The n-dimensional truncations of the QM Bosonic creation and annihilation operators: B!(n) and B(n), are considered. Defined on a finite dimensional Hilbert space Hilb(n), they emulate many of the physically desirable properties of the limiting creation and annihilation operators of infinite dimension. The case n=2 defines the Fermionic operators CAR, while CCR is a limit in the strong operator topology. Interesting features beyond those of CCR arise: The phase operator conjugate to the number operator (time operator conjugate to the energy of a quantum oscillator), impossible to define properly in infinite dimensions, becomes definable. Position is quantized; time is quantized; Energy is bounded. A General time operator must be defined by a physically reasonable Anstatz. Position eigenstates have intrinsic dispersion: they violate the uncertainty relation (as they do also in QM) and so may be considered unphysical. There is defined an indefinite form on the finite dimensional Hilbert spaces that leads to a causal ordering of vectors as symbols of quantal processes (not states), which ordering is preserved by a noncompact group of automorphisms conjugate in GL(n, C) to SU(n-1, 1), which contains the Homogeneous Lorentz group as a subgroup. The operators B!(n) and B(n), or Q(n) and P(n) generate by commutation, the Lie algebra su(n), as a kinematic algebra, rather than the nilpotent Heisenberg algebra. Uncertainty relations and diagonalizing transformations are derived. Fourier transforms are defined that connect position with momentum and time with energy. Structural theorems, and theorems regarding the limit of unbounded n are proved.
The analogs of both Schroedinger and Heisenberg equations of motion are presented, and shown to have the standard equational forms as limits as the dimension n becomes infinite. The time operator Anstatz determines these equations as identities.
Possibilities for finding the asymptotic 3+1 dimensionality of spacetime in the large are discussed in terms of operators acting on a Hilbert space of process that are related by naturally ocurring IRREPS of SU(2) rotations. A local, quantized, noncommutative geometry of spacetime can thus be defined, which allows that quantum gravity might be formulated in terms of a plexus of interlocking finite dimensional local algebras that are to define a quantum manifold.
An appendix-J constructs finite prime dimensional true IRREPS of CCR by n x n matricies over specifically constructed finite Galois fields of minimal order.