In recent work we presented an explicit and non-perturbative derivation of the classical radiation reaction force for a cut-off modelled by a special choice of tubes of finite radius around the charge trajectories. In this paper, we provide a further, simpler and so-called reduced radiation reaction model together with a systematic numerical comparison between both the respective radiation reaction forces and the Landau-Lifschitz force as a reference. We explicitly construct the numerical flow for the new forces and present the numerical integrator used in the simulations, a Gauss-Legendre method adapted for delay equations. For the comparison, we consider the cases of a constant electric field, a constant magnetic field, and a plane wave. In all these cases, the deviations between the three force laws are shown to be small. This excellent agreement is an argument for plausibility of both new equations but also means that an experimental differentiation remains hard. Furthermore, we discuss the effect of the tube radius on the trajectories, which turns out to be small in the regarded regimes. We conclude with a comparison of the numerical cost of the corresponding integrators and find that the integrator of the reduced radiation reaction to be numerically most and the integrator of Landau-Lifschitz least efficient.

The double-slit experiment is one of the quintessential quantum experiments. However, it tends to be overlooked that the build-up of the interference pattern involves single detection events with a random position and a random time of detection. Hence, a full theoretical account of this experiment requires the specification of the joint position and time distribution of detection at the screen, whose position marginal yields the famous interference pattern. The difficulty lies in what this distribution should be. While there are a variety of proposals for a quantum mechanical time observable , there is no consensus about the right choice. Here, we consider Bohmian mechanics, which allows for a natural and practical approach to this problem. We simulate this distribution in the case of an initial Gaussian wave packet passing through a double-slit potential. We also consider a more challenging setup in which one of the slits is closed during flight. To experimentally probe the quantum nature of the time distribution, a sufficient longitudinal spread of the initial wave packet is required, which has not been achieved so far. Without sufficient spread, the temporal aspect of the distribution can be treated classically. We illustrate this for the case of the double-slit experiment with helium atoms by Kurtsiefer et al. (1997), which reports the joint position and time distribution.

The Many-Interacting-Worlds (MIW) approach to a quantum theory without wave functions proposed in [8] leads naturally to numerical integrators of the Schrödinger equation on comoving grids. As yet, little is known about concrete MIW models for more than one spatial dimension and/or more than one particle. In honour of Detlef Dürr, we report on a further development of the MIW approach to treat arbitrary degrees of freedom and provide a numerical proof of concept for ground states in 2d. The latter is part of a systematic numerical study [22] that includes also 1d ground and excited states. With this step towards the treatment of higher degrees of freedom we hope to stimulate their further study.

In honour of Detlef Dürr, we report on a mathematical rigorous computation of the electric vacuum polarisation current and extract the well-known expression for the second order perturbation. Intermediate steps in the presented calculation demonstrate, to the knowledge of the authors for the first time, mathematical rigorous versions of the combined dimensional and Pauli-Villars regularisation schemes. These are employed as computational tools to infer convenient integral representations during the computation. The said second order expression is determined up to a remaining degree of freedom of a real number–without ill-defined terms from start to end.

Bohmian mechanics is an alternative to standard quantum mechanics that does not suffer from the measurement problem. While it agrees with standard quantum mechanics concerning its experimental predictions, it offers novel types of approximations not suggested by the latter. Of particular interest are semi-classical approximations, where part of the system is treated classically. Bohmian semi-classical approximations have been explored before for systems without electromagnetic interactions. Here, the Rabi model is considered as a simple model involving light-matter interaction. This model describes a single mode electromagnetic field interacting with a two-level atom. As is well-known, the quantum treatment and the semi-classical treatment (where the field is treated classically rather than quantum mechanically) give qualitatively different results. We analyze the Rabi model using a different semi-classical approximation based on Bohmian mechanics. In this approximation, the back-reaction from the two-level atom onto the classical field is mediated by the Bohmian configuration of the two-level atom. We find that the Bohmian semi-classical approximation gives results comparable to the usual mean field one for the transition between ground and first excited state. Both semi-classical approximations tend to reproduce the collapse of the population inversion, but fail to reproduce the revival, which is characteristic of the full quantum description. Also an example of a higher excited state is presented where the Bohmian approximation does not perform so well.

The Spin-Boson model describes a two-level quantum system that interacts with a second-quantized boson scalar field. Recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in [14] for the case of massless bosons. In the present work, we treat the massive case. On the one hand, one might rightfully expect that the massive case is easier to handle since, in contrast to the massless case, the corresponding Hamiltonian features a spectral gap. On the other hand, it turns out that the non-zero boson mass introduces a new complication as the spectrum of the complex dilated, free Hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies. This leads to an absence of decay of the corresponding complex dilated resolvent close to the real line, which, in [14], was a crucial ingredient to control the time evolution in the scattering regime. With the new strategy presented here, we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a Mourre theory argument combined with a suitable application of the Feshbach-Schur map instead of complex dilation.

In scattering experiments, physicists observe so-called resonances as peaks at certain energy values in the measured scattering cross sections per solid angle. These peaks are usually associate with certain scattering processes, e.g., emission, absorption, or excitation of certain particles and systems. On the other hand, mathematicians define resonances as poles of an analytic continuation of the resolvent operator through complex dilations. A major challenge is to relate these scattering and resonance theoretical notions, e.g., to prove that the poles of the resolvent operator induce the above mentioned peaks in the scattering matrix. In the case of quantum mechanics, this problem was addressed in numerous works that culminated in Simon's seminal paper [33] in which a general solution was presented for a large class of pair potentials. However, in quantum field theory the analogous problem has been open for several decades despite the fact that scattering and resonance theories have been well-developed for many models. In certain regimes these models describe very fundamental phenomena, such as emission and absorption of photons by atoms, from which quantum mechanics originated. In this work we present a first non-perturbative formula that relates the scattering matrix to the resolvent operator in the massless Spin-Boson model. This result can be seen as a major progress compared to our previous works [13] and [12] in which we only managed to derive a perturbative formula.

The Lorentz-Abraham-Dirac equations (LAD) may be the most commonly accepted equation describing the motion of a classical charged particle in its electromagnetic field. However, it is well known that they bare several problems. In particular, almost all solutions are dynamically unstable, and therefore, highly questionable. The question remains whether better equations of motion than LAD can be found to describe the dynamics of charges in the electromagnetic fields. In this paper we present an approach to derive such equations of motions, taking as input the Maxwell equations and a particular charge model only, similar to the model suggested by Dirac in his original derivation of LAD in 1938. We present a candidate for new equations of motion for the case of a single charge. Our approach is motivated by the observation that Dirac's derivation relies on an unjustified application of Stokes' theorem and an equally unjustified Taylor expansion of terms in his evolution equations. For this purpose, Dirac's calculation is repeated using an extended charge model that does allow for the application of Stokes' theorem and enables us to find an explicit equation of motion by adapting Parrott's derivation, thus avoiding a Taylor expansion. The result are second order differential delay equations, which describe the radiation reaction force for the charge model at hand. Their informal Taylor expansion in the radius of the charge model used in the paper reveals again the famous triple dot term of LAD but provokes the mentioned dynamical instability by a mechanism we discuss and, as the derived equations of motion are explicit, is unnecessary.

We establish the precise relation between the integral kernel of the scattering matrix and the resonance in the massless Spin-Boson model which describes the interaction of a two-level quantum system with a second-quantized scalar field. For this purpose, we derive an explicit formula for the two-body scattering matrix. We impose an ultraviolet cut-off and assume a slightly less singular behavior of the boson form factor of the relativistic scalar field but no infrared cut-off. The purpose of this work is to bring together scattering and resonance theory and arrive at a similar result as provided by Simon (Ann Math Sect Ser 97(2):247–274, 1973), where it was shown that the singularities of the meromorphic continuation of the integral kernel of the scattering matrix are located precisely at the resonance energies. The corresponding problem has been open in quantum field theory ever since. To the best of our knowledge, the presented formula provides the first rigorous connection between resonance and scattering theory in the sense of (Simon 1973) in a model of quantum field theory.

We show that the Bohmian approach in terms of persisting particles that move on continuous trajectories following a deterministic law can be literally applied to quantum field theory. By means of the Dirac sea model-exemplified in the electron sector of the standard model neglecting radiation-we explain how starting from persisting particles, one is led to standard QFT employing creation and annihilation operators when tracking the dynamics with respect to a reference state, the so-called vacuum. Since on the level of wave functions, both formalisms are mathematically equivalent, this proposal provides for an ontology of QFT that includes a dynamics of individual processes, solves the measurement problem, and explains the appearance of creation and annihilation events.

Dirac, Fock, and Podolsky [Ref. 1] devised a relativistic model in 1932 in which a fixed number of $N$ Dirac electrons interact through a second-quantized electromagnetic field. It is formulated with the help of a multi-time wave function $\psi(t_1,\mathbf{x}_1,...,t_N,\mathbf{x}_N)$ that generalizes the Schr\"odinger multi-particle wave function to allow for a manifestly relativistic formulation of wave mechanics. The dynamics is given in terms of $N$ evolution equations that have to be solved simultaneously. Integrability imposes a rather strict constraint on the possible forms of interaction between the $N$ particles and makes the rigorous construction of interacting dynamics a long-standing problem, also present in the modern formulation of quantum field theory. For a simplified version of the multi-time model, in our case describing $N$ Dirac electrons that interact through a relativistic scalar field, we prove well-posedness of the corresponding multi-time initial value problem and discuss the mechanism and type of interaction between the charges. For the sake of mathematical rigor we are forced to employ an ultraviolet cut-off in the scalar field. Although this again breaks the desired relativistic invariance, this violation occurs only on the arbitrary small but finite length-scale of this cut-off. In view of recent progress in this field, the main mathematical challenges faced in this work are, on the one hand, the unboundedness from below of the free Dirac Hamiltonians and the unbounded, time-dependent interaction terms, and on the other hand, the necessity of pointwise control of the multi-time wave function.

We study the two-body Dirac operator in a bounded external field and for a class of unbounded pair-interaction potentials, both repulsive and attractive, including the Coulomb type. Provided the coupling constant of the pair-interaction fulfills a certain bound, we prove existence of a self-adjoint extension of this operator which is uniquely distinguished by means of finite potential energy. In the case of Coulomb interaction, we require as a technical assumption the coupling constant to be bounded by $$2/\pi $$ .

We extend the method of Pizzo multiscale analysis for resonances introduced in [5] in order to infer analytic properties of resonances and eigenvalues (and their eigenprojections) as well as estimates for the localization of the spectrum of dilated Hamiltonians and norm-bounds for the corresponding resolvent operators, in neighborhoods of resonances and eigenvalues. We apply our method to the massless Spin-Boson model assuming a slight infrared regularization. We prove that the resonance and the ground-state eigenvalue (and their eigenprojections) are analytic with respect to the dilation parameter and the coupling constant. Moreover, we prove that the spectrum of the dilated Spin Boson Hamiltonian in the neighborhood of the resonance and the ground-state eigenvalue is localized in two cones in the complex plane with vertices at the location of the resonance and the ground-state eigenvalue, respectively. Additionally, we provide norm-estimates for the resolvent of the dilated Spin-Boson Hamiltonian near the resonance and the ground state eigenvalue. The topic of analyticity of eigenvalues and resonances has let to several studies and advances in the past. However, to the best of our knowledge, this is the first time that it is addressed from the perspective of Pizzo multiscale analysis. Once the multiscale analysis is set up our method gives easy access to analyticity: Essentially, it amounts to proving it for isolated eigenvalues only and use that uniform limits of analytic functions are analytic. The type of spectral and resolvent estimates that we prove are needed to control the time evolution including the scattering regime. The latter will be demonstrated in a forthcoming publication. The introduced multiscale method to study spectral and resolvent estimates follows its own inductive scheme and is independent (and different) from the method we apply to construct resonances. (C) 2019 Published by Elsevier Inc.

The paper sets out and defends against criticism the claims argued for in the book A minimalist ontology of the natural world.

Quantum field theory (QFT) may be considered one of the most fundamental frameworks of theoretical physics. Quantum Electrodynamics (QED) is the part of QFT that describes the interaction between matter and light. Although it is one of the experimentally best tested theories, it yet faces many open mathematical questions and challenges. The mathematical rigorous framework of QED and the implications deriving from it is the topic of Workshop 1737 held at MFO from September 11 through 15, 2017, bringing together mathematicians and theoretical physicists to discuss topics such as high- and low-energy QED, external field QED, quantum optics, many-boson and many-fermion systems, transport properties in condensed matter.

Recently the Many-Interacting-Worlds (MIW) approach to a quantum theory without wave functions was proposed. This approach leads quite naturally to numerical integrators of the Schr\"odinger equation. It has been suggested that such integrators may feature advantages over fixed-grid methods for higher numbers of degrees of freedom. However, as yet, little is known about concrete MIW models for more than one spatial dimension and/or more than one particle. In this work we develop the MIW approach further to treat arbitrary degrees of freedom, and provide a systematic study of a corresponding numerical implementation for computing one-particle ground and excited states in one dimension, and ground states in two spatial dimensions. With this step towards the treatment of higher degrees of freedom we hope to stimulate their further study.

Recently the Many-Interacting-Worlds (MIW) approach to a quantum theory without wave functions was proposed. This approach leads quite naturally to numerical integrators of the Schr\odinger equation. It has been suggested that such integrators may feature advantages over fixed-grid methods for higher numbers of degrees of freedom. However, as yet, little is known about concrete MIW models for more than one spatial dimension and/or more than one particle. In this work we develop the MIW approach further to treat arbitrary degrees of freedom, and provide a systematic study of a corresponding numerical implementation for computing one-particle ground and excited states in one dimension, and ground states in two spatial dimensions. With this step towards the treatment of higher degrees of freedom we hope to stimulate their further study.

The classical electrodynamic two-body problem has been a long standing open problem in mathematics. For motion constrained to the straight line, the interaction is similar to that of the two-body problem of classical gravitation. The additional complication is the presence of unbounded state-dependent delays in the Coulomb forces due to the finiteness of the speed of light. This circumstance renders the notion of local solutions meaningless, and therefore, straightforward ODE techniques cannot be applied. Here, we study the time-symmetric case, i.e., the Fokker–Schwarzschild–Tetrode (FST) equations, comprising both advanced and retarded delays. We extend the technique developed in Deckert and Hinrichs (J Differ Equ 260:6900–6929, 2016 ), where existence of FST solutions was proven on the half line, to ensure global existence—a result that had been obtained by Bauer (Ein Existenzsatz für die Wheeler–Feynman–Elektrodynamik, Herbert Utz Verlag, München, 1997 ). Due to the novel technique, the presented proof is shorter and more transparent but also relies on the idea to employ asymptotic data to characterize solutions.