Detection time of Dirac particles in one space dimension

Abstract: We consider particles emanating from a source point inside an interval in one-dimensional space and passing through detectors situated at the endpoints of the interval that register their arrival time. Unambiguous measurements of arrival or detection time are problematic in the orthodox narratives of quantum mechanics, since time is not a self-adjoint operator. By contrast, the arrival time at the boundary of a particle whose motion is being guided by a wave function through the deBroglie-Bohm guiding law is well-defined and unambiguous, and can be computationally feasible provided the presence of detectors can be modeled in an effective way that does not depend on the details of their makeup. We use an absorbing boundary condition for Dirac's equation (ABCD) proposed by Tumulka, which is meant to simulate the interaction of a particle initially inside a domain with the detectors situated on the boundary of the domain. By finding an explicit solution, we prove that the initial-boundary value problem for Dirac's equation satisfied by the wave function is globally well-posed, the solution inherits the regularity of the initial data, and depends continuously on it. We then consider the case of a pair of particles emanating from the source inside the interval, and derive explicit formulas for the distribution of first arrival times at each detector, which we hope can be used to study issues related to non-locality in this setup.

• The paper provides a constructive solution for the 1+1 dimensional Dirac equation using absorbing boundary conditions to yield explicit detection time distributions.
• It employs de Broglie-Bohm mechanics and Laplace transform techniques to ensure non-negative, globally well-posed results for both single and two-particle cases.
• The study confirms causality and operational consistency by demonstrating no-signaling between detectors, paving the way for experimental validation.

Explicit Detection Time Distributions for Dirac Particles in One Dimension

Introduction

The paper "Detection time of Dirac particles in one space dimension" (2112.07366) addresses the longstanding ambiguity in quantum mechanics regarding the operational and theoretical definition of arrival or detection times for quantum particles. The lack of a self-adjoint time operator complicates attempts to give a rigorous quantum prediction for time-of-arrival measurements. The authors adopt a de Broglie-Bohm framework combined with absorbing boundary conditions to provide explicit, unambiguous, and computationally tractable arrival time distributions for Dirac particles in one spatial dimension, including both single-particle and two-particle cases.

Mathematical Formulation and Solution of the One-Body Problem

The authors use the absorbing boundary condition for Dirac's equation (ABCD), following Tumulka's proposal, to simulate the action of detectors placed at the interval's endpoints. This boundary condition enforces that the Dirac current is always outgoing at the boundaries, precluding probability backflow and ensuring the non-negativity of the arrival time density.

They give an explicit solution to the initial-boundary value problem (IBVP) of the 1+1 dimensional Dirac equation subject to ABCDs. By decomposing the problem into its initial and boundary value parts and using Laplace transform techniques, they derive that the solution is a convergent (in fact, locally finite) infinite series, with each term corresponding to boundary reflections and transmissions encoded by the Dirac and Klein-Gordon dynamics. The regularity of the solution is determined by that of the initial data, and the solution depends continuously on said data, confirming global well-posedness.

This explicit construction fills a gap as previous results (Teufel & Tumulka) established existence and uniqueness via abstract functional analytic methods but lacked constructive solutions.

Arrival Time Distributions for Bohmian Trajectories

Within Bohmian mechanics, the particle trajectory is defined by the Dirac current, leading to a well-posed ODE for the position variable:

$$\frac{dQ(t)}{dt} = \frac{j_1(t, Q(t))}{j_0(t, Q(t))}$$

where $j_0$ and $j_1$ are the components of the Dirac current. The time of arrival at a detector corresponds to the first time $Q(t)$ reaches a boundary.

By integrating the outgoing current at each boundary, explicit and unambiguous distributions for the arrival times at the respective endpoints can be computed. The absorbing boundary ensures the probability density of detection time is always non-negative and consistent with the operational aspects of detection in time-of-flight experiments. Importantly, these arrival time distributions are generally unnormalized, reflecting the possibility of non-detection due to the initial state’s support.

Two-Body Dirac Problem and Nonlocality

The extension to two Dirac particles involves the multi-time formalism with tensor current conservation and absorbing boundary conditions imposed independently for each particle at both endpoints. The theoretical structure is adapted to study the arrival time distributions for each detector, accounting for label symmetry and the possibility of sequential detection and conditional wavefunction collapse, as prescribed in multi-time Bohm-Dirac theory.

The key technical result is the derivation of explicit, computable formulas for the joint and marginal arrival time distributions, accommodating both product and entangled initial states.

A critical consideration—directly related to nonlocality and relativity—is whether the distribution of arrival times at one detector can be influenced by operations at the other detector prior to the light-cone connection. The authors show, for the single-particle case and for multi-particle product and massless entangled cases, that signaling is excluded: the arrival distribution at one detector is unaffected by changes at the other detector before a relativistically allowed time elapses. For entangled massive states, future work is outlined to rigorously generalize this no-signaling result, but the provided formulae strongly suggest compatibility with causality.

Implications and Future Directions

The formalism developed provides a framework to robustly connect quantum theory (specifically the Bohmian interpretation with detectors) with empirical time-of-flight measurements. The explicit distributions derived circumvent negative probability densities and backflow artifacts that plague other proposals.

From a practical perspective, this approach permits direct calculation of predicted arrival time distributions for experimental setups involving relativistic spin-1/2 particles in one dimension. This creates a route for empirical validation—or falsification—of alternative arrival time formalisms using current quantum optical or cold atom setups.

Theoretically, the work clarifies and sharpens the role of boundaries and detection within quantum theory, providing a constructive approach that seamlessly aligns with relativistic causality. The two-particle entangled state analysis relates directly to issues in quantum nonlocality and measurement, and further work on the massive case will be important for a complete picture.

Conclusion

The paper provides an explicit, globally well-posed solution to the one-dimensional Dirac equation with absorbing boundary conditions, yielding computable arrival time distributions for both single and two-particle cases. The results confirm the operational adequacy and causal compatibility of the ABCD approach, adding concrete tools to the study of quantum detection times and laying a rigorous foundation for further investigation into relativistic quantum measurement and nonlocality.