International Journal of Modern Physics A

In the framework of a systematic approach, the present status of the problem of instanton-like transitions in high energy particle collisions is reviewed. The arguments in favor of the exponential form of the total two-particle cross section in the one-instanton sector, exp[1/g²F(E/E_sph)], are summarized. The perturbative technique for the evaluation of the function F(E/E_sph) is considered in detail. The nonperturbative semiclassical approach to the evaluation of F(E/E_sph), which may provide a reasonable basis for the numerical study of the problem, is discussed.

The temperature renormalization group equation (TRGE) is compared with a diagrammatic expansion for the (ϕ⁴)₄-theory. It is found that the one-loop TRGE resums the leading powers of temperature for the effective mass. A two-loop contribution to TRGE is required to do the leading resummation for the coupling constant. It is also shown that the higher order TRGE resums subleading powers of temperature.

Spontaneous compactification of Einstein–Cartan theory with quadratic Gauß–Bonnet term is investigated. For the homogeneous spaces SU(3)/U(1) × U(1), Sp(2)/SU(2) × U(1) and G₂/SU(3) we find the most general form of G-invariant metric and torsion and find solutions of spontaneous compactifying type. It turns out that these solutions admit a geometrical interpretation.

We investigate the Abelian bosonization procedure in the light of a path integral formulation. The correlation functions of the Bose and Fermi fields are directly related by use of several changes of variables including chiral rotations à la Fujikawa. Operator identities are replaced by statements on partition functions with suitably defined sources. The Thirring and sine–Gordon models as well as the Schwinger model are treated. We also discuss a generalized Thirring model.

We show that three-dimensional simplicial quantum gravity, as described by dynamically triangulated manifolds, is connected with a Gaussian model determined by the simple homotopy types of the underlying manifolds.

By exploiting this result it is shown that the partition function of three-dimensional simplicial quantum gravity is well defined in a convex region in the plane of the gravitational and cosmological coupling constants. Such a region is determined by the Reidemeister–Franz torsion invariants associated with orthogonal representations of the fundamental groups of the set of manifolds considered. The system shows a critical behavior and undergoes a first order phase transition at a well-defined value of the couplings, again determined by the torsion invariants. On the critical line the partition function can be explicitly related to a Gaussian measure on the general linear group GL(∞, R), showing evidence of a well-defined thermodynamical limit of the theory, with a stable (vacuum) configuration corresponding to three-dimensional (homology) manifolds. The first order nature of the transition yielding such a configuration seems to support the belief in the absence of a continuum limit of the theory. More generally, the approach presented here provides further analytical support for the picture of three-dimensional simplicial quantum gravity which has been abstracted from numerical simulations.

We describe the structure of models with effective many-fermion interaction and a finite cutoff in the symmetry-breaking regime. The large-N_c tadpole mechanism for the DSB and the fine-tuning condition in providing the scale invariance are used to classify the set of quasilocal vertices relevant for the dynamic mass formation. It is found that in such effective models the vertices with four, six and eight fermions only should be retained in four-dimensional space.