Phase-Induced Particle Creation in the Kappa-Gamma Vacuum

TL;DR

This paper develops a two-parameter κγ plane-wave framework for flat-spacetime quantum fields, extending prior κ-plane waves by introducing a phase degree of freedom that yields continuous-mode phase squeezing. It shows phase-induced particle creation: observers with different phase choices γ experience Bogoliubov mixing characterized by $N_Λ(γ',γ) = \frac{\sin^2(Δγ)}{\sinh^2(\frac{πΛ}{κ})}$, while the κ parameter sets squeezing strength and the associated thermal scale $T = κ/(2π)$. The authors construct a comprehensive set of generalized Bogoliubov transformations (the master map) linking κγ plane waves, κ'-Rindler modes, and Minkowski modes, and they compute the Wightman function $W_{κγ}(x,x')$, showing the vacuum $|0_{κ,γ}\rangle$ is Hadamard with no extra UV or horizon divergences. Interpreting $|0_{κ,γ}\rangle$ as a continuous-mode phase-squeezed state clarifies the role of γ as a global squeezing axis, and the boundary-modulated accelerating-mirror realization provides a physical route to realize these vacua and their observable phase-dependent effects.

Abstract

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

Figures (2)

• Figure 1: Structure of the $\kappa\gamma$-mode for a representative frequency of $\Lambda=1.0$. The panels show the real part of the wave function for different combinations of the deformation parameter $\kappa$ and the phase factor $\gamma$. The top row corresponds to a smaller deformation ($\kappa=0.5$), while the bottom row shows a larger deformation ($\kappa=5.0$). The columns represent phase shifts of $\gamma=0$ (left) and $\gamma=\pi/2$ (right).
• Figure 2: Phase-induced particle creation for different vacua. Each panel shows the particle number spectrum $N_\Lambda$ on a logarithmic scale as a function of frequency $\Lambda$ and phase difference $\Delta\gamma$ for a fixed value of $\kappa$. For small $\kappa$, particle creation is suppressed at all but the lowest frequencies; as $\kappa$ increases, the effect strengthens and extends to higher $\Lambda$, with the maximum always at $\Delta\gamma=\pi/2$.