Spontaneous particle creation in time-dependent overcritical fields of QED [Elektronische Ressource] / von Nikodem Szpak

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• Physik

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• Publié par: goethe_universitat_frankfurt_am_main
• Publié le: 01 janvier 2006
• Nombre de lectures: 21
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Spontaneous particle creation
in time-dependent
overcritical fields of QED
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Physik
der Johann Wolfgang Goethe – Universitat¨
in Frankfurt am Main
von
Nikodem Szpak
aus Bielsko-Bia la, Polen
Frankfurt 2005ii
vom Fachbereich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .der
Johann Wolfgang Goethe – Universit¨ at als Dissertation angenommen.
Dekan: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gutachter: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Datum der Disputation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Abstract
In the classical Dirac equation with strong potentials, called overcritical, a bound state
reaches the negative continuum. In QED the presence of a static overcritical external
electric field leads to a charged vacuum and indicates spontaneous particle creation when
the overcritical field is switched on. The goal of this work is to clarify whether this
effect exists, i.e. if it can be uniquely defined and proved, in time-dependent physical
processes. Starting from a fundamental level of the theory we check all mathematical and
interpretational steps from the algebra of fields to the very effect.
In the first, theoretical part of this thesis we introduce the mathematical formulation of
the classical and quantized Dirac theory with their most important results. Using this lan-
guage we define rigorously the notion of spontaneous particle creation in overcritical fields.
First, we give a rigorous definition of resonances as poles of the resolvent or the Green’s
function and show how eigenvalues become resonances under Hamiltonian perturbations.
In particular, we consider essential for overcritical potentials perturbation of eigenvalues at
the edge of the continuous spectrum. Next, we gather various adiabatic theorems and dis-
cuss well-posedness of the scattering in the adiabatic limit. Then, we construct Fock space
representations of the field algebra, study their equivalence and give a unitary implementer
of all Bogoliubov transformations induced by unitary transformations of the one-particle
Hilbert space as well as by the projector (or vacuum vector) changes as long as they lead
to unitarily equivalent Fock representations. We implement in Fock space self-adjoint and
unitary operators from the one-particle space, discussing the charge, energy, evolution
and scattering operators. Then we introduce the notion of particles and several particle
interpretations for time-dependent processes with a different Fock space at every instant
of time. We study how the charge, energy and number of particles change in consequence
of a change of representation or in implemented evolution or scattering processes, what is
especially interesting in presence of overcritical potentials. Using this language we define
rigorously the notion of spontaneous particle creation. Then we look for physical processes
which show the effect of vacuum decay and spontaneous particle creation exclusively due
to the overcriticality of the potential. We consider several processes with static as well
as suddenly switched on (and off) static overcritical potentials and conclude that they
are unsatisfactory for observation of the spontaneous particle creation. Next, we consider
properties of general time-dependent scattering processes with continuous switch on (and
iiiiv
off) of an overcritical potential and show that they also fail to produce stable signatures
of the particle creation due to overcriticality. Further, we study and successfully define
the spontaneous particle creation in adiabatic processes, where the spontaneous antipar-
ticle is created as a result of a resonance (wave packet) decay in the negative continuum.
Unfortunately, they lead to physically questionable pair production as the adiabatic limit
is approached. Finally, we consider extension of these ideas to non-adiabatic processes in-
volving overcritical potentials and argue that they are the best candidate for showing the
spontaneous pair creation in physical processes. Demanding creation of the spontaneous
antiparticle in the state corresponding to the overcritical resonance rather quick than slow
processes should be considered, with a possibly long frozen overcritical period.
In the second part of this thesis we concentrate on a class of spherically symmetric
square well potentials with a time-dependent depth. First, we solve the Dirac equation
and analyze the structure and behaviour of bound states and appearance of overcritical-
ity. Then, by analytic continuation we find and discuss the behaviour of resonances in
overcritical potentials. Next, we derive and solve numerically (introducing a non-uniform
continuum discretization for a consistent treatment of narrow peaks) a system of differen-
tial equations (coupled channel equations) to calculate particle and antiparticle produc-
tion spectra for various time-dependent processes including sudden, quick, slow switch on
and off of a sub- and overcritical potentials. We discuss in detail how and under which
conditions an overcritical resonance decays during the evolution giving rise to the spon-
taneous production of an antiparticle. We compare the antiparticle production spectrum
with the shape of the resonance in the overcritical potential. We study processes, where
the overcritical potentials are switched on at different speed and are possibly frozen in
the overcritical phase. We prove, in agreement with conclusions of the theoretical part,
that the peak (wave packet) in the negative continuum representing a dived bound state
partially follows the moving resonance and partially decays at every stage of its evolu-
tion. This continuous decay is more intensive in slow processes, while in quick processes
the wave packet more precisely follows the resonance. In the adiabatic limit, the whole
decay occurs already at the edge of the continuum, resulting in production of antipar-
ticles with vanishing momentum. In contrast, in quick switch on processes with delay
in the overcritical phase, the spectrum of the created antiparticles agrees best with the
shape of the resonance. Finally, we address the question how much information about the
time-dependent potential can be reconstructed from the scattering data, represented by
the particle production spectrum. We propose a simple approximation method (master
equation) basing on an exponential, decoherent decay of time-dependent resonances for
prediction of particle creation spectra and obtain a good agreement with the results of full
numerical calculations.
Additionally, we discuss various sources of errors introduced by the numerical dis-
cretization, find estimations for them and prove convergence of the numerical schemes.“Everything should be made as simple as possible, but not simpler.”
A. Einstein
Summary and Preface
The goal of this work is to clarify whether the effect of spontaneous particle creation
exists in physical processes. More precisely, if it can be uniquely defined and proved.
Since the considerations are made on the fundamental level, it is necessary to check all
mathematical as well as interpretational steps in the construction of the theory from the
algebra of fields to the very effect. We have observed in the literature that most of the
discrepancies in opinions are due to the different frameworks chosen (e.g. Greiner, Scharf),
like definition and measure of the effect, particle interpretation, and choice of the algebra
representation (projectors).
In construction of the theory of a quantum Dirac field, describing electrons and
positrons in an external electromagnetic field, one encounters the following questions and
difficulties. Which of all possible representations of the canonical anticommutation rela-
tions (CAR) are physically plausible and which of them are unitarily equivalent? Rep-
resentations via operators in a Hilbert space seem quite natural and the addition of a
reference state (vacuum) makes them irreducible what is physically plausible. Despite
these restrictions, there remains a freedom in the choice of a projector defining the dis-
tinction between particles and antiparticles. Although it can be chosen freely in the theory,
not all choices lead to the same physics. Some choices keep the physical results untouched
(unitarily equivalent representations) and some not (nonequivalent representations). It
is the point – unfortunately very rarely discussed in the textbooks – where physical ar-
guments must be used in order to guarantee the uniqueness of the constructed theory,
e.g. that vacuum is chosen as an energetic ground state in Fock space. This definition
is unique if the Hamiltonian, and thus the electromagnetic field, is static. If it is not,
then there is no unique vacuum state and hence no unique particle interpretation. They
can be restored asymptotically if the Hamiltonian is asymptotically (far past or future)
static. Although the particle definition can then be extended to the whole time axis, it is
a