Proceedings of the International Conference on Geometry, Integrability and Quantization

On New Ideas of Nonlinearity in Quantum Mechanics

Vasyl Kovalchuk

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Abstract

Our main idea is to suggest a new model of non-perturbative and geometrically motivated nonlinearity in quantum mechanics. The Schrödinger equation and corresponding relativistic linear wave equations derivable from variational principles are analyzed as usual self-adjoint equations of mathematical physics. It turns out that introducing the second-order time derivatives to dynamical equations, even as small corrections, can help to obtain the regular Legendre transformation. Following the conceptual transition from the special to general theory of relativity, where the metric tensor loses its status of the absolute geometric object and becomes included into degrees of freedom (gravitational field), in our treatment the Hilbert-space scalar product becomes a dynamical quantity which satisfies together with the state vector the system of differential equations. The structure of obtained Lagrangian and equations of motion is very beautiful, as usually in high-symmetry problems.

Article information

Source
Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, Andrei Ludu and Akira Yoshioka, eds. (Sofia: Avangard Prima, 2015), 195-206

Dates
First available in Project Euclid: 13 July 2015

Permanent link to this document
https://projecteuclid.org/ euclid.pgiq/1436815744

Digital Object Identifier
doi:10.7546/giq-16-2015-195-206

Mathematical Reviews number (MathSciNet)
MR3363845

Zentralblatt MATH identifier
1348.81041

Citation

Kovalchuk, Vasyl. On New Ideas of Nonlinearity in Quantum Mechanics. Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization, 195--206, Avangard Prima, Sofia, Bulgaria, 2015. doi:10.7546/giq-16-2015-195-206. https://projecteuclid.org/euclid.pgiq/1436815744


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