X iv : h ep - t h / 06 03 12 8 v 1 1 6 M ar 2 00 6 Exotic Oscillator , Noncommutative Spacetime and an Associated Non - linear σ - Model

A nonlinear oscillator is studied that enjoys periodic motion with constant amplitude and energy dependent frequency. Relativistic Nambu-Goto type of Lagrangian model with above properties has been proposed. Direct connection to various forms of Noncommutative spacetime has been demonstrated. An underlying non-linear σ-model is discussed in detail. The current algebra has an interesting non-abelian form, with field dependent structure functions. 1 Introduction: In the present paper we will study an unusual form of non-linear " harmonic " oscillator (in a non-relativistic framework), termed as " Exotic Oscillator " in our previous works [1, 2], and its associated relativistic field theory, which happens to be a non-linear σ-model. More generalized form of oscillator model of a similar kind has been studied before [3] but we will focus on some particular aspects of the oscillator which was not discussed. Besides the energy and angular momentum, the model has a set of conserved quantities. These are shown to yield a operatorial form of non-commutaive spacetime, which has appeared elsewhere in our work [4, 5]. The field theoretic extension is new. Its structure is that of a general form of non-linear σ-model, constructed from the multiplet of real scalar fields φ a (x) in arbitrary dimensions but (at least in the present work) no specific transformation property of the multiplet is imposed. It is interesting to note that there is a complete correspondence between the behavior of the Exotic oscillator and the non-linear σ-model at the level of equations of motion and conserved quantities. Indeed, this is not unexpected but from a practical point of view, the latter mapping has helped us to reveal a global " gauge " invariance of the σ-model and its associated conserved Noether current. we have used the term " global " to stress that there is an uncanny similarity between this invariance and a global non-abelian gauge invariance. However, elevating this global " gauge " invariance to a local one poses problem as we explicitly show. We will compare our results regarding local current algebra structures of the model with existing works that discuss current algebra in a generic σ-model setting [6]. We find a very nice mapping between our results and [6] once field dependent structure functions are introduced in our results. Note that [6] uses a conventional form of non-abelian gauge structure with constant structure functions. The field theory studied here is worthwhile for the following …