Finite dimensional quantum mechanics

The basic properties of nonrelativistic finite-dimensional quantum mechanics are presented. Position, momentum and Hamiltonian operators are introduced. The space of states in the case is the space C/sup r/. Properties of position and momentum operators are studied. Dynamics of the particles is considered. An Approximation Theorem is proved: finite dimentional quantum mechanics approximates ordinary quantum mechanics and the approximation gets better as the dimension increases. Second quantization, the symmetric and antisymmetric Fock spaces are discussed. It is shown that the Hilbert space L2(R3) of a single nonrelativistic particle p is the second quantization of the finite-dimensional space C3. Creation, annihilation and field operators are introduced and represented in terms of different operators. A Theorem is proved which states that there is no conventional scattering in finite dimensional quantum mechanics. An isomorphism between SV and L2(R/sup r/) is shown to be the decomposition by generalized eigenfunctions of field operators. Corresponding spaces of these and generalized functions are constructed. A functional space isomorphism is obtained for the complete tensor product of a finite dimensional space V = C/sup r/. Irreducible representations of the symmetry group in a complete tensor are constructed. Physical applications to elementary particles and quarks are briefly discussed