Gauge fields on truncated Heisenberg space

Abstract

Our aim in this talk was to describe and analyze possible relations between renormalizability, noncommutative gravity and matrix models. The initial idea was to explain the fact that the renormalizable Grosse-Wulkenhaar model is a model of scalar field on a curved noncommutative space. The space itself is closely related to the oscillator representation of coordinate and momentum in quantum mechanics, or more precisely, to its finite matrix truncations. Truncated Heisenberg space has highly nontrivial geometry. It is three-dimensional and axially symetric (around the z-axis; the generator of rotations is M = 1/2 ((μx)2+(μy)2+μ′z)). The space is curved and components of the curvature tensors can be calculated. The exterior algebra has interesting properties: it is well defined but quite different from the corresponding commutative algebra: For example, for value of noncommutativity parameter ε = 1, the rules of multiplication imply that θ1θ2θ3 = 0 while at the same time θ2θ3θ2 ≠ 0 which is the reason why we choose the latter to define the volume form. For the same value e = 1 the truncated Heisenberg algebra has finite representations, while the contraction μ′ →0 gives the Heisenberg algebra with its infinite-dimensional representation. We use this contraction, or dimensional reduction, to define models of scalar and gauge fields. We discuss in some detail how to construct classical U1 gauge theory following the same noncommutative-geometric logic as for the scalar field, in order to obtain a candidate for renormalizable gauge theory. As a result we obtain the action SYM =1/2 ∫ ((1-ε2)(F12)2-2(1- ε2)μF12φ +(5-ε2) μ2φ2+4iεF12φ2 +(D1φ)2+(D2φ)2- ε2{p1+A1,φ}2- ε2{p2+A2,φ}2) which consists of a gauge field and a scalar field coupled in a particular way. Both fields propagate except in the case e = 1, in which the action reduces to SYM|ε=1 =1/2 ∫ (D1φ)2+(D2φ) 2+4μ2φ2+4iF12φ 2-{p1+A1,φ}2-{p 2+A2,φ}2. (9.1) In general, this action implies mixed gauge-scalar propagators but all of them are related to or expressible in terms of the Mehler kernel, [8]. The next step would be to quantize the presented model. Preliminary results in this direction are encouraging: First, the model has as a solution to the equations of motion the usual vacuum A1 = 0, A2 = 0, φ = 0. Further, a nilpotent BRST transformation s can be defined and it leaves the quantum action invariant. The 'only' remaining thing to perform is the explicit quantization, and this we will try to do in our future work. © Copyright owned by the author(s).