Phase-Induced Particle Creation in the Kappa-Gamma Vacuum

We develop a two-parameter family of flat-spacetime modes labeled by a deformation scale $\kappa$ and a phase angle $\gamma$, extending the $\kappa$-plane wave framework to include complex squeezing. The resulting $\kappa \gamma$ basis provides a globally well-defined mode decomposition whose associated vacuum $\mathrm{\mid }{0}_{\kappa ,\gamma }⟩$ is a continuous-mode phase-squeezed state: $\kappa$ fixes the squeezing magnitude, while $\gamma$ sets the squeezing angle in phase space. We identify phase-induced particle creation, in which a relative phase mismatch $\mathrm{\Delta }\gamma$ between observers generates a nontrivial particle spectrum governed by $(\kappa ,\mathrm{\Delta }\gamma )$ even when $\kappa$ is held fixed. We then derive the two reciprocal Bogoliubov maps: the $\kappa \gamma$ plane-wave operators in terms of ${\kappa }^{\mathrm{\prime }}$-Rindler operators, and conversely the ${\kappa }^{\mathrm{\prime }}$-Rindler operators in terms of $\kappa \gamma$ plane-wave operators, providing a closed algebraic bridge between these bases. Finally, by analyzing the Wightman function we show that $\mathrm{\mid }{0}_{\kappa ,\gamma }⟩$ is globally regular, with no singularities beyond those of the standard Minkowski vacuum.