The Logico-Algebraic Approach to Quantum Mechanics

The twentieth century has witnessed a striking transformation in the un­ derstanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi­ tional opinion that one should look to the specific applications of a theory in order to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that struc­ ture, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical maneuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrödinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation the elementary theory moved, flanked even at the later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic altemative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical struc­ tures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manip­ ulation of purely abstract structures.

Inhalt

The Logic of Quantum Mechanics (1936).- The Logic of Complementarity and the Foundation of Quantum Theory (1972).- Mathematics as Logical Syntax - A Method to Formalize the Language of a Physical Theory (1937-38).- Three-Valued Logic and the Interpretation of Quantum Mechanics (1944).- Three-Valued Logic (1957).- Reichenbach's Interpretation of Quantum Mechanics (1958).- Measures on the Closed Subspaces of a Hilbert Space (1957).- The Logic of Propositions Which are not Simultaneously Decidable (1960).- Baer -Semigroups (1960).- Axioms for Non-Relativistic Quantum Mechanics (1961).- Probability in Physics and a Theorem on Simultaneous Observability (1962).- Semantic Representation of the Probability of Formulas in Formalized Theories (1963).- The Structure of the Propositional Calculus of a Physical Theory (1964).- Boolean Embeddings of Orthomodular Sets and Quantum Logic (1965).- Logical Structures Arising in Quantum Theory (1965).- The Calculus of Partial Propositional Functions (1965).- The Problem of Hidden Variables in Quantum Mechanics (1967).- Logics Appropriate to Empirical Theories (1965).- The Probabilistic Argument for a Non-Classical Logic of Quantum Mechanics (1966).- Foundations of Quantum Mechanics (1967).- Baer -Semigroups and the Logic of Quantum Mechanics (1968).- Semimodularity and the Logic of Quantum Mechanics (1968).- On the Structure of Quantum Logic (1969).- On the Structure of Quantal Proposition Systems (1969).- The Current Interest in Orthomodular Lattices (1970).- Integration Theory of Observables (1970).- Probabilistic Formulation of Classical Mechanics (1970).- Atomicity and Determinism in Boolean Systems (1971).- Survey of General Quantum Physics (1972).- Quantum Logics (1974).- The Labyrinth of Quantum Logics (1974).