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 $q \rightarrow 1$. 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.
Highlights
The Proca equation [18] constitutes, along with the Dirac and the Klein–Gordon equations, one of the fundamental relativistic wave equations [19]
We proposed a generalized Lagrangian that leads to a nonlinear extension of the Proca field equation
We discussed some of the main features of this nonlinear field theory, focusing on the existence of exact time-dependent, localized solutions of the q-plane wave form
Summary
The Proca equation [18] constitutes, along with the Dirac and the Klein–Gordon equations, one of the fundamental relativistic wave equations [19]. The nonlinear Schrödinger, Dirac, and Klein–Gordon equations investigated in [1,2,3,4,5,6,7] share the physically appealing property of admitting (in the case of vanishing interactions) exact soliton-like localized solutions that behave as free particles, in the sense of complying with the celebrated Einstein–Planck-de Broglie relations connecting frequency and wave number, respectively, with energy and momentum Given these previous developments, it is natural to ask if the corresponding nonlinear extension can be implemented for massive vector bosons.
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