Maxwell’s equations and Lorentz force in doubly special relativity

Abstract

On the basis of all commutation relations of the $\kappa$-deformed phase space incorporating the $\kappa$-Minkowski space-time, we have derived in this paper an extended first approximation of both Maxwell’s equations and Lorentz force in doubly (or deformed) special relativity (DSR). For this purpose, we have used our approach of the special relativistic version of Feynman’s proof by which we have established the explicit formulations of electric and magnetic fields. As in Fock’s nonlinear relativity (FNLR), the laws of electrodynamics depend on the particle mass which therefore constitutes a common point between the two extended forms of special relativity. As one consequence, the corresponding equation of motion contains two different types of contributions. In addition to the usual type, another one emerges as a consequence of the coexistence of mass and charge which are coupled with the $\kappa$-deformation and electromagnetic field. This new effect completely induced by the $\kappa$-deformed phase space is interpreted as the gravitational-type Lorentz force. Unlike FNLR, the corrective terms all depend on the electromagnetic field in DSR.

Appendix

To find the explicit first-order form of the electromagnetic tensor in DSR, we first use Eqs. (26) and (33) to get

$$\begin{aligned} \left[ x^{\mu }, F^{\nu \lambda } \right] _{(1)}& = \left[ x^{\mu } + \delta x^{\mu }, F^{\nu \lambda } (x) + \delta F^{\nu \lambda }(x,{\dot{x}})\right] _{(1)}\nonumber \\& = \left[ x^{\mu }, {\dot{x}}^{\beta }\right] _{(0)}\frac{\partial \delta F^{\nu \lambda }}{\partial {\dot{x}}^{\beta }} + \frac{m}{\kappa }\left[ {\dot{x}}^{0}, x^{\alpha }\right] _{(0)}\frac{\partial F^{\nu \lambda }}{\partial x^{\alpha }} x^{\mu } \nonumber \\& = -i\hbar \left( \frac{1}{m}\frac{\partial \delta F^{\nu \lambda }}{\partial {\dot{x}}_{\mu }} - \frac{1}{\kappa }\partial ^{0} F^{\nu \lambda } x^{\mu }\right) . \end{aligned}$$

(63)

Furthermore, making use of Jacobi identity involving one coordinate and two velocities, we can make sure that Eq. (29) gives

$$\begin{aligned} \left[ x^{\mu }, F^{\nu \lambda } \right] _{(1)} = \frac{m^{2}}{i\hbar q}\left( \left[ {\dot{x}}^{\lambda }, \left[ x^{\mu }, {\dot{x}}^{\nu } \right] \right] _{(1)}- \left[ {\dot{x}}^{\nu }, \left[ x^{\mu }, {\dot{x}}^{\lambda } \right] \right] _{(1)}\right) . \end{aligned}$$

(64)

Taking into account Eq. (55) and the zeroth-order form of (29), we find

$$\begin{aligned}&\left[ {\dot{x}}^{\lambda }, \left[ x^{\mu }, {\dot{x}}^{\nu } \right] \right] _{(1)} = \frac{\hbar ^{2}q}{m^{2}\kappa }\Big \{-\eta ^{\mu 0} F^{\lambda \nu }\nonumber \\&\quad +\,\left( 2\eta ^{\mu \nu } \partial ^{0}A^{\lambda } + \eta ^{\lambda \nu } \partial ^{0}A^{\mu } + \eta ^{\lambda \mu } \partial ^{0}A^{\nu }\right) \nonumber \\&\quad +\,\left( \partial ^{\lambda }\partial ^{0} A^{\mu }x^{\nu } + \partial ^{\lambda }\partial ^{0} A^{\nu }x^{\mu }\right) \Big \}, \end{aligned}$$

(65)

which, after interchanging $\lambda$ with $\nu$, gives

$$\begin{aligned}&\left[ {\dot{x}}^{\nu }, \left[ x^{\mu }, {\dot{x}}^{\lambda } \right] \right] _{(1)} = \frac{\hbar ^{2}q}{m^{2}\kappa }\Big \{-\eta ^{\mu 0} F^{\nu \lambda } \nonumber \\&\quad +\,\left( 2\eta ^{\mu \lambda } \partial ^{0}A^{\nu } + \eta ^{\nu \lambda } \partial ^{0}A^{\mu } + \eta ^{\nu \mu } \partial ^{0}A^{\lambda }\right) \nonumber \\&\quad +\,\left( \partial ^{\nu }\partial ^{0} A^{\mu }x^{\lambda } + \partial ^{\nu }\partial ^{0} A^{\lambda }x^{\mu }\right) \Big \}. \end{aligned}$$

(66)

Substituting Eqs. (65) and (66) into (64), we obtain

$$\begin{aligned}&\left[ x^{\mu }, F^{\nu \lambda } \right] _{(1)} = - \frac{i\hbar }{\kappa }\Big \{2\eta ^{\mu 0} F^{\nu \lambda } -\partial ^{0}F^{\nu \lambda }x^{\mu } \nonumber \\&\quad +\,\left( \eta ^{\mu \nu } \partial ^{0}A^{\lambda } -\eta ^{\mu \lambda } \partial ^{0}A^{\nu }\right) \nonumber \\&\quad +\,\left( \partial ^{\lambda }\partial ^{0} A^{\mu }x^{\nu } - \partial ^{\nu }\partial ^{0} A^{\mu }x^{\lambda }\right) \Big \}. \end{aligned}$$

(67)

Finally, after identification of (67) with (63) and integration of the resulting equation, the generalized electromagnetic tensor takes the following form

$$\begin{aligned}&F^{\nu \lambda }_{(1)}(x, {\dot{x}}) = F^{\nu \lambda }(x) + \frac{m}{\kappa }\left\langle \Big \{ 2\eta ^{\mu 0}F^{\nu \lambda } + \left( \eta ^{\mu \nu }\partial ^{0}A^{\lambda } - \eta ^{\mu \lambda }\partial ^{0}A^{\nu }\right) \right. \nonumber \\&\quad +\,\left. \left( \partial ^{\lambda }\partial ^{0}A^{\mu }x^{\nu } - \partial ^{\nu }\partial ^{0}A^{\mu }x^{\lambda }\right) \Big \}{\dot{x}}_{\mu } \right\rangle . \end{aligned}$$

(68)

From above, it is obvious that $F^{\nu \lambda }_{(1)}(x, {\dot{x}})$ is antisymmetric with respect to the permutation of indices $\nu$ and $\lambda$. Apart from the doubled corrective term, we remark also that the other terms vanish in the static case.