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The new situation in the case of noncommutative quantum mechanics is conceptually different from that encountered in noncommutative quantum field theory. Indeed, while in the relativistic case what becomes noncommutative are the parameters in the theory (time and space), in the nonrelativistic one (a single parameter theory) the noncommutativity just modifies the algebra of the basic observables.
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Everywhere in this paper the Einstein summation convention is used. According to it, repeated indices in a monomial expression are summed up from 1 to N unless otherwise is indicated.
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One is, of course, also familiar with the problem of quantizing a system which does not possess a classical analogue, such as the spin 1/2 system. All that is required here is the equal-time algebra obeyed by the basic observables leading to a complete specification of the physical system, together with the structure of the Hamiltonian operator.
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