Logic Of Quantum Mechanics Research Articles

An orthomodular poset which does not admit a normed orthovaluation

It is of relevance to studies in the logic of quantum mechanics whether or not every separable completely orthomodular poset admits a normed σ-ortho-valuation. A finite orthomodular poset is constructed which is a counter-example to this proposition.

Semimodularity and the logic of quantum mechanics

If (ℰ, ℒ,P, Ω) is an event-state-operation structure, then the events form an orthomodular ortholattice (ℰ, ≦, ′) and the operations, mappings from the set of states ℒ into ℒ, form a Baer *-semigroup (SΩ, ℴ, *, ′). Additional axioms are adopted which yield the existence of a homomorphism ϑ from (SΩ, ℴ, *, ′) into the Baer *-semigroup (S(ℰ), ℴ, *, ′) of residuated mappings of (ℰ, ≦, ′) such thatx ∈ SΩ maps states while ϑ x ∈S (ℰ) maps supports of states. If (ℰ, ≦, ′) is atomic and there exists a correspondence between atoms and pure states, then the existence of ϑ provides the result: (ℰ, ≦, ′) is semimodular if and only if every operationx ∈ SΩ is a pure operation (maps pure states into pure states).

Birkhoff and von Neumann's Interpretation of Quantum Mechanics

Birkhoff and von Neumann's famous paper, “The Logic of Quantum Mechanics”, culminates in a proposal which clashes with each of a number of assumptions made by the authors. A thought experiment, intended to show the need for the proposal, is rejected.

Baser*-semigroups and the logic of quantum mechanics

The theory of orthomodular ortholattices provides mathematical constructs utilized in the quantum logic approach to the mathematical foundations of quantum physics. There exists a remarkable connection between the mathematical theories of orthomodular ortholattices and Baer *-semigroups; therefore, the question arises whether there exists a phenomenologically interpretable role for Baer *-semigroups in the context of the quantum logic approach. Arguments, involving the quantum theory of measurements, yield the result that the theory of Baer *-semigroups provides the mathematical constructs for the discussion of “operations” and conditional probabilities.

The Logic of Quantum Mechanics

One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).