1,658 research outputs found
Ultraviolet Properties of the Spinless, One-Particle Yukawa Model
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 . 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
and the coupling constant. More importantly, we show that the model
considered here becomes essentially trivial in the limit
regardless of any (nucleon) mass and self-energy renormalization. Our results
hold in the small coupling regime.Comment: 30 pages, typos corrected, references extende
Dirac Equation with External Potential and Initial Data on Cauchy Surfaces
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
On the spontaneous emission of electromagnetic radiation in the CSL model
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, LaTe
The Mass Shell of the Nelson Model without Cut-Offs
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 correcte
On the Existence of Dynamics of Wheeler-Feynman Electromagnetism
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 adde
Quantum Dynamics with Bohmian Trajectories
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 extende
Dynamics of Sound Waves in an Interacting Bose Gas
We consider a non-relativistic quantum gas of bosonic atoms confined to a
box of volume in physical space. The atoms interact with each other
through a pair potential whose strength is inversely proportional to the
density, , of the gas. We study the time evolution of
coherent excitations above the ground state of the gas in a regime of large
volume and small ratio . 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 . 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 . 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 page
Maxwell-Lorentz Dynamics of Rigid Charges
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 correcte
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