It is argued that the familiar algebra of non-commutative space-time with c-number θμν is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting θμν to an anti-symmetric tensor operator θ̂μν. The simplest among them is the Doplicher-Fredenhagen-Roberts (DFR) algebra, in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional θ-space of the hermitian operators, θ̂μν . It is shown that we then recover the Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need for the compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to the division of the θ-space into two disjoint spaces not connected by any Lorentz transformation, so that the CCZ covariant moment formula holds in each space separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on M4 × Z2 obtained in the commutative limit.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy (miscellaneous)