# Relativistic Time-of-Arrival Measurements: Predictions, Post-Selection and Causality Problems

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## Abstract

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## 1. Introduction

## 2. Relativistic Time-of-Arrival Probabilities

#### 2.1. Detection Probability from a Von Neumann-Type Measurement

#### 2.2. Post Selection with Respect to Recorded Events

#### 2.3. Wigner Representation

#### 2.4. Probability Dependence on the Detector Kernel

## 3. Causality Issues

#### 3.1. Apparent Causality Violation

#### 3.2. Retarded Propagator versus Feynman Propagator

#### 3.3. Causal Propagation versus Restricted Propagation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Probability density ${P}_{c}\left(\tau \right)$ as a function of $\tau :=t/\left(2m{a}^{2}\right)$ for different values of localization length $\sigma $ and for different distances, as well as for pseudo-probability density (21). The plots correspond to the non-relativistic regime. The insets show the early-time behavior of ${P}_{c}\left(t\right)$.

**Figure 3.**The probability density ${P}_{c}\left(t\right)$ for maximal localization as a function of $\tau :=t/T$ for $L/T=10$ and $mT={10}^{3}$. The non-causal behavior is manifested in the jump of the probability density prior to $t=L-T$.

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**MDPI and ACS Style**

Anastopoulos, C.; Plakitsi, M.-E.
Relativistic Time-of-Arrival Measurements: Predictions, Post-Selection and Causality Problems. *Foundations* **2023**, *3*, 724-737.
https://doi.org/10.3390/foundations3040041

**AMA Style**

Anastopoulos C, Plakitsi M-E.
Relativistic Time-of-Arrival Measurements: Predictions, Post-Selection and Causality Problems. *Foundations*. 2023; 3(4):724-737.
https://doi.org/10.3390/foundations3040041

**Chicago/Turabian Style**

Anastopoulos, Charis, and Maria-Electra Plakitsi.
2023. "Relativistic Time-of-Arrival Measurements: Predictions, Post-Selection and Causality Problems" *Foundations* 3, no. 4: 724-737.
https://doi.org/10.3390/foundations3040041