A.P. Demichev

Quantum Poincaré group

Abstract We construct a quantum deformation of the Poincare group using the twistor approach to the definition of Minkowski space ant the multiparametric deformation of the GL (4, C ) group.

q-deformed path integral

Abstract Using differential and integral calculi on the quantum plane which are invariant with respect to quantum inhomogeneous Euclidean group E(2) q , we construct the path integral representation for the quantum mechanical evolution operator kernel of the q -oscillator.

The Casimir operators of inhomogeneous groups

We have found the number of invariant operators for the inhomogeneous groups IGL(n, R), ISL(n, R), ISO(p, q), IU(p, q), ISU(p, q), ISp(2n), i.e. the inhomogeneous groups with the classical homogeneous subgroups, and also for the Weyl group W(p, q). For some special cases explicit forms of the invariant operators are obtained. We also discuss the methods applied, together with problems in some cases, possible further developments and relevance for the supersymmetric theories.

Noncommutative quantum field theory: unitarity and discrete time

Abstract The unitarity condition for noncommutative field theories on compact space–times is considered. Although such theories are free of ultraviolet divergences, the compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in absence of the star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. We also discuss violation of causality in terms of commutators for observables and possibility for a violation of the spin–statistics theorem.

Manifestations of space-time multidimensionality in scattering of scalar particles

Abstract We analyze a possibility of experimental detection of the contribution of the Kaluza-Klein tower of heavy particles to the scattering cross-section in a six-dimensional scalar model with two dimensions being compactified to the torus with the radii R . It is shown that there is a noticeable effect even for the energies of colliding particles below R −1 which may be observed in future collider experiments if R −1 is of the order of 1 TeV.

BRST anomaly in string theory

Abstract We show that BRST transformations in string theory on Riemann surfaces of genus h ⩽2 are anomalous. The anomaly is due to the non-invariance of the integration domains over antighosts and auxiliary fields under BRST transformations.

Gauge fixing: chiral model versus string theory

Abstract The chiral model is discussed as an analog to the string theory to study anomalies in BRST Ward identities (WI). We construct a counterpart of the conformal gauge rendering the anomalous WI and also another gauge with non-anomalous WI. The source of the anomaly in the chiral model is shown to be the same as in the string theory.

Anomalous BRST ward identity in string theory

BRST transformations are studied in the path integral approach to string theory of Riemann surfaces of genus h ⩾ 2. The BRST Ward identity (WI) is shown to be anomalous, the anomaly being due to non-invariance of the functional integration domain under BRST transformations. The distinction between complete Lagrange BRST transformations including the metric and the auxiliary field and the commonly used “truncated” BRST transformation is discussed in detail. The problem of decoupling of spurions from physical operators is investigated.

QUANTUM FIELD THEORY ON NONCOMMUTATIVE SPACE–TIME: THE RELATION BETWEEN ULTRAVIOLET BEHAVIOUR, TOPOLOGY AND SYMMETRY

We study properties of a scalar quantum field theory on two-dimensional noncommutative space–times. We show that ultraviolet behaviour of a field theory on noncommutative spaces is sensitive to the topology of the space–time, namely to its compactness. The compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in the absence of a star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time.

Fields on Noncommutative Manifolds

The essence of the noncommutative geometry consists in reformulating the geometry in terms of commutative algebras and modules of smooth functions, and then generalizing them to their noncommutative analogs, [1]. Here we shall describe models on a Fuzzy cylinder R S 1={τ on real line interpreted as the time, x ±=ρe ±iϕ identified with the circle S1}, for details see [2], [3].