The Logico-Algebraic Approach to Quantum Mechanics

The twentieth century has witnessed a striking transformation in the understanding 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 orrter 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 structure, 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 manoeuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrodinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation, the elementary theory moved, flanked even at this later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic alternative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical structures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manipulation of purely abstract structures.

Inhalt
I. Formal Development of the Theory of Quantum Logic.- Spectral Theory in Quantum Logics.- Semantics of the Minimal Logic of Quantum Mechanics.- Representations of Groups as Automorphisms on Orthomodular Lattices and Posets.- The Conditional in Abstract and Concrete Quantum Logic.- On the Logical Structure of Quantum Mechanics.- II. Philosophy of Quantum Mechanics and Philosophy of Logic.- Matter, Space and Logic.- The Physics of Logic.- Complementarity, Context Dependence, and Quantum Logic.- Is Logic Empirical?.- III. Probability Theory and Quantum Logic.- Conditional Probabilities in Non-Boolean Possibility Structures.- Foundations of a Quantum Probability Theory.- Two Concepts of Probability in Physics.- IV. *-Algebras and Quantum Logic.- Foundations for Quantum Mechanics.- A Survey of Axiomatic Quantum Mechanics.- V. Quaternions, Quantification and Quantum Logic.- Notes On Quaternion Quantum Mechanics.- The Leibniz Project.