Abstract
The main aspect of this thesis was to extend the range of applicability for
functional integrals in quantum statistics and quantum field theory. Distributed over
four parts, this thesis combines the formal justification of dealing with continuous
path integrals from a perturbative point of view and a general solution for Gaussian
path integrals in phase space with variational perturbation theory as a powerful
resummation method which is also applicable for strongly coupled systems, where perturbative
methods fail. The perturbative column on the one hand and the nonperturbative one on the other hand
are bridged by a recursive graphical construction method which permits a systematic
generation of all topologically different Feynman diagrams contributing to any order of
perturbation with their correct multiplicities. As an interesting detail, the applicability
of this method in quantum field theory is demonstrated for quantum electrodynamical scattering
processes.
Motivated by the partial nonexistence of analytic results we have applied variational perturbation theory
for atomic systems at arbitrary temperature and thermodynamical properties of fluctuating membranes.
To this end, we have extended and generalized variational perturbation theory in a manifold way.
For calculating density matrices, we generalized the
smearing formula which accounts for the effects of thermal and quantum fluctuations.
This was essential
for the treatment of nonpolynomial interactions. We applied the theory to calculate the
particle density in the double-well potential, and the electron density in the Coulomb
potential, the latter as an example for nonpolynomial application. In both cases, the
approximations were satisfactory.
We have also calculated the effective classical potential for the hydrogen atom
in a magnetic field. For this we have extended variational perturbation theory
to phase space to make it applicable to physical systems with uniform external magnetic field.
The effective classical potential containing the complete
quantum statistical information of the system was determined in first-order
variational perturbation theory. For zero-temperature, it gave the binding
energy of the system. Our result consists of a single analytic expression which is quite accurate
at all temperatures and magnetic field strengths. The different asymptotic behavior of the
perturbation series for the binding energy for weak and strong magnetic fields has been investigated
in detail. In the weak-field case, we confirmed the power series character of the expansion, while for
strong magnetic field strengths a deeply structured logarithmic behavior occurs.
As an application for strong-coupling theory in membrane physics,
we have calculated the universal constant $\alpha$ occurring in the pressure
law of a membrane fluctuating between two walls. This has been
done by replacing the walls by a smooth potential.
The anharmonic part of the smooth potential was treated perturbatively
and the strong-coupling limit of the power series was
calculated by variational perturbation theory. Extrapolating the lowest four approximations
to infinity yields a pressure constant, which is in very good agreement with Monte Carlo values.
We have also calculated the pressure constants for a stack of different numbers
of membranes between two walls
in excellent agreement with results from Monte Carlo simulations.
The requirement that the membranes cannot penetrate each other was accounted for
by introducing a repulsive potential and going to the strong-coupling limit
of hard repulsion.
We have used the similarity of the membrane system to a stack of strings enclosed by
line-like walls, which is exactly solvable,
to determine the potential parameters in such a way that the two-loop result is exact.
This minimizes the neglected terms in the variational perturbation expansion, when
applying the same potential to membranes. |
