Quantum Mechanical Clock and Classical Relativistic Clock

A cyclic nature of quantum mechanical clock is discussed as “quantization of time.” Quantum mechanical clock is seen to be equivalent to the relativistic classical clock. John von Neumann proposed a problem [1] in 1932 in the footnote on page 6 of the English translation [2] of his book “Die Mathematische Grundlagen der Quantenmechanik,” Springer-Verlag, Berlin: – in all attempts to develop a general and relativistic theory of electromagnetism, in spite of noteworthy partial successes, the theory (of Quantum Mechanics) seems to lead to great difficulties, which apparently cannot be overcome without the introduction of wholly new ideas. I will give a solution below in the case of one particle in 3 dimensional space R first. After then I will add a sketchy explanation of general N -particle case, which is almost obvious by the first part. I. One particle case with mass m We consider this one particle in the universe expressed as a vector bundle X×R6 with base space X being the curved Riemannian space where relativistic CM (classical mechanics) holds and with a fibre R (a phase space associated to each point x ∈ X) on which QM (quantum mechanics) holds as in [3], [4]. Then inside the QM system of the one particle (which we call a “local system” of the particle), the particle follows the Schrödinger equation when the local time t of the system is given as in definition 1 below, and in