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Abstract. We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.

Generalized coherent states (GCs) under deformed quantum mechanics which exhibit intrinsic minimum length and maximum momentum have been well studied following Gazeau-Klauder approach in Refs. [1][2][3][4]. In this paper, as an extension to the study of quantum deformation, we investigate the famous Schrödinger cat states (SCs) under these two classes of quantum deformation. Following the concept of generalized Gazeau-Klauder Schrödinger cat states (GKSCs) in [5], we construct the deformed-GKSCs for both phenomenological models that exhibit intrinsic minimum length and/or maximum momentum. All comparisons between minimum length and maximum momentum deformations are illustrated and plots are done in even and odd cat states since they are one of the most important classic statistical characteristics of SCs. Probability distribution and entropies are studied. In general, deformed cat states do not possess the original even and odd states statistical properties. Nonclassical properties of the deformed-GKSCs are explored in terms of Mandel Q parameter, quadrature squeezing (∆X q ) · (∆Y q ) as well as Husimi quasi-probability distribution Q. Some of these distinguishing quantum-gravitational features may possibly be realized qualitatively and even be measured quantitatively in future experiments with the advanced development in quantum atomic and optics technology.

We investigate potential quantum nonlinear corrections to Dirac's equation through its sub-leading effect on neutrino oscillation probabilities. Working in the plane-wave approximation and in the µ − τ sector, we explore various classes of nonlinearities, with or without an accompanying Lorentz violation. The parameters in our models are first delimited by current experimental data before they are used to estimate corrections to oscillation probabilities. We find that only a small subset of the considered nonlinearities have the potential to be relevant at higher energies and thus possibly detectable in future experiments. A falsifiable prediction of our models is an energy dependent effective mass-squared, generically involving fractional powers of the energy. *

We examine the nonperturbative effect of instrinsic maximum momentum on the relativistic wave equations. Using the momentum representation, we obtain the exact eigen-energies and wavefunctions of one-dimensional Klein-Gordon and Dirac equation with linear confining potentials, and Dirac oscillator. Similar to the undeformed case, bound state solutions are only possible when the strength of scalar potential are stronger than vector potential. The energy spectrum of the systems studied are bounded from above, whereby classical characteristics are observed in the uncertainties of position and momentum operators. Also, there is a truncation in the maximum number of bound states that is allowed. Some of these quantum-gravitational features may have relevant applications in quarkonium confinement and quantum gravity phenomenology.