# Transformation of intermediate times in the decays of moving unstable quantum systems via the exponential modes

## TL;DR

This paper investigates how the decay laws of moving unstable quantum systems transform at intermediate times, using superpositions of exponential modes and Prony analysis to approximate survival probabilities across reference frames.

## Contribution

It introduces a method to approximate intermediate-time decay laws of moving quantum systems via exponential mode superpositions and analyzes their transformation under reference frame changes.

## Key findings

- Survival probability transforms approximately according to a scaling law.
- Relativistic time dilation holds approximately over certain time windows.
- The effective mass at rest influences the time dilation of decay laws.

## Abstract

The transformation of canonical decay laws of moving unstable quantum systems is studied by approximating, over intermediate times, the decay laws at rest with superpositions of exponential modes via the Prony analysis. The survival probability $\mathcal{P}_p(t)$, which is detected in the laboratory reference frame where the unstable system moves with constant linear momentum $p$, is represented by the transformed form $\mathcal{P}_0\left(\varphi_p(t)\right)$ of the survival probability at rest $\mathcal{P}_0(t)$. The transformation of the intermediate times, which is induced by the change of reference frame, is obtained by evaluating the function $\varphi_p(t)$. Under determined conditions, this function grows linearly and the survival probability transforms, approximately, according to a scaling law over an estimated time window. The relativistic dilation of times holds, approximately, over the time window if the mass of resonance of the mass distribution density is considered to be the effective mass at rest of the moving unstable quantum system.