 627–629). (My translation).
5. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Bell, J.S. On the Einstein Podolsky Rosen Paradox. Physics 1964, 1, 195–200. [Google Scholar] [CrossRef]
- Bell, J.S. La Nouvelle Cuisine. In Between Science and Technology; Sarlemijn, A., Kroes, P., Eds.; Elsevier Science: Amsterdam, The Netherlands, 1990. [Google Scholar]
- Kochen, S.; Specker, E.P. The Problem of Hidden Variables in Quantum Mechanics. J. Math. Mech. 1967, 17, 59–87. [Google Scholar] [CrossRef]
- Bell, J.S. Speakable and Unspeakable in Quantum Mechanics, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Norsen, T. John S. Bell’s Concept of Local Causality. Am. J. Phys. 2011, 79, 1261–1275. [Google Scholar] [CrossRef]
- Maudlin, T. What Bell Did. J. Phys. A Math. Theor. 2014, 47, 424010. [Google Scholar] [CrossRef]
- Mermin, N.D. Hidden Variables and the Two Theorems of John Bell. Rev. Mod. Phys. 1993, 65, 803–815. [Google Scholar] [CrossRef]
- Bell, J.S. On the Problem of Hidden Variables in Quantum Mechanics. Rev. Mod. Phys. 1966, 38, 447–452. [Google Scholar] [CrossRef]
- Maudlin, T. Three Measurement Problems. Topoi 1995, 14, 7–15. [Google Scholar] [CrossRef]
- Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II. Phys. Rev. 1952, 85, 180–193. [Google Scholar] [CrossRef]
- Heisenberg, W. Physics and Beyond: Encounters and Conversations; Harper & Row: New York, NY, USA, 1971. [Google Scholar]
- Lazarovici, D.; Oldofredi, A.; Esfeld, M. Observables and Unobservables in Quantum Mechanics: How the No-Hidden-Variables Theorems Support the Bohmian Particle Ontology. Entropy 2018, 20, 381. [Google Scholar] [CrossRef]
- Schilpp, P.A. (Ed.) Albert Einstein: Philosopher-Scientist, 1st ed.; Number VII in the Library of Living Philosophers; The Library of Living Philosophers Inc.: Evanston, IL, USA, 1949. [Google Scholar]
- Goldstein, S. Boltzmann’s Approach to Statistical Mechanics. In Chance in Physics: Foundations and Perspectives; Bricmont, J., Dürr, D., Galavotti, M.C., Ghirardi, G., Petruccione, F., Zanghì, N., Eds.; Springer: Berlin/Heidelberg, Germany, 2001; pp. 39–54. [Google Scholar]
- Lazarovici, D. Typicality Reasoning in Probability, Physics, and Metaphysics; New Directions in the Philosophy of Science; Palgrave Macmillan: Cham, Switzerland, 2023. [Google Scholar] [CrossRef]
- Spekkens, R.W. Contextuality for Preparations, Transformations, and Unsharp Measurements. Phys. Rev. A 2005, 71, 052108. [Google Scholar] [CrossRef]
- Dürr, D.; Teufel, S. Bohmian Mechanics: The Physics and Mathematics of Quantum Theory; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Dürr, D.; Goldstein, S.; Zanghì, N. Quantum Physics without Quantum Philosophy; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Hemmo, M.; Shenker, O.R. The Road to Maxwell’s Demon: Conceptual Foundations of Statistical Mechanics; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar] [CrossRef]
- Frauchiger, D.; Renner, R. Quantum Theory Cannot Consistently Describe the Use of Itself. Nat. Commun. 2018, 9, 3711. [Google Scholar] [CrossRef] [PubMed]
- Lazarovici, D.; Hubert, M. How Quantum Mechanics Can Consistently Describe the Use of Itself. Sci. Rep. 2019, 9, 470. [Google Scholar] [CrossRef]
- Feynman, R.P. Space-Time Approach to Non-Relativistic Quantum Mechanics. Rev. Mod. Phys. 1948, 20, 367–387. [Google Scholar] [CrossRef]
- Fine, A. Joint Distributions, Quantum Correlations, and Commuting Observables. J. Math. Phys. 1982, 23, 1306–1310. [Google Scholar] [CrossRef]
- Beck, C. Local Quantum Measurement and Relativity; Fundamental Theories of Physics; Springer International Publishing: Cham, Switzerland, 2021; Volume 201. [Google Scholar] [CrossRef]
- von Neumann, J. Mathematische Grundlagen der Quantenmechanik; Springer: Berlin/Heidelberg, Germany, 1932. [Google Scholar] [CrossRef]
- Bell, J.S. Interview: John Bell. Omni 1988, 1988, 85–92. [Google Scholar]
- Bub, J. Von Neumann’s `No Hidden Variables’ Proof: A Re-Appraisal. Found. Phys. 2010, 40, 1333–1340. [Google Scholar] [CrossRef]
- Dieks, D. Von Neumann’s Impossibility Proof: Mathematics in the Service of Rhetorics. Stud. Hist. Philos. Sci. Part B Stud. Hist. Philos. Mod. Phys. 2017, 60, 136–148. [Google Scholar] [CrossRef]
- Acuña, P. Von Neumann’s Theorem Revisited. Found. Phys. 2021, 51, 73. [Google Scholar] [CrossRef]
- Daumer, M.; Dürr, D.; Goldstein, S.; Zanghì, N. Naive Realism about Operators. Erkenntnis 1996, 45, 379–397. [Google Scholar] [CrossRef]
- Norsen, T. The Pilot-Wave Perspective on Spin. Am. J. Phys. 2014, 82, 337–348. [Google Scholar] [CrossRef]
- Dürr, D.; Goldstein, S.; Zanghì, N. Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory. J. Stat. Phys. 2004, 116, 959–1055. [Google Scholar] [CrossRef]
- Greenberger, D.M.; Horne, M.A.; Zeilenger, A. Going Beyond Bell’s Theorem. In Bell’s Theorem, Quantum Theory and Conceptions of the Universe; Kafatos, M., Ed.; Kluwer Academic Publishers: Dodrecht, The Netherlands, 1989; pp. 69–72. [Google Scholar]
- Mermin, N.D. What is Wrong with These Elements of Reality? Phys. Today 1990, 43, 9–11. [Google Scholar] [CrossRef]
- Bohm, D.; Hiley, B.J. The Undivided Universe. An Ontological Interpretation of Quantum Theory; Routledge: London, UK, 1993. [Google Scholar]
- Allori, V.; Goldstein, S.; Tumulka, R.; Zanghì, N. On the Common Structure of Bohmian Mechanics and the Ghirardi-Rimini-Weber Theory. Br. J. Philos. Sci. 2008, 59, 353–389. [Google Scholar] [CrossRef]
- Esfeld, M. From the Measurement Problem to the Primitive Ontology Programme. In Do Wave Functions Jump? Perspectives of the Work of GianCarlo Ghirardi; Allori, V., Bassi, A., Dürr, D., Zanghi, N., Eds.; Springer Nature: Cham, Switzerland, 2020; pp. 95–108. [Google Scholar]
- Lazarovici, D.; Reichert, P. The Point of Primitive Ontology. Found. Phys. 2022, 52, 120. [Google Scholar] [CrossRef]
- Dürr, D.; Goldstein, S.; Zanghì, N. Quantum Equilibrium and the Origin of Absolute Uncertainty. J. Stat. Phys. 1992, 67, 843–907. [Google Scholar] [CrossRef]
- Cowan, C.W.; Tumulka, R. Epistemology of Wave Function Collapse in Quantum Physics. Br. J. Philos. Sci. 2016, 67, 405–434. [Google Scholar] [CrossRef]
- Wilce, A. Quantum Logic and Probability Theory. In The Stanford Encyclopedia of Philosophy; Fall 2021 ed.; Zalta, E.N., Ed.; Metaphysics Research Lab, Stanford University: Stanford, CA, USA, 2021. [Google Scholar]
- Pitowsky, I. Quantum Mechanics as a Theory of Probability. In Physical Theory and Its Interpretation: Essays in Honor of Jeffrey Bub; Demopoulos, W., Pitowsky, I., Eds.; Springer: Dordrecht, The Netherlands, 2006; pp. 213–240. [Google Scholar] [CrossRef]
- von Meyenn, K. (Ed.) Eine Entdeckung von ganz Außerordentlicher Tragweite: Schrödingers Briefwechsel zur Wellenmechanik und zum Katzenparadoxon; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar] [CrossRef]
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Lazarovici, D. Hidden Variables in Quantum Mechanics from the Perspective of Boltzmannian Statistical Mechanics. Quantum Rep. 2024, 6, 465-481. https://doi.org/10.3390/quantum6030031
Lazarovici D. Hidden Variables in Quantum Mechanics from the Perspective of Boltzmannian Statistical Mechanics. Quantum Reports. 2024; 6(3):465-481. https://doi.org/10.3390/quantum6030031
Chicago/Turabian StyleLazarovici, Dustin. 2024. "Hidden Variables in Quantum Mechanics from the Perspective of Boltzmannian Statistical Mechanics" Quantum Reports 6, no. 3: 465-481. https://doi.org/10.3390/quantum6030031
APA StyleLazarovici, D. (2024). Hidden Variables in Quantum Mechanics from the Perspective of Boltzmannian Statistical Mechanics. Quantum Reports, 6(3), 465-481. https://doi.org/10.3390/quantum6030031