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Dieter Schuch
J. W. Goethe Universität, Frankfurt a. M., DE

Is Quantum Mechanics Emerging from a Nonlinear Theory?

Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated. This also leads to comparisons with susy quantum mechanics, dynamical invariants and generalized creation/annihilation operators with corresponding coherent states. Furthermore, the time-independent Schrödinger equation can also be rewritten as complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, from nonlinear dynamics via statistical thermodynamics to cosmology.

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