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Significance of zero modes in path-integral quantization of solitonic theories with BRST invariance
(1996)

The significance of zero modes in the path-integral quantization of some solitonic models is investigated. In particular a Skyrme-like theory with topological vortices in (1 + 2) dimensions is studied, and with a BRST invariant gauge fixing a well defined transition amplitude is obtained in the one loop approximation. We also present an alternative method which does not necessitate evoking the time-dependence in the functional integral, but is equivalent to the original one in dealing with the quantization in the background of the static classical solution of the non-linear field equations. The considerations given here are particularly useful in - but also limited to -the one-loop approximation.

The constraint structure of the induced 2D-gravity with the Weyl and area-preserving diffeomorphism invariances is analysed in the ADM formulation. It is found that when the area-preserving diffeomorphism constraints are kept, the usual conformal gauge does not exist, whereas there is the possibility to choose the so-called "quasi-light-cone" gauge, in which besides the area-preserving diffeomorphism invariance, the reduced Lagrangian also possesses the SL(2,R) residual symmetry. This observation indicates that the claimed correspondence between the SL(2,R) residual symmetry and the area-preserving diffeomorphism invariance in both regularisation approaches does not hold. The string-like approach is then applied to quantise this model, but a fictitious non-zero central charge in the Virasoro algebra appears. When a set of gauge-independent SL(2,R) current-like fields is introduced instead of the string-like variables, a consistent quantum theory is obtained, which means that the area-preserving diffeomorphism invariance can be maintained at the quantum level.

A new look at the RST model
(1996)

The RST model is augmented by the addition of a scalar field and a boundary term so that it is well-posed and local. Expressing the RST action in terms of the ADM formulation, the constraint structure can be analysed completely. It is shown that from the view point of local field theories, there exists a hidden dynamical field 1 in the RST model. Thanks to the presence of this hidden dynamical field, we can reconstruct the closed algebra of the constraints which guarantee the general invariance of the RST action. The resulting stress tensors TSigma Sigma are recovered to be true tensor quantities. Especially, the part of the stress tensors for the hidden dynamical field 1 gives the precise expression for tSigma . At the quantum level, the cancellation condition for the total central charge is reexamined. Finally, with the help of the hidden dynamical field 1, the fact that the semi-classical static soluti on of the RST model has two independent parameters (P,M), whereas for the classical CGHS model there is only one, can be explained.

A new approach with BRST invariance is suggested to cure the degeneracy problem of ill defined path integrals in the path- integral calculation of quantum mechanical tunneling effects in which the problem arises due to the occurrence of zero modes. The Faddeev-Popov procedure is avoided and the integral over the zero mode is transformed in a systematic way into a well defined integral over instanton positions. No special procedure has to be adopted as in the Faddeev-Popov method in calculating the Jacobian of the transformation. The quantum mechanical tunneling for the Sine-Gordon potential is used as a test of the method and the width of the lowest energy band is obtained in exact agreement with that of WKB calculations.

It is shown that nonvacuum pseudoparticles can account for quantum tunneling and metastability. In particular the saddle- point nature of the pseudoparticles is demonstrated, and the evaluation of path-integrals in their neighbourhood. Finally the relation between instantons and bounces is used to derive a result conjectured by Bogomolny and Fateyev.

Skyrme Sphalerons of an O(3)-oe Model and the Calculation of Transition Rates at Finite Temperature
(1997)

The reduced O(3)-oe model with an O(3) ! O(2) symmetry breaking potential is considered with an additional Skyrmionic term, i. e. a totally antisymmetric quartic term in the field derivatives. This Skyrme term does not affect the classical static equations of motion which, however, allow an unstable sphaleron solution. Quantum fluctuations around the static classical solution are considered for the determination of the rate of thermally induced transitions between topologically distinct vacua mediated by the sphaleron. The main technical effect of the Skyrme term is to produce an extra measure factor in one of the fluctuation path integrals which is therefore evaluated using a measure-modified Fourier-Matsubara decomposition (this being one of the few cases permitting this explicit calculation). The resulting transition rate is valid in a temperature region different from that of the original Skyrme-less model, and the crossover from transitions dominated by thermal fluctuations to those dominated by tunneling at the lower limit of this range depends on the strength of the Skyrme coupling.

A formula suitable for a quantitative evaluation of the tunneling effect in a ferromagnetic particle is derived with the help of the instanton method. The tunneling between n-th degenerate states of neighboring wells is dominated by a periodic pseudoparticle configuration. The low-lying level-splitting previously obtained with the LSZ method in field theory in which the tunneling is viewed as the transition of n bosons induced by the usual(vacuum) instanton is recovered.The observation made with our new result is that the tunneling effect increases at excited states. The results should be useful in analyzing results of experimental tests of macroscopic quantum coherence in ferromagnetic particles.

The tunneling splitting of the energy levels of a ferromagnetic particle in the presence of an applied magnetic field - previously derived only for the ground state with the path integral method - is obtained in a simple way from Schr"odinger theory. The origin of the factors entering the result is clearly understood, in particular the effect of the asymmetry of the barriers of the potential. The method should appeal particularly to experimentalists searching for evidence of macroscopic spin tunneling.