Effective field theory for non-commutative spacetime: a toy model

Abstract

A novel geometric model of a non-commutative plane has been constructed. We demonstrate that it can be construed as a toy model for describing and explaining the basic features of physics in a non-commutative spacetime from a field theory perspective. The non-commutativity is induced internally through constraints and does not require external interactions. We show that the non-commutative spacetime is to be interpreted as having an internal angular momentum throughout. Subsequently, the elementary excitations—i.e., point particles—living on this plane are endowed with a spin. This is explicitly demonstrated for the zero-momentum Fourier mode. The study of these excitations reveals in a natural way various stringy signatures of a non-commutative quantum theory, such as dipolar nature of the basic excitations [Phys. Lett. B 455 (1999) 129] and momentum dependent shifts in the interaction point [Phys. Rev. D 62 (2000) 066004]. The observation [JHEP 9909 (1999) 032] that non-commutative and ordinary field theories are alternative descriptions of the same underlying theory, is corroborated here by showing that they are gauge equivalent.Also, treating the present model as an explicit example, we show that, even classically, in the presence of additional constraints, (besides the usual ones due to reparameterization invariances), the equivalence between Nambu–Goto and Polyakov formulations is subtle.