Classical and Quantum probabilities

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SUMMARY

The discussion centers on the distinctions between classical (Kolmogorov axiomatization) and quantum probabilities, particularly in the context of statistical results from experiments like the double-slit experiment. Participants assert that while classical probabilities arise from a lack of information, quantum probabilities are fundamental characteristics of nature. The conversation emphasizes that the mathematical frameworks for both types of probabilities are valid, but the physical assumptions underlying them differ significantly, particularly regarding the interpretation of events and measurements. The consensus is that classical probability does not invalidate quantum mechanics but rather highlights the limitations of classical assumptions in quantum contexts.

PREREQUISITES
  • Understanding of Kolmogorov probability theory
  • Familiarity with quantum mechanics principles, particularly the double-slit experiment
  • Knowledge of Bayesian vs. frequentist statistics
  • Basic grasp of Bell's inequalities and their implications
NEXT STEPS
  • Explore the implications of Bell's inequalities in both classical and quantum contexts
  • Study the differences between Bayesian and frequentist statistics in depth
  • Investigate the role of measurement in quantum mechanics and its impact on probability
  • Learn about the foundational aspects of quantum mechanics, focusing on the interpretation of probabilities
USEFUL FOR

Physicists, mathematicians, statisticians, and anyone interested in the foundational differences between classical and quantum probability frameworks.

  • #31
slyboy said:
Bohmian mechanics is a good example of the kind of thing I am talking about. There your ontic state (state of reality) is just the particle positions and the quantum state vector. You just have an ordinary Kolmogorv distribution over particle positions.

Yes, this is the part of Bohmian mechanics that is sufficient to show that you do not NEED any "quantum probability rules" that go beyond Kolmogorov probability ; the very fact that it is possible to construct a model that gives you the same predictions as QM using Kolmogorov probability is sufficient to show that. Don't understand me wrong: I didn't want to say that Bohmian mechanics must therefore be "right" or whatever ; it is just that the very existence of that model, giving QM probabilities, and obeying Kolmogorov probabilities, disproves the statement that you cannot have QM probabilities satisfying Kolmogorov's axioms.

However, there are perfectly good Kolmogorov distributions over position that are not allowed in this theory, since they have to obey the ``equilibrium hypothesis'' in order to agree with the predictions of QM, i.e. the prob. distribution must be the one coming from the Born rule applied to the state vector.

Yes, this is correct ; however, it doesn't invalidate the claim that QM probabities CAN be generated by a system obeying Kolmogorov's axioms. If that system can also generate other probabilities (not those of QM) doesn't matter. The very fact that with the right distributions, you DO get out the QM predictions, is enough.

Just as a side note: this doesn't mean I endorse Bohmian mechanics. I have mixed feelings towards it, my main complaint being that the guiding equations do not obey the imposed symmetries on the wave dynamics (like Lorentz invariance). But I do think that it is very useful to study Bohmian mechanics because it is a model that disproves a lot of claims about QM, like the one we're discussing here, namely that "quantum probabilities" are somehow not "normal probabilities as we know them" (Kolmogorov axioms).

cheers,
Patrick.
 

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