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 $|0_{\kappa,\gamma}\rangle$ 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 $\Delta\gamma$ between observers generates a nontrivial particle spectrum governed by $(\kappa,\Delta\gamma)$ even when $\kappa$ is held fixed. We then derive the two reciprocal Bogoliubov maps: the $\kappa\gamma$ plane-wave operators in terms of $\kappa'$-Rindler operators, and conversely the $\kappa'$-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 $|0_{\kappa,\gamma}\rangle$ is globally regular, with no singularities beyond those of the standard Minkowski vacuum.