abstract

The canonical Hamiltonian form of the Einstein–Cartan theory with gauge group SO(3,1) is developed. The connection of the field variables with those of Arnowitt, Deser, and Misner and with Ashtekar’s canonical variables is worked out.

abstract

A rigorous theory of semiclassical limit of the one-dimensional Schrödinger equation based on the Balian–Bloch representation is developed. It is shown that for a large class of potentials a global fundamental solution to the Schrödinger equation can be constructed which can be Laplace transformed with respect to \(\hbar ^{-1}\) (or to some other relevant variable). This global solution has a definite asymptotic series expansion for \(\hbar ^{-1} \to +\infty \). The series is shown to be Borel summable to the global solution itself. Primitive coefficients — some other quantities basic for the quantum one-dimensional theory — are shown to be Borel summable, too. An efficient technique is developed to show both the analytic properties and the Borel summability of energy levels for a large class of potentials. The technique combines the analytic properties of the Stokes graphs and the primitive coefficient identities and is used together with the Bender–Wu method to determine the large order behaviour of the semiclassical series coefficients. The method is extended to a class of perturbing potentials which admit semiclassical treatment. The cubic- quartic single- and double-well potentials are studied in details. Our approach is generalized to \(\hbar ^{-1}\)-dependent potentials.

abstract

Topological arguments suggest that the Weinberg–Salam model possesses unstable solutions, sphalerons, representing the top of energy barriers between inequivalent vacua of the gauge theory. In the limit of vanishing Weinberg angle, such unstable solutions are known: the sphaleron of Klinkhamer and Manton and at large values of the Higgs mass in addition the deformed sphalerons. Here a systematic study of the discrete normal modes about these sphalerons for the full range of the Higgs mass is presented. The emergence of deformed sphalerons at critical values of the Higgs mass is seen to be related to the crossing of zero of the eigenvalue of the particular normal modes about the sphaleron.