Generalized Nonlinear Proca Equation and its Free-Particle Solutions

Abstract

We introduce a non-linear 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 Schroedinger, 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 $q\to 1$. We derive the nonlinear Proca equation from a Lagrangian that, 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 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.

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Cited by in corpus (2)

• Lorentzian Solitary Wave in a Generalised Nonlinear Schrödinger Equation
• Symplectic quantization of multi-field Generalized Proca electrodynamics