International Journal of Modern Physics A

Results of the conference, “String Quantum Gravity and Physics at the Planck Energy Scale,” organized by Dr. Norma Sanchez, in Erice, Sicily, June 21–28, 1992 will appear in Proceedings of the International Workshop of Theoretical Physics, 6th Session, World Scientific Publishing Co. Pte. Ltd., Singapore, 1993. The conference has given new occasion to ask what linkage, if any, ties together particle physics and the quantum theory of gravitation, where this fantastically short distance first displayed its relevance.

A method for nonperturbative path integral calculation is proposed. Quantum mechanics as the simplest example of a quantum field theory is considered. All modes are decomposed into hard (with frequencies ) and soft (with frequencies ) ones, where ω₀ is some parameter. Hard mode contribution is considered through weak coupling expansion. A low energy effective Lagrangian for soft modes is used. In the case of soft modes we apply a strong coupling expansion. To realize this expansion a special basis in functional space of trajectories is considered. A good convergency for the pro posed procedure in the case of potential V(x)=λx⁴ is demonstrated. The ground state energy of the unharmonic oscillator is calculated.

The structure of the free field representation and some four-point correlation functions of the SU(3) conformal field theory are considered.

When a Hamiltonian system is subject to constraints which depend explicitly on time, difficulties can arise in attempting to reduce the system to its physical phase space. Specifically, it is nontrivial to restrict the system in such a way that one can find a Hamiltonian time-evolution equation involving the Dirac bracket. Using a geometrical formulation, we derive an explicit condition which is both necessary and sufficient for this to be possible, and we give a formula defining the resulting Hamiltonian function. Some previous results are recovered as special cases.

We investigate the spectrum of physical states in the W₃ string theory, up to level 2 for a multiscalar string, and up to level 4 for the two-scalar string. The (open) W₃ string has a photon as its only massless state. By using screening charges to study the null physical states in the two-scalar W₃ string, we are able to learn about the gauge symmetries of the states in the multiscalar W₃ string.

We present an example of a coset conformal field theory which cannot be described by the identification current method. To study such examples we determine formulae for the characters of coset conformal field theories.

The supersymmetric WKB quantization condition is used to study the so-called spectral inverse problem. Wavefunctions for the harmonic oscillator and hydrogen atom are obtained from the knowledge of their bound-state energy spectra. The analysis presented is based essentially on a repackaging of the conventional theory of integral equations.

By viewing the sine-Gordon and massive Thirring models as perturbed conformal field theories, one sees that they are different (the difference being observable, for instance, in finite-volume energy levels). The UV limit of the former (SGM) is a Gaussian model, that of the latter (MTM) a so-called fermionic Gaussian model, the compactification radius of the boson underlying both theories depending on the SG/MT coupling. (These two families of conformal field theories are related by a “twist”.) Corresponding SG and MT models contain a subset of fields with identical correlation functions, but each model also has fields the other one does not have; for example, the fermion fields of MTM are not contained in SGM, and the bosonic soliton fields of SGM are not in MTM. Our results imply, in particular, that the SGM at the so-called “free-Dirac point” β²=4π is actually a theory of two interacting bosons with diagonal S-matrix S=−1, and that for arbitrary couplings the overall sign of the accepted SG S-matrix in the soliton sector should be reversed. More generally, we draw attention to the existence of new classes of quantum field theories, analogs of the (perturbed) fermionic Gaussians models, whose partition functions are invariant only under a subgroup of the modular group. One such class comprises “fermionic versions” of the Virasoro minimal models.

The action of an SU(2) multimatrix model can always be rewritten as that of an SO(3) multivector model. In the context of a zero-dimensional model, we argue that all symmetry invariants can be expressed in terms of O(3) invariants and develop a systematic, vector-like approximation. This approximation has no interpretation in terms of subsets of Feynman diagrams of the original model.