q -plane-wave solutions of q -Weyl gravity
- PDF / 580,666 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 17 Downloads / 179 Views
ELEMENTARY PARTICLES AND FIELDS
q -Plane-Wave Solutions of q -Weyl Gravity* V. K. Dobrev1)** and S. G. Mihov2)*** Received August 20, 2007
Abstract—The solutions of the q-deformed equations of quantum conformal Weyl gravity in terms of q-deformed plane waves are given. PACS numbers: 04.50.+h, 04.60.-m DOI: 10.1134/S1063778808050141
1. INTRODUCTION One of the purposes of quantum deformations is to provide an alternative to the regularization procedures of quantum field theory. Applied to Minkowski space–time, the quantum-deformation approach is also an alternative to Connes’ noncommutative geometry [1]. The first step in such an approach is to construct a noncommutative quantum deformation of Minkowski space–time. There are several possible such deformations; cf. [2–6]. We shall follow the deformation of [6] which is different from the others, the most important aspect being that it is related to a deformation of the conformal group. The first problem to tackle in a noncommutative deformed setting is to study the q-deformed analogs of the conformally invariant equations. Here we continue the study of hierarchies of deformed equations derived in [6–8] with the use of quantum conformal symmetry. We give now a description of our setting starting from the simplest example. It is well known that the d’Alembert equation, ϕ(x) = 0,
= ∂ µ ∂µ = (∂0 )2 − (∂)2 ,
(1)
is conformally invariant, cf., e.g., [9]. Here, ϕ is a scalar field of fixed conformal weight, x = (x0 , x1 , x2 , x3 ) denotes the Minkowski space–time coordinates. Not known was the fact that (1) may be interpreted as a conditionally conformally invariant equation and thus may be rederived from a subsingular ∗
The text was submitted by the authors in English. Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria, and Abdus Salam International Center for Theoretical Physics, Trieste, Italy. 2) Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria. ** E-mail: [email protected] *** E-mail: [email protected] 1)
vector of a Verma module of the algebra sl(4), the complexification of the conformal algebra su(2, 2) [7]. The same idea was used in [7] to derive a q-d’Alembert equation, namely, as arising from a subsingular vector of a Verma module of the quantum algebra Uq (sl(4)). The resulting equation is a q-difference equation, and the solution spaces are built on the noncommutative q-Minkowski space– time of [6]. Besides the q-d’Alembert equation, in [7] were derived a whole hierarchy of equations corresponding to the massless representations of the conformal group and parametrized by a nonnegative integer r [7]. The case r = 0 corresponds to the q-d’Alembert equation, while for each r > 0 there are two pairs of equations involving fields of conjugated Lorentz representations of dimension r + 1. For instance, the case r = 1 corresponds to the massless Dirac equation, one pair of equations describing the neutrino, the other pair of equations describing the antineutrino, while the c
Data Loading...
