Abstract
All quantum theories have one thing in common: they agree that chance (or randomness) should feel at home in the theory. It is sometimes claimed that quantum randomness, or equivalently quantum probability, is irreducible, by which it is meant that there is just no way out of it. Quantum mechanical probability is expressed in Born’s statistical interpretation of the wave function and this is frequently taken as an axiom. Then nothing more can be said, unless one feels unhappy about the axiom. And indeed, there are reasons for not being happy about it.
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Notes
- 1.
From: “Remarks concerning the essays brought together in this co-operative volume”. In P.A. Schilpp (Ed.), Albert Einstein Philosopher–Scientist. MJF books, New York, 1949, p. 672.
- 2.
M. Kac, Statistical Independence in Probability, Analysis and Number Theory. The Carus Mathematical Monographs, 1959, S. 22.
- 3.
λ is a standard notation for Lebesgue measure, but we do not need the full story here.
- 4.
In classical physics, the symbol Γ is used for phase space, following Boltzmann. In the mathematical foundations of probability theory, Andrey Nikolaevich Kolmogorov (1903–1987) introduced instead the symbol Ω, which may be more suggestive of the “fundamental set”.
- 5.
Introduced by William Rowan Hamilton (1805–1865) and denoted by H in honour of Christiaan Huygens (1629–1695).
- 6.
Flow lines (integral curves) can never cross each other since they are defined by a vector field. If they did, then at a crossing point, there would be two tangential “velocity” vectors, which is impossible for a vector field unless the vector field is zero at that point.
- 7.
That does not mean that a theory which is considered fundamental at the present time cannot be superseded by another one. For example, classical physics gets incorporated into quantum physics, and the realm of validity of the old theory is understood in the context of the new fundamental theory (see Sect. 1.6).
- 8.
We know that the stationarity requirement need not yield a unique measure. This should not be viewed as a failure of the typicality idea per se. In more technical terms, a typicality measure is really a representative of an equivalence class of measures, which are absolutely continuous with respect to each other. For stationary measures which are not absolutely continuous with respect to each other (like the microcanonical and canonical measures), we need to apply further insights, for example, that the universe is a “closed” system which does not exchange energy with some “outside” system, so that the total energy of the universe is fixed. In quantum mechanics, by the way, we shall have uniqueness of the typicality measure.
- 9.
Compare the instructive computation in D. Dürr and S. Teufel, Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, 2009.
- 10.
See, for instance, D. Lazarovici and P. Reichert, Arrow(s) of Time without a Past Hypothesis, in: V. Allori (ed.), Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature, World Scientific, 2020 and S. Carroll: From Eternity to Here, Dutton, New York, 2010.
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Dürr, D., Lazarovici, D. (2020). Chance in Physics. In: Understanding Quantum Mechanics . Springer, Cham. https://doi.org/10.1007/978-3-030-40068-2_3
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DOI: https://doi.org/10.1007/978-3-030-40068-2_3
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