arXiv subjects

Maik Reddiger

Publications and source records attributed to Maik Reddiger.

7 recordsLinked to original sources

arXiv2025

A solution of the quantum time of arrival problem via mathematical probability theory

Time of arrival refers to the time a particle takes after emission to impinge upon a suitably idealized detector surface. Within quantum theory, no generally accepted solution exists so far for the corresponding probability distribution of arrival times. In this work we derive a general solution for a single body without spin impacting on a so called ideal detector in the absence of any other forces or obstacles. A solution of the so called screen problem for this case is also given. After discussing the shortcomings of the so called "absorbing boundary condition", which is arguably the natural approach within quantum mechanics, we construct the ideal detector model via mathematical probability theory. This detector model assures that the probability flux through the detector surface is always positive, so that the corresponding distributions can be derived via an approach originally suggested by Daumer, D\"urr, Goldstein, and Zangh\`i. The resulting dynamical model is based on an adaption of the Madelung equations and is, strictly speaking, not compatible with quantum mechanics. Still, it is well-described within geometric quantum theory. Geometric quantum theory is a novel adaption of quantum mechanics, which makes the latter consistent with mathematical probability theory. Implications to the general theory of measurement and avenues for future research are also provided. Future mathematical work should focus on finding an appropriate distributional formulation of the evolution equations and studying the well-posedness of the corresponding Cauchy problem.

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arXiv2024

On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part I: foundations

By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a "quantum probability theory". As such, it is regarded a generalization of the "classical probability theory" due to Kolmogorov. Outside of quantum physics, however, Kolmogorov's axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann's axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic $N$-body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov's axioms and physically natural random variables. Though this theory is provably distinct from its quantum mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.

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arXiv2022

Towards a mathematical Theory of the Madelung Equations

Even though the Madelung equations are central to many 'classical' approaches to the foundations of quantum mechanics such as Bohmian and stochastic mechanics, no coherent mathematical theory has been developed so far for this system of partial differential equations. Wallstrom prominently raised objections against the Madelung equations, aiming to show that no such theory exists in which the system is well-posed and in which the Schr\"odinger equation is recovered without the imposition of an additional 'ad hoc quantization condition'--like the one proposed by Takabayasi. The primary objective of our work is to clarify in which sense Wallstrom's objections are justified and in which sense they are not, with a view on the existing literature. We find that it may be possible to construct a mathematical theory of the Madelung equations which is satisfactory in the aforementioned sense, though more mathematical research is required.

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arXiv2020

Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime

In this work we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of quantum field theory (QFT) such as the `problem of infinities' (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work we consider the $1$-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the $1$-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the `problem of time' in this approach and therefore serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved (example given for the latter).

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arXiv2019

The Differentiation Lemma and the Reynolds Transport Theorem for Submanifolds with Corners

The Reynolds Transport Theorem, colloquially known as 'differentiation under the integral sign', is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and statistical physics. In this work we state and prove generalizations thereof to submanifolds with corners evolving in a manifold via the flow of a smooth time-independent or time-dependent vector field. Thereby we close a practically important gap in the mathematical literature, as related works require various 'boundedness conditions' on domain or integrand that are cumbersome to satisfy in common modeling situations. By considering manifolds with corners, a generalization of manifolds and manifolds with boundary, this work constitutes a step towards a unified treatment of classical integral theorems for the 'unbounded case' for which the boundary of the evolving set can exhibit some irregularity.

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arXiv2018

An Observer's View on Relativity: Space-Time Splitting and Newtonian Limit

We motivate and construct a mathematical theory for the separation of space and time in general relativity. The formalism only requires a single observer and an optional choice of reference frame at each instant. As the splitting is done via the observer's past light cone, it is both closer to the experimental situation and mathematically less restrictive than the splitting via observer vector fields or spacelike hypersurfaces. Indeed, the theory can in principle be applied to all spacetimes and adapted to other `metric' theories of gravity. Instructive examples are developed along with the general theory. In particular, we obtain an alternative description for accelerated frames of reference in Minkowski spacetime. Further, we use the splitting formalism to motivate a new mathematical approach to the Newtonian limit of the motion of mass points. This employs a general formula for their observed motion, distinguishing between `actual' forces (i.e. those detectable via an accelerometer) and pseudo-forces. Via this formula we show that for inertial frames of reference in Minkowski spacetime the essential laws of non-gravitational Newtonian mechanics can be derived. Physically relevant, related, open problems are indicated throughout the text. These include the proof, that the Newtonian limit gives rise to the central pseudo-forces known from Newtonian mechanics (`constant gravity', Euler, Coriolis and centrifugal force) for non-inertial frames of reference in Minkowski spacetime, as well as the derivation of Newton's law of gravitation in the Schwarzschild spacetime under said limit. This is a slightly corrected version of a master's thesis in mathematical relativity, written at TU Berlin in 2016/2017. Comments by the reviewers have been taken into account. If there are any remaining errors, they are solely due to the author.

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arXiv2017

The Madelung Picture as a Foundation of Geometric Quantum Theory

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recoursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we argue that the Schroedinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by E. Madelung, naturally ground the Schroedinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modelling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.

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