In this monograph, we shall present a new mathematical formulation
of quantum theory, clarify a number of discrepancies within the
prior formulation of quantum theory, give new applications to
experiments in physics, and extend the realm of application of
quantum theory well beyond physics. Here, we motivate this new
formulation and sketch how it developed. Since the publication of
Dirac's famous book on quantum mechanics [Dirac, 1930] and von
Neumann's classic text on the mathematical foundations of quantum
mechanics two years later [von Neumann, 1932], there have appeared
a number of lines of development, the intent of each being to
enrich quantum theory by extra polating or even modifying the
original basic structure. These lines of development have seemed to
go in different directions, the major directions of which are
identified here: First is the introduction of group theoretical
methods [Weyl, 1928; Wigner, 1931] with the natural extension to
coherent state theory [Klauder and Sudarshan, 1968; Peremolov,
1971]. The call for an axiomatic approach to physics [Hilbert,
1900; Sixth Problem] led to the development of quantum logic
[Mackey, 1963; Jauch, 1968; Varadarajan, 1968, 1970; Piron, 1976;
Beltrametti & Cassinelli, 1981], to the creation of the
operational approach [Ludwig, 1983-85, 1985; Davies, 1976] with its
application to quantum communication theory [Helstrom, 1976;
Holevo, 1982), and to the development of the C* approach [Emch,
1972]. An approach through stochastic differential equations
("stochastic mechanics") was developed [Nelson, 1964, 1966, 1967].
General
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