Analysis over Discrete spaces

Let

be a σ-finite nonatomic measure space. We think of the customary analysis based upon

as Continuum Analysis. In contrast, we regard Discrete Analysis as being based upon a countable subset of X rather than upon X itself. The particular version of Discrete Analysis introduced here—we shall call it Poisson Analysis—treats simultaneously all countable subsets of X and distinguishes between them probabalisticly via a counting process whose values are random variables with Poisson distributions. The present paper demonstrates the viability of this idea by applying it to the formal theory of a free Boson field. The customary version of this theory may be phrased in terms of

and the related normal Wiener process N. The alternate version introduced here uses, in place of N, what we shall call a Gaussian jump process. We interpret the resulting theory as Discrete Analysis and compare it with the customary theory. In particular we show that the discrete theory already contains the usual one inside it (see Theorem 1), and that, in the discrete case, symmetries are much more readily implementable by unitary transformations.