O ct 1 99 8 Deformation quantization of the n - tuple point

Contrary to the classical methods of quantum mechanics, the deformation quan-tization can be carried out on phase spaces which are not even topological manifolds. In particular, the Moyal star product gives rise to a canonical functor F from the category of affine analytic spaces to the category of associative (in general, non-commutative) C-algebras. Curiously, if X is the n-tuple point, x n = 0, then F (X) is the algebra of n × n matrices. 1. Introduction. This short note, which is largely about an entertaining interpretation of the classical algebra of n × n-matrices as a quantized n-tuple point, is almost a mathematical anecdote. This is also an attempt to understand what a quantum mechanical system may be on spaces like the " cross " X 1 = {(x, y) ∈ R 2 | xy = 0}, the " tick " X 2 = {(x, y) ∈ R 2 | y 2 − x 3 = 0} or the real line with one double point X 3 = {(x, y) ∈ R 2 | xy = 0, y 2 = 0} which either fail to be topological manifolds or/and have nilpotents in their structure sheaves. Contrary to the standard methods of quantum mechanics, the deformation quantization [2] (see also [5] for an up-to-date overview) easily sustains the introduction of this type of singularities and equippes the (complexified) structure sheaves of the associated phase spaces with well-defined one-parameter non-commutative associa-tive star products * which, however, depend meromorphically on the Planck constant. Their physical interpretation is left to the imagination of the reader.