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We study the oscillations of a scalar field on a noncommutative disc implementing the boundary as the limit case of an interaction with an appropriately chosen confining background. The space of quantum fluctuations of the field is finite dimensional and displays the rotational and parity symmetry of the disc. We perform a numerical evaluation of the (finite) Casimir energy and obtain similar results as for the fuzzy sphere and torus.
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We revise and extend the algorithm provided in [1] to compute the finite Connes distance between normal states. The original formula in [1] contains an error and actually only provides a lower bound. The correct expression, which we provide here, involves the computation of the infimum of an expression which involves the transverse component of the algebra element in addition to the longitudinal component of [1]. This renders the formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a non-trivial task, but under rather generic conditions it turns out that the Connes distance is proportional to the trace norm of the difference in the density matrices, leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [2] in our Hilbert-Schmidt operatorial formulation. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere is never reached for any finite $n$-representation of $SU(2)$. Indeed, for the case of maximal non-commutativity ($n = 1/2$), the finite distance is shown to coincide exactly with the above mentioned lower bound, with the transverse component playing no role. This, however starts changing from $n=1$ onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our new and modified algorithm.
Recent work [hep-th/0504183,hep-th/0508002] indicates an approach to the formulation of diffeomorphism invariant quantum field theories (qfts) on the Groenewold-Moyal (GM) plane. In this approach to the qfts, statistics gets twisted and the S-matrix in the non-gauge qfts becomes independent of the noncommutativity parameter theta^{mu u}. Here we show that the noncommutative algebra has a commutative spacetime algebra as a substructure: the Poincare, diffeomorphism and gauge groups are based on this algebra in the twisted approach as is known already from the earlier work of [hep-th/0510059]. It is natural to base covariant derivatives for gauge and gravity fields as well on this algebra. Such an approach will in particular introduce no additional gauge fields as compared to the commutative case and also enable us to treat any gauge group (and not just U(N)). Then classical gravity and gauge sectors are the same as those for theta^{mu u}=0, but their interactions with matter fields are sensitive to theta^{mu u}. We construct quantum noncommutative gauge theories (for arbitrary gauge groups) by requiring consistency of twisted statistics and gauge invariance. In a subsequent paper (whose results are summarized here), the locality and Lorentz invariance properties of the S-matrices of these theories will be analyzed, and new non-trivial effects coming from noncommutativity will be elaborated. This paper contains further developments of [hep-th/0608138] and a new formulation based on its approach.
Twisted quantum field theories on the Groenewold-Moyal plane are known to be non-local. Despite this non-locality, it is possible to define a generalized notion of causality. We show that interacting quantum field theories that involve only couplings between matter fields, or between matter fields and minimally coupled U(1) gauge fields are causal in this sense. On the other hand, interactions between matter fields and non-abelian gauge fields violate this generalized causality. We derive the modified Feynman rules emergent from these features. They imply that interactions of matter with non-abelian gauge fields are not Lorentz- and CPT-invariant.
We study all the symmetries of the free Schrodinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.
We show how to get a non-commutative product for functions on space-time starting from the deformation of the coproduct of the Poincare group using the Drinfeld twist. Thus it is easy to see that the commutative algebra of functions on space-time (R^4) can be identified as the set of functions on the Poincare group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincare group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the dffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.
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