Time-Space Noncommutativity: Quantised Evolutions

In previous work, we developed quantum physics on the Moyal plane withtime-space noncommutativity, basing ourselves on the work of Doplicher et al..Here we extend it to certain noncommutative versions of the cylinder,$\mathbb{R}^{3}$ and $\mathbb{R}\times S^{3}$. In all these models, onlydiscrete time translations are possible, a result known before in the first twocases. One striking consequence of quantised time translations is that eventhough a time independent Hamiltonian is an observable, in scatteringprocesses, it is conserved only modulo $\frac{2\pi}{\theta}$, where $\theta$ isthe noncommutative parameter. (In contrast, on a one-dimensional periodiclattice of lattice spacing $a$ and length $L=Na$, only momentum mod$\frac{2\pi}{L}$ is observable (and can be conserved).) Suggestions for furtherstudy of this effect are made. Scattering theory is formulated and an approachto quantum field theory is outlined.Comment: 25 pages, LaTex; minor corrections, references correcte