Exponentially accelerated mirrors as a physical realization of the kappa plane-wave vacuum

TL;DR

The κ-plane-wave vacuum is a kinematic family of states with thermal properties that lacked a clear dynamical origin. This paper provides a concrete realization by showing that the state on I^+ produced by a Carlitz–Willey moving mirror is physically equivalent to the κ–plane-wave vacuum, evidenced through identical Bogoliubov squeezes, matching nonlocal thermal kernels in Wightman functions obeying the KMS condition, and Planckian responses of an Unruh–DeWitt detector at temperature $T=\frac{κ}{2π}$. It further demonstrates, via Mellin diagonalization, that CW trajectories implement a per-frequency single-mode SU(1,1) transform with the same Boltzmann factor as the κ-vacuum, and derives the full class of trajectories that reproduce the same thermal kernel on $\mathscr I^{+}_R$, highlighting that only purely exponential trajectories yield a constant flux. The identification of $κ$ with the physical scale of the CW trajectory bridges kinematic and dynamic pictures of quantum thermality, providing a robust dynamical origin for κ vacua and offering a framework for exploring more general thermality in moving-mirror and horizon-related settings.

Abstract

The kappa plane-wave vacuum is a family of kinematically defined quantum states whose thermal properties are well understood, but whose physical origin has remained obscure. In this paper we provide a concrete dynamical realization of this vacuum, showing that it is physically and operationally equivalent to the quantum state produced on future null infinity by a mirror following the Carlitz-Willey (CW) trajectory. The equivalence is established through a three-pronged analysis: we demonstrate that the two constructions share identical Bogoliubov squeeze parameters, identical nonlocal thermal kernels in their Wightman functions, and identical Planckian responses of an Unruh-DeWitt detector. This result anchors an abstract kinematic construction in a well-understood dynamical model, identifying the parameter $κ$ with the physical scale that governs the Carlitz-Willey trajectory. In the final part of the paper we characterize, within the moving-mirror framework, the complete class of mirror trajectories that reproduce the same asymptotic thermal kernel on $\mathscr I^+_R$, and show that only the purely exponential CW trajectory generates a constant, stationary flux.

Figures (4)

• Figure 1: Penrose diagram of (1+1)D Minkowski spacetime. The conformal diagram brings the asymptotic boundaries of spacetime to a finite location, illustrating the global causal structure. The Bogoliubov transformation relates the "in" quantum state, defined on past null infinity ($\mathscr{I}^-_{\!L} \cup \mathscr{I}^-_{\!R}$), to the "out" state measured on future null infinity ($\mathscr{I}^+_{\!L} \cup \mathscr{I}^+_{\!R}$). The diagram also shows lines of constant time ($t$) and constant position ($x$), which become hyperbolae in these conformal coordinates.
• Figure 2: Causal structure and ray tracing for the Carlitz--Willey trajectory on a compactified Minkowski diagram. The curved worldline represents the mirror, which emerges from past timelike infinity $i^{-}$ and asymptotically approaches the null line $v=v_{H}$ (dashed), defining a future event horizon. Incoming right–moving null rays from $\mathscr I^-_{\!R}$ reflect from the mirror and are bunched toward $\mathscr I^+_{\!R}$, producing a steady thermal flux; rays that cross the horizon never reach $\mathscr I^+_{\!R}$ and instead end on $\mathscr I^+_{\!L}$.
• Figure 3: Structure of the Wightman function on $\mathscr{I}^+_R$ for the $\kappa$-plane-wave vacuum ($\kappa=1$). All plots use a symmetric logarithmic color scale to handle the singularity on the diagonal and reveal off-diagonal features. (Left) The full Wightman function $\mathrm{Re}\,W(u,u')$. Its structure shows dependence on both the non-local separation $\Delta u = u-u'$ (features parallel to the main diagonal) and the local coordinate sum $\Sigma = u+u'$ (features perpendicular to the diagonal). (Center) The mixed derivative $\partial_u \partial_{u'} W$. In this plot, the local, non-stationary part survives, visible as the cross-like structure that depends on $u+u'$. (Right) The universal thermal kernel $\partial_{\Delta u}^2 W$. This derivative operator eliminates the local, $\Sigma$-dependent part, leaving a pure function of $\Delta u$ that is constant along lines perpendicular to the diagonal. This visually confirms the isolation of the stationary, non-local thermal correlations.
• Figure 4: Minkowski spacetime diagram of the accelerating Carlitz-Willey mirror. The mirror's worldline (red curve) undergoes eternal acceleration, asymptotically approaching the speed of light and creating an event horizon (dashed line) behind it. Uniformly spaced incoming null rays (blue lines from the top-left), representing the empty Minkowski vacuum, reflect off the mirror. Due to the mirror's increasing velocity, the outgoing reflected rays are compressed, or "bunched up." An asymptotic observer on $\mathscr{I}^+_R$ (at the far right) intercepts this compressed wave pattern, which has the power spectrum of thermal radiation at a temperature $T=\kappa/(2\pi)$ set by the mirror's acceleration.

Theorems & Definitions (1)

• proof