 2  Author
 F. H. Fröhner  Requires cookie*   Title
 Missing Link Between Probability Theory and Quantum Mechanics: the RieszFejer Theorem    Abstract
 Quantum mechanics is spectacularly successful on the technical level but the meaning of its rules remains shrouded in mystery even more than seventy years after its inception. Quantummechanical probabilities are often considered as fundamentally different from classical probabilities, in disregard of the work of Cox (1946) and of Schrödinger (1947) on the foundations of probability theory. One central question concerns the superposition principle, i. e. the need to work with interfering wave functions, the absolute squares of which are probabilities. Other questions concern the relationship between spin and statistics or the collapse of the wave function when new data become available. These questions are reconsidered from the Bayesian point of view. The superposition principle is found to be a consequence of an apparently littleloiown mathematical theorem for nonnegative Fourier polynomials published by Fejer in 1915 that implies wavemechanical interference already for classical probabilities. Combined with the classical Hamiltonian equations for free and accelerated motion, gauge invariance and particle indistinguishability, it yields all basic quantum features waveparticle duality, operator calculus, uncertainty relations, Schrödinger equation, CPT invariance and even the spinstatistics relationship which demystifies quantum mechanics to quite some extent.   
Reference
 Z. Naturforsch. 53a, 637—654 (1998); received January 28 1998   
Published
 1998   
Keywords
 Superposition Principle, Wave Packets, Logical Inference, WaveParticle Duality, Quantum Mechanics   
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3  Author
 B. H. Lavenda  Requires cookie*   Title
 Is Relativistic Quantum Mechanics Compatible with Special Relativity?    Abstract
 The transformation from a timedependent random walk to quantum mechanics converts a modi fied Bessel function into an ordinary one together with a phase factor e,ir/2 for each time the electron flips both direction and handedness. Causality requires the argument to be greater than the order o f the Bessel function. Assuming equal probabilities for jumps ± 1 , the normalized modified Bessel function o f an imaginary argument is the solution o f the finite difference differential Schrödinger equation whereas the same function o f a real argument satisfies the diffusion equation. In the nonrelativistic limit, the stability condition o f the difference scheme contains the mass whereas in the ultrarelativistic limit only the velocity of light appears. Particle waves in the nonrelativistic limit become elastic waves in the ultrarelativistic limit with a phase shift in the frequency and wave number of 7r/2. The ordinary Bessel function satisfies a second order recurrence relation which is a finite difference differential wave equation, using nonnearest neighbors, whose solutions are the chirality components o f a freeparticle in the zero fermion mass limit. Reintroducing the mass by a phase transformation transforms the wave equation into the KleinGordon equation but does not admit a solution in terms o f ordinary Bessel functions. However, a sign change of the mass term permits a solution in terms o f a modified Bessel function whose recurrence formulas produce all the results of special relativity. The Lorentz transformation maximizes the integral o f the modified Bessel function and determines the paths o f steepest descent in the classical limit. If the definitions of frequency and wave number in terms o f the phase were used in special relativity, the condition that the frame be inertial would equate the superluminal phase velocity with the particle velocity in violation o f causality. In order to get surfaces o f constant phase to move at the group velocity, an integrating factor is required which determines how the intensity decays in time. The phase correlation between neighboring sites in quantum mechanics is given by the phase factor for the electron to reverse its direction, whereas, in special relativity, it is given by the Doppler shift.   
Reference
 Z. Naturforsch. 56a, 347—365 (2001); received February 14 2001   
Published
 2001   
Keywords
 Random Walks, Quantum Mechanics, Special Relativity, Ordinary and Modified Bessel Functions   
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