5,070 research outputs found
Degrees of Propositionality in Construals of Time Quantities1
The paper investigates the possible conceptual bases of differences between seemingly synonymous and easily definable temporal expressions. Looking at the usage patterns of nominal temporal phrases in reference corpora of English and Polish we attempt to relate these subtleties to the different granularity of the cognitive scales on which construals of time quantities in general are based. More specifically, we focus on a subset of nominal temporal expressions which adhere to the “number + time unit” pattern, matching what Haspelmath (1997: 26) describes as “culture-bound artificial time units”. Using the British National Corpus (BNC) and the National Corpus of Polish (NCP), we first analyse both the variation and the regularity found in naturally-occurring samples of Polish and English. Finally, we compare the patterns of use emerging from the two corpora and arrive at cross-linguistic generalisations about the conceptualisation of time quantities
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
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
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
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