P.A.M. Dirac (1930) and J. Von Neumann (1932), books on quantum mechanics

This chapter discusses P.A.M. Dirac's and J. Von Newmann's works on classical mechanics. These works are the classic physicist's treatise on the quantum mechanical transformation theory, which is a generalization of the matrix-mechanical and wave-mechanical quantum theories of Werner Heisenberg and Erwin Schrödinger, respectively. These works formulated an alternative mathematical theory of quantum mechanics, known as quantum algebra. Dirac generalized the matrix formulation of quantum mechanics, in which he called the non-commuting quantum variables q-numbers, distinguishing them from the classical commuting quantities that he called c-numbers. In 1927, Dirac published a quantum theory of the electromagnetic field interacting with electrons, the so-called quantum electrodynamics (QED). Dirac's first chapter considers the principle of superposition, which is different from the classical notion of superposition of waves. While Hilbert had shown in 1906 that every bounded symmetrical bilinear form possessed a unique representation with a unique set of eigenvalues, this was not yet clear for the unbounded symmetrical operators occurring in von Neumann's space H for quantum mechanics. Nature agrees with quantum mechanics as the proper description of atomic processes, and preserves the important contributions contained in von Neumann's Mathematical foundations.