In this PhD thesis classical and quantum dynamics of tachyon systems are
examined. Those systems are described by spatially homogeneous scalar fields,
in the limit of classical and quantum mechanics and cosmological applications.
Understanding and modeling of these systems are of particular importance in the
development of the field and string theory. In this thesis its application in the
modern cosmology, inflationary, classical and quantum phase in the evolution of
the Universe will be discussed.
The main part of the thesis is devoted to study of the dynamics of scalar
tachyonic field with non-standard Lagrangian of DBI type, or Lagrangian used in
so-called effective field theories, with various potentials. The interest for this
research originates from Sen's assumptions about the essential role of tachyonic
fields in fundamental interactions of brains, or string theory and their direct
implications on the pre-classical phase of cosmic inflation. It is suitable to use
lo...wer-dimensional models, including zero-dimensional classical-mechanical
analogue, to better understand the dynamics of tachyon systems and their
modeling.
In this thesis original calculations for several specific and important
tachyon potentials are presented in detail. Those potentials are also exactly
solvable in the framework of Friedmann cosmology, and they have their great
physical motivation in the current system of string theory and inflationary
cosmology. The procedure of transition to lower-dimensional theory is the
standard one in field theory. Systems with an infinite number of degrees of
freedom in the limits of classical mechanics transit to the systems with fini te
number of degrees of freedom. The equations of motion, their solutions,
applications of solutions in calculation of the classical Lagrangians and actions
are essential elements of this part of the thesis.
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Besides, the locally equivalent Lagrangians of the standard type are
considered. Those Lagrangians lead to the same equations of motion as well as
initial (DBI) tachyonic Lagrangian. Two methods of finding a standard
Lagrangian are shown. In the first method, one starts from the equations of
motion and the standard Lagrangian is constructed by standard procedure. In this
procedure the field has to be rescaled in appropriate way. The second method is
based on the classical canonical transformations. In this method it is not
necessary to rescale the field. For a certain class of tachyon potentials, by the
correct choice of generating function of canonical transformation, the standard
Lagrangian and quadratic actions can be found. In addition, it was shown that
additional degrees of freedom can be introduced in the theory, by which observed
phase space expands, and the system can now be equivalently described by the
Lagrangian of a standard form, with the identical classic action of initial and
“new” Lagrangian.
A part of the thesis is devoted to the consideration of the previous systems
and the models not only on real, but also on nonarcimedian and ultrametric
spaces. The physical processes under consideration, the formation of the
Universe and the beginning of the inflationary phase, essentially are related to
the quantum effects and the Planck scale, where the standard - Archimedean
geometry and real numbers , and their algebraic extension - complex
numbers , are inadequate or, at least, incomplete. One of the most promising
alternatives is to use nonarchimedean geometry and the fields of p-adic numbers
p . The considered models are based on several tachionic potentials on
Archimedean and nonarchimedean geometries, and fields of numbers and
p on adelic approach. The procedure of quantization of tachyon system in real,
p-adic and adelic case in the framework of the Feynman path integral is shown.
The possible application of the results obtained to describe the period of
cosmological inflation using the tachyon fields are discussed in the final part.
Besides, open problems in the respective quantum cosmological models - the
wave function of the universe, and classic models, with Minkowski metric,
especially with FRW metric, are discussed.
