arXiv subjects

Lawrence Frolov

Publications and source records attributed to Lawrence Frolov.

5 recordsLinked to original sources

arXiv2026

Arbitrarily precise arrival time measurements in quantum mechanics

The quantum Zeno effect is often regarded as an obstruction to precise arrival time measurements in quantum mechanics. Here, an arbitrarily precise arrival time measurement procedure is constructed using a localized detection process and a suitably chosen boundary condition. The arrival time of an incoming particle is recorded in the position of a clock particle that is emitted by the apparatus upon detection. A non-zero probability of arrival is shown to survive even in the limit as exact arrival time measurement precision is achieved. In this limit, it is also found that the interaction between the incoming particle and the detector is described by an absorbing boundary condition. This justifies the claim of Tumulka that absorbing boundary conditions may be used to model idealized detectors capable of registering particles at the instant of their arrival.

quant-ph↗

arXiv2025

Detecting screens modeled by Schr\"odinger operators that generate $C_0$ contraction semigroups

Consider a non-relativistic quantum particle with wave function $\psi$ in a bounded $C^2$ region $\Omega \subset \mathbb{R}^n$, and suppose detectors are placed along the boundary $\partial \Omega$. Assume the detection process is irreversible, its mechanism is time independent and also hard, i.e., detections occur only along the boundary $\partial \Omega$. Under these conditions Tumulka informally argued that the dynamics of $\psi$ must be governed by a $C_0$ contraction semigroup that weakly solves the Schr\"odinger equation and proposed modeling the detector by a time-independent local absorbing boundary condition at $\partial \Omega$. In this paper, we apply the newly discovered theory of boundary quadruples to parameterize all $C_0$ contraction semigroups whose generators extend the Schr\"odinger Hamiltonian, and prove a variant of Tumulka's claim: all such evolutions are generated by the placement of a linear absorbing boundary condition on $\psi$ along $\partial \Omega$. We combine this result with the work of Werner to show that each $C_0$ contraction semigroup naturally admits a Born rule for the time of detection along $\partial \Omega$, and we prove that a detection will almost surely occur in finite time if detectors have been placed everywhere along $\partial \Omega$.

math-ph↗

arXiv2024

On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions

A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac's formalism of multi-time wave functions, i.e, wave functions $Ψ(\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_1},\textbf{x}_{\text{e}_2})$ where $\textbf{x}_{\text{ph}},\textbf{x}_{\text{e}_1},\textbf{x}_{\text{e}_2}$ are generic spacetime events of the photon and two electrons respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds $\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_1}\}$ and $\{\textbf{x}_{\text{ph}}=\textbf{x}_{\text{e}_2}\}$ compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times.

math-ph↗

arXiv2024

Joint evolution of a Lorentz-covariant massless scalar field and its point-charge source in one space dimension

In this paper we prove that the static solution of the Cauchy problem for a massless real scalar field that is sourced by a point charge in $1+1$ dimensions is asymptotically stable under perturbation by compactly-supported radiation. This behavior is due to the process of back-reaction. Taking the approach of Kiessling, we rigorously derive the expression for the force on the particle from the principle of total energy-momentum conservation. We provide a simple, closed form for the particle's self-action, and show that it is restorative in this model , i.e. proportional to negative velocity, and causes the charge to return to rest after the radiation passes through. We establish these results by studying the joint evolution problem for the particle-scalar field system, and proving its global well-posedness and the claimed asymptotic behavior.

math.AP↗

arXiv2019

Existence of Schrodinger Evolution with Absorbing Boundary Condition

Consider a non-relativistic quantum particle with wave function inside a region $\Omega\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial \Omega$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the \emph{absorbing boundary rule}, involves a time evolution for the particle's wave function $\psi$ expressed by a Schr\"odinger equation in $\Omega$ together with an ``absorbing'' boundary condition on $\partial \Omega$ first considered by Werner in 1987, viz., $\partial \psi/\partial n=i\kappa\psi$ with $\kappa>0$ and $\partial/\partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $\psi$; we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure.

math-ph↗