On the Operator Algebra for the Space-time Uncertainty Relations

The purpose of this note is to show that the construction of the C-algebra for the space-time uncertainty relations which was introduced by Doplicher, Fredenhagen and Roberts [2,3,4] fits comfortably into the deformation quantization framework developed in [5]. This has the mild advantages that one can work directly with functions on space-time rather than with their Fourier transforms, the treatment of the unbounded space-time operators is fairly smooth, and the sense in which one has a deformation of commutative space-time is made technically precise. Possibly our techniques will be useful in some related situations. We will not repeat here the physical motivation and the treatment of the uncertainty relations themselves given in [2,3,4]. We will deal only with the construction of the C-algebra and the affiliated operators having the desired properties. To the extent that there is no additional complication, we will work in slightly greater generality than immediately needed, both because this might be useful at some later time, and because it clarifies which aspects of structure are involved. We now recall from [4] the desired mathematical properties, in the form most convenient for our purposes. Let M0 denote Minkowski space, and let L denote the full Lorentz group. Let L act on 4× 4 matrices by similarity, that is, Λ ∈ L sends the matrix σ to ΛσΛ (where t = transpose). Let σ0 denote the standard symplectic matrix (as in 3.28 of [4]), and let Σ denote its orbit under L. Then Σ is a manifold consisting of certain invertible skew-symmetric matrices, and is a homogeneous space for L. (See 3.24 of [4].) We want unbounded selfadjoint operators qμ, μ = 0, . . . , 3, corresponding to the standard coordinates on space-time. Let Qμν = −i[qμ,qν ],