Generalizations of the Klein–Gordon and the Dirac Equations from Non-standard Lagrangians

Abstract

We present various generalizations of the Klein–Gordon and Dirac formalisms based on non-standard Lagrangians within the framework of the calculus of variations characterized by a power-law Lagrangian \( {\mathsf{L}}^{1 + \gamma},\,\gamma \) being a free parameter. In the case of the bosonic scalar field, the modified dispersion relation has been derived and based on this, it was observed that for a particular choice of non-standard Lagrangians, the new field theory forbid the presence of massless particles. Besides, the Klein–Gordon equation is modified and becomes similar to the Barut equation which is a second order equation in the \( ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2},0) \oplus (0,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}) \) representation of the Lorentz group, which explains the splitting of leptons. For the case of the spinor scalar field, the Barut-like equation was derived from a non-standard Lagrangian as well. For some specific class of non-standard Lagrangians, the modified dispersion is modified and prohibits the presence of massless particles.