Unstable mode and the Unruh-DeWitt detector

TL;DR

The paper addresses the quantization of a single unstable mode arising from a Robin boundary condition in (1+1)-dimensional half-Minkowski spacetime, using a rigged Hilbert space framework to treat the inverted harmonic oscillator-like state as a decaying quantum state with lifetime set by the boundary parameter $ extgamma$. It then couples this unstable mode to an Unruh–DeWitt detector and analyzes the detector's transition probability for static, inertial, and uniformly accelerated trajectories, obtaining a Breit–Wigner resonance for static motion with width $ extGamma_0=2 extgamma$, a Doppler-shifted width for inertial motion, and a complex, incomplete-Gamma–based structure for acceleration. The key findings show that Dirichlet limits suppress the unstable mode, Neumann limits introduce infrared divergences in inertial frames but yield finite, oscillatory responses under acceleration, and the RHS quantization provides a mathematically consistent interpretation of unstable field configurations; overall, acceleration can act as an infrared regulator in this setting, revealing observer-dependent aspects of instability in quantum field theory. These results contribute a rigorous, physically meaningful description of unstable modes in non-globally hyperbolic spacetimes and offer insight into how detectors perceive such states.

Abstract

We investigate the quantization of a single unstable mode in a real scalar field subject to a Robin boundary condition in (1+1)-dimensional half-Minkowski spacetime. The instability arises from an imaginary frequency mode - analogous to that of the inverted harmonic oscillator - requiring the rigged Hilbert space formalism for consistent quantization. Within this framework, the unstable mode is naturally described as a well-defined decaying (or growing) quantum state with a characteristic mean lifetime. We investigate its physical consequences via the response of an Unruh-DeWitt detector along static, inertial, and uniformly accelerated trajectories. For static and inertial observers, the detector response exhibits a Breit-Wigner resonance profile, with a decay width determined by the unstable frequency and a Doppler factor. In the Neumann limit, infrared divergences emerge from arbitrarily low-frequency modes. Interestingly, for accelerated detectors, the response acquires a nontrivial dependence on acceleration, and the Neumann limit yields a finite, oscillatory signal rather than a divergence, suggesting that acceleration can act as an effective infrared regulator.

Figures (6)

• Figure 1: Penrose diagram of the two-dimensional half-Minkowski spacetime. The vertical line in the region $t < 0$ denotes the boundary at $x = 0$, where the Dirichlet boundary condition is imposed. After $t = 0$, the vertical wavy line in the region $t>0$ represents the emergence of a naked singularity, where the Robin boundary condition is imposed.
• Figure 2: Conformal diagram illustrating static (black), inertial (blue), and uniformly accelerated (red) detector trajectories in (1+1)-dimensional half-Minkowski spacetime. Solid lines indicate regions where the detector interacts with the unstable mode ($t>0$).
• Figure 3: Transition probability $\mathcal{P}^{+}(\Omega)$ over $\lambda^2$, given by Eq. \ref{['probability3']}, as a function of detector velocity $v$. We consider $x_0 = 0$ and $\gamma = 1$. As $v \to 1$, the probability sharply decreases, indicating the suppression of detection in the ultrarelativistic regime.
• Figure 4: Transition probability $\mathcal{P}^{+}(\Omega)/\lambda^2$ for a uniformly accelerated detector, as given by Eq. \ref{['probability4']}. Here we consider different accelerations for a fixed boundary parameter $\gamma=1$.
• Figure 5: Transition probability $\mathcal{P}^{+}(\Omega)/\lambda^2$ given by Eq. \ref{['probability4']} as a function of the proper acceleration. We are setting $\gamma=1$.