International Journal of Modern Physics A

This theoretical review is intended to give nontheorists a flavor of the ideas driving the current efforts to experimentally find supersymmetry. We discuss the main reasons behind the expectation that supersymmetry may be "just around the corner" and may be discovered in the near future. We use simple quantum-mechanical examples to illustrate the concept — and the power — of supersymmetry, the possible ways to break supersymmetry, and the dynamical generation of small scales. We then describe how this theoretical machinery helps shape our perception of what physics beyond the electroweak scale might be.

In the covariant-gauge two-dimensional quantum gravity, various derivations of the critical dimension D=26 of the bosonic string are critically reviewed, and their interrelations are clarified. It is shown that the string theory is not identical with the proper framework of the two-dimensional quantum gravity, but the former should be regarded as a particular aspect of the latter. The appearance of various anomalies is shown to be explainable in terms of a new type of anomaly in a unified way.

A previously proposed generalized BRST quantization on inner product spaces for second class constraints is further developed through applications. This BRST method involves a conserved generalized BRST charge Q which is not nilpotent, Q²≠0, but which satisfies Q=δ+δ^†, δ²=0, and by means of which physical states are obtained from the projection δ|ph>=δ^†|ph>=0. A simple model is analyzed in detail from which some basic properties and necessary ingredients are extracted. The method is then applied to a massive vector field. An effective theory is derived which is close to that of the Stückelberg model. However, since the scalar field here is introduced in order to have inner product solutions, a massive Yang–Mills theory with polynomial interaction terms might be possible to cosntruct.

In SU(2) Seiberg–Witten theory, it is known that the dual pair of fields are expressed by hypergeometric functions. As for the theory with SU(3) gauge symmetry without matters, it was shown that the dual pairs of fields can be expressed by means of the Appell function of type F₄. These expressions are convenient for analyzing analytic properties of fields. We investigate the relation between the Seiberg–Witten theory of rank two gauge group without matters and hypergeometric series of two variables. It is shown that the relation between gauge theories and Appell functions can be observed for other classical gauge groups of rank two. For B₂ and C₂, the fields are written in terms of Appell functions of type H₅. For D₂, we can express fields by Appell functions of type F₄ which can be decomposed to two hypergeometric functions, corresponding to the fact SO(4)~SU(2)×SU(2). We also consider the integrable curve of type C₂ and show how the fields are expressed by Appell functions. However in the case of exceptional group G₂, our examination shows that they can be represented by the hypergeometric series which does not correspond to the Appell functions.

The supersymmetric generalization of a recently proposed Abelian axial gauge model with antisymmetric tensor matter fields is presented.

We study the nonrelativistic limit of the quantum theory of a Chern–Simons field minimally coupled to a scalar field with quartic self-interaction. The renormalization of the relativistic model, in the Coulomb gauge, is discussed. We employ a procedure to calculate scattering amplitudes for low momenta that generates their |p|/m expansion and separates the contributions coming from high and low energy intermediary states. The two-body scattering amplitude is calculated up to order p²/m². It is shown that the existence of a critical value of the self-interaction parameter for which the two-particle scattering amplitude reduces to the Aharonov–Bohm one is a strictly nonrelativistic feature. The subdominant terms correspond to relativistic corrections to the Aharonov–Bohm scattering. A nonrelativistic reduction scheme and an effective nonrelativistic Lagrangian to account for the relativistic corrections are proposed.

Causal anomalies in two Kaluza–Klein gravity theories are examined, particularly as to whether these theories permit solutions in which the causality principle is violated. It is found that similarly to general relativity the field equations of the space–time–mass Kaluza–Klein (STM-KK) gravity theory do not exclude violation of causality of Gödel type, whereas the induced matter Kaluza–Klein (IM-KK) gravity rules out noncausal Gödel-type models. The induced matter version of general relativity is shown to be an efficient therapy for causal anomalies that occurs in a wide class of noncausal geometries. Perfect fluid and dust Gödel-type solutions of the STM-KK field equations are studied. It is shown that every Gödel-type perfect fluid solution is isometric to the unique dust solution of the STM-KK field equations. The question as to whether 5D Gödel-type noncausal geometries induce any physically acceptable 4D energy–momentum tensor is also addressed.

A variational calculation scheme of SU(N) lattice gauge Wilson models for large but finite N is presented. The order parameter E_P and the phase transition points of lattice SU(N) gauge systems for N=4~6 are calculated. The results are in good agreement with Monte Carlo simulations. An extrapolation procedure for determination of the phase transition point in the large N limit is proposed.

Exploiting the Lax form of the Korteweg-de Vries equation, we determine the whole set of its symmetry operators and relate them to conserved quantities. We also study a KdV supersymmetric version and put in evidence its even (bosonic) and odd (fermionic) symmetries. We then get the Lie superalgebra characterizing this supersymmetric nonlinear equation.