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On new ideas of nonlinearity in quantum mechanics. (English) Zbl 1348.81041

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 16th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 6–11, 2014. Sofia: Avangard Prima. 195-206 (2015).
Summary: 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.
For the entire collection see [Zbl 1317.00020].

MSC:

81P05 General and philosophical questions in quantum theory
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
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