Quantum Logic and Probability Theory First published Mon Feb 4, 2002; substantive revision Mon Aug 27, 2012 At its core, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic. More specifically, in quantum mechanics each probability-bearing proposition of the form “the value of physical quantity A lies in the range B” is represented by a projection operator on a Hilbert space H. These form a non-Boolean—in particular, non-distributive—orthocomplemented lattice. Quantum-mechanical states correspond exactly to probability measures (suitably defined) on this lattice. What are we to make of this? Some have argued that the empirical success of quantum mechanics calls for a revolution in logic itself. This view is associated with the demand for a realistic interpretation of quantum mechanics, i.e., one not grounded in any primitive notion of measurement. Against this, there is a long tradition of interpreting quantum mechanics operationally, that is, as being precisely a theory of measurement. On this latter view, it is not surprising that a “logic” of measurement-outcomes, in a setting where not all measurements are compatible, should prove not to be Boolean. Rather, the mystery is why it should have the particular non-Boolean structure that it does in quantum mechanics. A substantial literature has grown up around the programme of giving some independent motivation for this structure—ideally, by deriving it from more primitive and plausible axioms governing a generalized probability theory. 1. Quantum Mechanics as a Probability Calculus Supplement: The Basic Theory of Ordering Relations 2. Interpretations of Quantum Logic 3. Generalized Probability Theory 4. Logics Associated to Probabilistic Models 5. Pirons Theorem 6. Classical Representations 7. Composite Systems Bibliography Academic Tools Other Internet Resources Related Entries Read more:
Posted on: Tue, 14 Oct 2014 01:32:55 +0000
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