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International Journal of Theoretical Physics

On the Mathematical Basis of the Dirac Formulation of Quantum Mechanics

Manuel Gadella, F. Gómez-Cubillo

Published Oct 1, 2003• 49 citations
Publisher
DOI
10.1023/b:ijtp.0000005956.11617.e9

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Enseignement

CERN

In an attempt to give an exact mathematical formulation of Bohr*s Correspondence Principle, Heisenberg (June 1925) discovered the rules governing the behaviour of quantum-theoretical magnitudes.In fall 1925 Born, Heisenberg and Jordan and, independently, Dirac, formulated consistent algebraic schemes of quantum mechanics.Early in 1926 Schrodinger developed wave mechanics.In quick succession were discovered : Bora's probability interpretation of the wave function, the transformation theory of Dirac, Jordan and E. London, Heisenberg's uncertainty relations and Bohr's principle of complementarity.By September 1927 the basis of a complete theory of atomic phenomena had been established.Aspects of this development, in which Heisenberg played a central role, will be presented as a tribute to his memory.

•Jan 1, 1976•0 citations
In this paper the results from various areas of mathematical research which are necessary for a consistent unification of the Dirac and von Neumann formulations of quantum mechanics are collected and presented as a single synthesis. For this purpose, direct integral decompositions of Hilbert space must be introduced into Dirac's formulation of spectral theory and representation theory; true unit vectors in the direct integral decomposition spaces replace unnormalizable vectors of infinite length. It then becomes clear that families of modified Dirac projection operators are simply related to the Radon-Nikodym derivative of von Neumann spectral measures. In terms of these mathematical preliminaries a second paper will present the more physical aspects of the resulting unified formulation of quantum mechanics.
In an attempt to give an exact mathematical formulation of Bohr*s Correspondence Principle, Heisenberg (June 1925) discovered the rules governing the behaviour of quantum-theoretical magnitudes.In fall 1925 Born, Heisenberg and Jordan and, independently, Dirac, formulated consistent algebraic schemes of quantum mechanics.Early in 1926 Schrodinger developed wave mechanics.In quick succession were discovered : Bora's probability interpretation of the wave function, the transformation theory of Dirac, Jordan and E. London, Heisenberg's uncertainty relations and Bohr's principle of complementarity.By September 1927 the basis of a complete theory of atomic phenomena had been established.Aspects of this development, in which Heisenberg played a central role, will be presented as a tribute to his memory.