Abstract. In previous work [7], we developed quantum physics on the Moyal plane with time-space noncommutativity, basing ourselves on the work of Doplicher et al. [1]. Here we extend it to certain noncommutative versions of the cylinder, ³ and × S³. In all these models, only discrete time translations are possible, a result known before in the first two cases [2]–[6]. One striking consequence of quantised time translations is that even though a time independent hamiltonian is an observable, in scattering processes, it is conserved only modulo 2π/θ, where θ is the noncommutative parameter. (In contrast, on a one-dimensional periodic lattice of lattice spacing a and length L = Na, only momentum mod 2π/L is observable (and can be conserved).) Suggestions for further study of this effect are made. Scattering theory is formulated and an approach to quantum field theory is outlined.

Key words: Field Theories in Lower Dimensions; Non-Commutative Geometry

E-print number: hep-th/0410067
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Received 18 October 2004, accepted for publication 24 November 2004
Published 4 January 2005