Generalized nonlinear Proca equation and its free-particle solutions

We introduce a nonlinear extension of Proca’s field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schrödinger, Dirac, and Klein–Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit $$q \rightarrow 1$$. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field $$\Psi ^{\mu }(\vec {x},t)$$, involves an additional field $$\Phi ^{\mu }(\vec {x},t)$$. We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation $$E^{2} = p^{2}c^{2} + m^{2}c^{4}$$ for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed