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Will Cavendish

Publications and source records attributed to Will Cavendish.

5 recordsLinked to original sources
arXiv2026

The arrival position problem in quantum mechanics

The problem of making unambiguous probabilistic predictions about experiments involving waiting "always on" detectors remains a challenge for quantum theory. While most research on this problem studies arrival time, i.e., predicting the distribution of when detection events occur, this paper studies the arrival position problem, which is the complementary challenge of predicting the distribution of where detection events occur. Despite the widespread recognition of the arrival time problem, the inability of standard quantum theory to address the arrival position problem remains a pervasive theoretical blind spot. In this paper, we compare quantitative arrival position predictions derived from prominent proposed solutions to the screen problem. As we show, these models yield distinguishable predictions even in relatively simple experiments achievable with current technology. Notably, many of these discrepancies persist even in the far-field limit, where standard semiclassical approximations are typically assumed to be valid.

quant-ph↗
arXiv2025

Absorbing detectors meet scattering theory

Any proposed solution to the "screen problem" in quantum mechanics -- the challenge of predicting the joint distribution of particle arrival times and impact positions -- must align with the extensive data obtained from scattering experiments. In this paper, we conduct a direct consistency check of the Absorbing Boundary Condition (ABC) proposal, a prominent approach to address the screen problem, against the predictions derived from scattering theory (ST). Through a series of exactly solvable one- and two-dimensional examples, we demonstrate that the ABC proposal's predictions are in tension with the well-established results of ST. Specifically, it predicts sharp momentum- and screen-orientation-dependent detection probabilities, along with secondary reflections that contradict existing experimental data. We conclude that while it remains possible that physical detectors described by the ABC proposal could be found in the future, the proposal is empirically inadequate as a general solution to the screen problem, as it is inconsistent with the behavior of detectors in standard experimental settings.

quant-ph↗
arXiv2021

Towards an Elementary Formulation of the Riemann Hypothesis in Terms of Permutation Groups

This paper investigates the relationship between the Riemann hypothesis and the statement $\forall n, ~g(n) \le e^{\sqrt{p_n}}$, where $g(n)$ is the maximum order of an element of $S_n$, the symmetric group on $n$ elements, and $p_n$ is the $n$-th prime. We show this inequality holds under the Riemann Hypothesis. We also make progress towards establishing the converse by proving $\exists n,~g(n)>e^{\sqrt{p_n}}$ if the Riemann Hypothesis is false and the supremum of the set of the real parts of the Riemann zeta function's zeros $\sup \{\Re(\rho)~|~\zeta(\rho) = 0\}$ is not equal to 1.

math.NT↗
arXiv2014

On finite derived quotients of 3-manifold groups

This paper studies the set of finite groups appearing as $π_1(M)/π_1(M)^{(n)}$, where $M$ is a closed, orientable 3-manifold and $π_1(M)^{(n)}$ denotes the $n$-th term of the derived series of $π_1(M)$. Our main result is that if $M$ is a closed, orientable 3-manifold, $n\ge 2$, and $G\cong π_1(M)/π_1(M)^{(n)}$ is finite, then the cup product pairing $H^2(G)\otimes H^2(G)\to H^4(G)$ has cyclic image $C$, and the pairing $H^2(G)\otimes H^2(G)\stackrel{\smile}{\longrightarrow} C$ is isomorphic to the linking pairing $H_1(M)_{\textrm{Tors}}\otimes H_1(M)_{\textrm{Tors}}\to \mathbb{Q}/\mathbb{Z}$.

math.GT↗