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arXiv:2509.04320 · Symmetry

A Generalized Nonlinear Extension of Quantum Mechanics

Alan Chodos, Fred Cooper

Published Nov 3, 2025• 2 citations
arXivHTMLPDF

Abstract

We construct the most general form of our previously proposed nonlinear extension of quantum mechanics that possesses three basic properties. Unlike the simpler model, the new version is not completely integrable, but it has an underlying Hamiltonian structure. We analyze a particular solution in detail, and we used a natural extension of the Born rule to compute particle trajectories. We find that closed particle orbits are possible.

DOI
10.3390/sym17111850

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We construct the most general form of our previously proposed nonlinear extension of quantum mechanics that possesses three basic properties. Unlike the simpler model, the new version is not completely integrable, but it has an underlying Hamiltonian structure. We analyze a particular solution in detail, and we used a natural extension of the Born rule to compute particle trajectories. We find that closed particle orbits are possible.
We uncover a previously unrecognized hierarchical algebraic structure in the Bose-Hubbard model that enables a systematic construction of its spectrum without relying on integrability. We develop a recursive framework in which, within each fixed particle-number sector, the eigenproblem of a p -site system is reduced to a closed set of algebraic relations determined entirely by the eigenpairs and single-particle transfer amplitudes of the ( p − 1 ) -site subsystem. This reduction yields two complementary sets of Bethe-type equations whose solutions exhaust the spectrum. The underlying structure originates from the locality of the Hamiltonian, which constrains the coupling between subsystems and gives rise to an emergent sparse, near-tridiagonal representation of the many-body problem. As an explicit demonstration, we construct the complete spectrum of the three-site Bose-Hubbard model from that of the two-site system for particle numbers n ≤ 3 , obtaining exact agreement with direct diagonalization. The resulting formulation substantially reduces computational cost and reveals a form of hierarchical solvability in a nonintegrable quantum many-body system. More broadly, our results suggest that recursive algebraic constructions may provide a general framework for analyzing lattice models beyond the conventional integrable paradigm.
The possibility to recover the which-way information, for example in the two slit experiment, is based on a natural but implicit assumption about the position of a particle {\it before} a position measurement is performed on it. This assumption cannot be deduced from the standard postulates of quantum mechanics. In the present paper this assumption is made explicit and formally postulated as a new rule, the {\it quantum typicality rule}. This rule correlates the positions of the particles at two different times, thus defining their trajectories. Unexpectedly, this rule is also equivalent to the Born rule with regard to the explanation of the results of statistical experiments. For this reason it can be considered a generalization of the Born rule. The existence of the quantum typicality rule strongly suggests the possibility of a new trajectory-based formulation of quantum mechanics. According to this new formulation, a closed quantum system is represented as a {\it quantum process}, which corresponds to a canonical stochastic process in which the probability measure is replaced by the wave function and the usual frequentist interpretation of probability is replaced by the quantum typicality rule.
About 15 years ago, we (Heinz-Dietrich Doebner and I) proposed a special type of nonlinear modification of the usual Schrödinger time-evolution equation in quantum mechanics. Our equation was motivated by certain unitary representations of the group of diffeomorphisms of physical space, in the framework of either nonrelativistic local current algebra or quantum Borel kinematics. Subsequently, we developed this and related approaches to nonlinearity in quantum mechanics considerably further, to incorporate theories of measurement, groups of nonlinear gauge transformations, symmetry and invariance properties, unification of a large family of nonlinear perturbations, and possible physical contexts for quantum nonlinearity. Some of our results and highlights of some open questions are summarized.
A lattice version of the quantum nonlinear Schrodinger (NLS) equation is considered, which has a significant simple form and fulfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS equations, the present model belongs to a class which does not exhibit the usual symmetry properties. However, this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectra. The model allows exact quantum solution through the algebraic Bethe ansatz and also a straightforward and natural generalization to the vector case, thus giving a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff-like q-boson operators is constructed.