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We consider a non-relativistic quantum gas of $N$ bosonic atoms confined to a box of volume $\Lambda$ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, $\rho=\frac{N}{\Lambda}$, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume $\Lambda$ and small ratio $\frac{\Lambda}{\rho}$. The initial state of the gas is assumed to be close to a \textit{product state} of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of $\Lambda$. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the ratio $\frac{\Lambda}{\rho}$. We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials., Comment: 42 pages

With this paper we provide a mathematical review on the initial-value problem of the one-particle Dirac equation on space-like Cauchy hypersurfaces for compactly supported external potentials. We, first, discuss the physically relevant spaces of solutions and initial values in position and mass shell representation; second, review the action of the Poincar\'e group as well as gauge transformations on those spaces; third, introduce generalized Fourier transforms between those spaces and prove convenient Paley-Wiener- and Sobolev-type estimates. These generalized Fourier transforms immediately allow the construction of a unitary evolution operator for the free Dirac equation between the Hilbert spaces of square-integrable wave functions of two respective Cauchy surfaces. With a Picard-Lindel\"of argument this evolution map is generalized to the Dirac evolution including the external potential. For the latter we introduce a convenient interaction picture on Cauchy surfaces. These tools immediately provide another proof of the well-known existence and uniqueness of classical solutions and their causal structure.

We investigate whether quantum theory can be understood as the continuum limit of a mechanical theory, in which there is a huge, but finite, number of classical 'worlds', and quantum effects arise solely from a universal interaction between these worlds, without reference to any wave function. Here a `world' means an entire universe with well-defined properties, determined by the classical configuration of its particles and fields. In our approach each world evolves deterministically; probabilities arise due to ignorance as to which world a given observer occupies; and we argue that in the limit of infinitely many worlds the wave function can be recovered (as a secondary object) from the motion of these worlds. We introduce a simple model of such a 'many interacting worlds' approach and show that it can reproduce some generic quantum phenomena---such as Ehrenfest's theorem, wavepacket spreading, barrier tunneling and zero point energy---as a direct consequence of mutual repulsion between worlds. Finally, we perform numerical simulations using our approach. We demonstrate, first, that it can be used to calculate quantum ground states, and second, that it is capable of reproducing, at least qualitatively, the double-slit interference phenomenon., Comment: Published version (including further discussion of interpretation and quantum limit)

Spontaneous photon emission in the Continuous Spontaneous Localization (CSL) model is studied one more time. In the CSL model each particle interacts with a noise field that induces the collapse of its wave function. As a consequence of this interaction, when the particle is electrically charged, it radiates. As discussed in [1], the formula for the emission rate, to first perturbative order, contains two terms: One is proportional to the Fourier component of the noise field at the same frequency as that of the emitted photon and one is proportional to the zero Fourier component of the noise field. As discussed in previous works, this second term seems unphysical. In [1], it was shown that the unphysical term disappears when the noises is confined to a bounded region and the final particle's state is a wave packet. Here we investigate the origin of the unphysical term and why it vanishes according to the previous prescription. For this purpose, the electrodynamic part of the equation of motion is solved exactly while the part due to the noise is treated perturbatively. We show that the unphysical term is connected to exponentially decaying function of time which dies out in the large time limit, however, approximates to 1 in the first perturbative order in the electromagnetic field., Comment: 10 pages, 1 figure, LaTeX

The direct interaction theory of electromagnetism, also known as Wheeler-Feynman electrodynamics, is often misinterpreted and found unappealing because of its reference to the absorber and, more importantly, to the so-called absorber condition. Here we remark that the absorber condition is indeed questionable and presumably not relevant for the explanation of irreversible radiation phenomena in our universe. What is relevant and deserves further scrutiny is the emergent effective description of a test particle in an environment. We therefore rephrase what we consider the relevant calculation by Wheeler and Feynman and comment on the status of the theory.

We present an approximate model of Wheeler-Feynman electrodynamics for which uniqueness of solutions is proved. It is simple enough to be instructive but close enough to Wheeler-Feynman electrodynamics such that we can discuss its natural type of initial data, constants of motion and stable orbits with regard to Wheeler-Feynman electrodynamics., Comment: 16 pages

We study a system consisting of a heavy quantum particle, called tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density $\to\infty$). Our estimates allow us to interchange the thermodynamic with the mean-field limit., Comment: 27 pages, typos corrected, a few more explanations added

We consider the one-particle sector of the spinless Yukawa model, which describes the interaction of a nucleon with a real field of scalar massive bosons (neutral mesons). The nucleon as well as the mesons have relativistic dispersion relations. In this model we study the dependence of the nucleon mass shell on the ultraviolet cut-off $\Lambda$. For any finite ultraviolet cut-off the nucleon one-particle states are constructed in a bounded region of the energy-momentum space. We identify the dependence of the ground state energy on $\Lambda$ and the coupling constant. More importantly, we show that the model considered here becomes essentially trivial in the limit $\Lambda\to\infty$ regardless of any (nucleon) mass and self-energy renormalization. Our results hold in the small coupling regime., Comment: 30 pages, typos corrected, references extended

The massless Nelson model describes non-relativistic, spinless quantum particles interacting with a relativistic, massless, scalar quantum field. The interaction is linear in the field. We analyze the one particle sector. First, we construct the renormalized mass shell of the non-relativistic particle for an arbitrarily small infrared cut-off that turns off the interaction with the low energy modes of the field. No ultraviolet cut-off is imposed. Second, we implement a suitable Bogolyubov transformation of the Hamiltonian in the infrared regime. This transformation depends on the total momentum of the system and is non-unitary as the infrared cut-off is removed. For the transformed Hamiltonian we construct the mass shell in the limit where both the ultraviolet and the infrared cut-off are removed. Our approach is constructive and leads to explicit expansion formulae which are amenable to rigorously control the S-matrix elements., Comment: explanations added, typos corrected

Resolving the tension between quantum superpositions and the uniqueness of the classical world is a major open problem. One possibility, which is extensively explored both theoretically and experimentally, is that quantum linearity breaks above a given scale. Theoretically, this possibility is predicted by collapse models. They provide quantitative information on where violations of the superposition principle become manifest. Here we show that the lower bound on the collapse parameter lambda, coming from the analysis of the human visual process, is ~ 7 +/- 2 orders of magnitude stronger than the original bound, in agreement with more recent analysis. This implies that the collapse becomes effective with systems containing ~ 10^4 - 10^5 nucleons, and thus falls within the range of testability with present-day technology. We also compare the spectrum of the collapsing field with those of known cosmological fields, showing that a typical cosmological random field can yield an efficient wave function collapse., Comment: 13 pages, LaTeX, 3 figures

We establish global existence and uniqueness of the dynamics of classical electromagnetism with extended, rigid charges and fields which need not to be square integrable. We consider also a modified theory of electromagnetism where no self-fields occur. That theory and our results are crucial for approaching the as yet unsolved problem of the general existence of dynamics of Wheeler Feynman electromagnetism, which we shall address in the follow up paper., Comment: 32 pages, revised Introduction, typos corrected

We study the equations of Wheeler-Feynman electrodynamics which is an action-at-a-distance theory about world-lines of charges that interact through their corresponding advanced and retarded Li\'enard-Wiechert field terms. The equations are non-linear, neutral, and involve time-like advanced as well as retarded arguments of unbounded delay. Using a reformulation in terms of Maxwell-Lorentz electrodynamics without self-interaction, which we have introduced in a preceding work, we are able to establish the existence of conditional solutions. These are solutions that solve the Wheeler-Feynman equations on any finite time interval with prescribed continuations outside of this interval. As a byproduct we also prove existence and uniqueness of solutions to the Synge equations on the time half-line for a given history of charge trajectories., Comment: 45 pages, introduction revised, typos corrected, explanations added

We construct the time-evolution for the second quantized Dirac equation subject to a smooth, compactly supported, time dependent electromagnetic potential and identify the degrees of freedom involved. Earlier works on this (e.g. Ruijsenaars) observed the Shale-Stinespring condition and showed that the one-particle time-evolution can be lifted to Fock space if and only if the external field had zero magnetic components. We scrutinize the idea, observed earlier by Fierz and Scharf, that the time-evolution can be implemented between time varying Fock spaces. In order to define these Fock spaces we are led to consider classes of reference vacua and polarizations. We show that this implementation is up to a phase independent of the chosen reference vacuum or polarization and that all induced transition probabilities are well-defined and unique., Comment: 60 pages, 1 figure, revised introduction, summary of results added, typos corrected

A major problem in exploiting microscopic systems for developing a new technology based on the principles of Quantum Information is the influence of noise which tends to work against the quantum features of such systems. It becomes then crucial to understand how noise affects the evolution of quantum circuits: several techniques have been proposed among which stochastic differential equations (SDEs) can represent a very convenient tool. We show how SDEs naturally map any Markovian noise into a linear operator, which we will call a noise gate, acting on the wave function describing the state of the circuit, and we will discuss some examples. We shall see that these gates can be manipulated like any standard quantum gate, thus simplifying in certain circumstances the task of computing the overall effect of the noise at each stage of the protocol. This approach yields equivalent results to those derived from the Lindblad equation; yet, as we show, it represents a handy and fast tool for performing computations, and moreover, it allows for fast numerical simulations and generalizations to non Markovian noise. In detail we review the depolarizing channel and the generalized amplitude damping channel in terms of this noise gate formalism and show how these techniques can be applied to any quantum circuit., Comment: 10 pages, 4 figures: journal reference added + some typos corrected

We describe the advantages and disadvantages of numerical methods when Bohmian trajectory-grids are used for numerical simulations of quantum dynamics. We focus on the crucial non crossing property of Bohmian trajectories, which numerically must be paid careful attention to. Failure to do so causes instabilities or leads to false simulations., Comment: 17 pages, 18 figures; some typos corrected, 4 figures added, some paragraphs extended, source code extended