Generalized nonlinear Proca equation and its free-particle solutions

Abstract

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 → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field μ ( x, t), involves an additional field μ ( 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 qgeneralized 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.

Tsallis’ quantum q-fields

“…We construct the CFTs corresponding to the q-Schrödinger, q-Klein-Gordon, and q-Dirac equations introduced in [1,2,3,4]. We do the same for the q-Proca y q-Yang-Mills (Abelian) defined in [5]. Also, and for the first time ever, we deal with the equation and q-QFT corresponding to a non Abelian q-Yang-Mills field.…”

“…Classical field theories (CFT) associated with Tsallis' q-scenarios have received much attention recently [1][2][3][4][5][6]. Associated q-quantum field theories (q-QFTs) have also been discussed [2,3].…”

“…We do the same for the q-Proca and q-Yang-Mills (Abelian) defined in Ref. [5]. Also, and for the first time ever, we deal with the equation and q-QFT corresponding to a non-Abelian q-Yang-Mills field.…”

Quantum q-field theory: q-Schrödinger and q-Klein-Gordon fields

“…Here we evaluate the selfenergy and propagator up to ssecond order, thus generalizing results of [24]. In this respect, note also recent work on Proca-de Broglies' classical field theory [22].…”

Non linear spin 1 field equation in curved space-time: separation and solution in static RW metric

“…Another open problem of interest is of determining explicit solution of the field equation, at least for the lowest spin values, in some specific space time model. (For recent models see e.g., [8]; for the non linear Dirac equation see e.g., [15,16,17]).…”