  We construct a class of representations of the Heisenberg algebra in terms of
the complex shift operators subject to the proper continuous limit imposed by
the correspondence principle. We find a suitable Hilbert space formulation of
our construction for two types of shifts: (1) real shifts, (2) purely imaginary
shifts. The representations involving imaginary shifts are free of spectrum
doubling. We determine the corresponding coordinate and momentum operators
satisfying the canonical commutation relations. The eigenvalues of the
coordinate operator are in both cases discrete.
