The construction of tensor operators in a D=10 noncommutative space

Full text. In a recent work a new version of noncommutative quantum mechanics (NCQM) has been presented by one of us, where not only the coordinates xμ and their canonical momenta pμ are considered as operators in Hilbert space H, but also the objects of noncommutativity θμν and their canonical conjugate momenta πμν. All these operators belong to the same algebra and have the same hierarchical level. This enlargement of the usual set of Hilbert space operators allows the theory to be invariant under the rotation group SO(D). Rotation invariance in a nonrelativistic theory, is fundamental if one intends to describe any physical system in a consistent way. In other words it was proposed a minimal canonical extension of the DFR algebra, which permits to implement Poincare invariance as a dynamical symmetry in NCQM. The main motivation of DFR to study the noncommutative relations was the belief that exact measurements of space-time localization could confine photons through energy yield to test particles in order to create a gravitational field. In this work we construct new operators in order to formulate a D=10 Fock space using a new framework where the object of noncommutativity θμν represents independent degrees of freedom. Namely, θμν is an operator as well as its canonical momentum πμν and both live in an augmented D = 10 Hilbert space. This structure comprises the minimal canonical extension of the Doplicher-Fredenhagen-Roberts (DFR) algebra and consequently a modified Poincare group of symmetry. (author)