Foundations for Quantum Mechanics

A discussion is given of the structure of a physical theory and an ``ideal form'' for such a theory is proposed. The essential feature is that all concepts should be defined in operational terms. Quantum (and classical) mechanics is then formulated in this way (the formulation being, however, restricted to the kinematical theory). This requires the introduction of the concept of a mixed test, related to a pure test (or ``question'') just as a mixed state is related to a pure state. In the new formulation, the primitive concepts are not states and observables but certain operationally accessible mixed states and tests called physical. The notion of a C*‐system is introduced; each such system is characterized by a certain C*‐algebra. The structure of a general C*‐system is then studied, all concepts being defined in terms of physical states and tests. It is shown first how pure states and tests can be so defined. The quantum analog of the phase space of classical mechanics is then constructed and on it is b...

[1]  H. Araki.  On representations of the canonical commutation relations , 1971 .

[2]  M. Berger.  On one parameter families of real solutions of nonlinear operator equations , 1969 .

[3]  R. Morrow,et al.  Foundations of Quantum Mechanics , 1968 .

[4]  R. Plymen.  C*-algebras and Mackey's axioms , 1968 .

[5]  E. Davies.  On the Borel structure ofC*-algebras , 1968 .

[6]  Edwin Hewitt,et al.  Real And Abstract Analysis , 1967 .

[7]  R. Kadison,et al.  Derivations and automorphisms of operator algebras , 1967 .

[8]  Richard V. Kadison,et al.  Transformations of states in operator theory and dynamics , 1965 .

[9]  R. Giles,et al.  Mathematical Foundations of Thermodynamics , 1964 .

[10]  R. Haag,et al.  An Algebraic Approach to Quantum Field Theory , 1964 .

[11]  Leonard Gillman,et al.  Rings of continuous functions , 1961 .

[12]  J. Glimm,et al.  Unitary operators in $C^{\ast}$-algebras. , 1960 .

[13]  R. Kadison.  Unitary Invariants for Representations of Operator Algebras , 1957 .

[14]  R. Kadison.  A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras , 1952 .

[15]  George W. Mackey,et al.  A theorem of Stone and von Neumann , 1949 .

[16]  I. Segal.  Postulates for General Quantum Mechanics , 1947 .

[17]  I. Segal.  Irreducible representations of operator algebras , 1947 .

[18]  R. Schatten,et al.  Norm Ideals of Completely Continuous Operators , 1970 .

[19]  V. Varadarajan.  Geometry of quantum theory , 1968 .

[20]  R. Plymen.  A modification of Piron's axioms , 1968 .

[21]  F. Lurçat.  Application of Mathematics to Problems in Theoretical Physics , 1967 .

[22]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[23]  Garrett Birkhoff,et al.  Lattice Theory Revised Edition , 1948 .