A Generalized Nonlinear Extension of Quantum Mechanics

TL;DR

This work develops a generalized nonlinear extension of quantum mechanics based on a two-state-vector formalism with eight real parameters governing the nonlinear coupling, all embedded in an underlying Hamiltonian structure. By recasting the dynamics in terms of a tau-delta pair, the authors derive a one-dimensional Hamiltonian system with potential $V(\kappa)=4e^{\kappa}+ c e^{p\kappa}$ and energy $h=N^2$, revealing exact and simple-solution regimes, including a solvable case with $p=-1$ yielding Jacobi-elliptic solutions and a stationary-$\tau$ branch. The analysis shows that the extended density matrix acquires nonlinear evolution, leading to non-purity in general and to trajectory physics for observables via $\langle X(t)\rangle=\mathrm{Tr}(\rho X)$, including planar, harmonic closed orbits with frequency $2\sigma$. The paper discusses implications for gravity-inspired physics, potential no-signaling concerns, and proposes practical approximations to guide future analytic progress toward a fuller solution of the generalized model.

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 use a natural extension of the Born rule to compute particle trajectories. We find that closed particle orbits are possible.

Figures (3)

• Figure 1: $\kappa (s)$ for $N^2=25,c=4$
• Figure 2: $\tau (s)$ for $N^2=25,c=4$
• Figure 3: $\delta (s)$ for $N^2=25,c=4$