Finite Quantum Theory of the Harmonic Oscillator
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We apply the Segal process of group simplification to the linear harmonic oscillator. The result is a finite quantum theory with three quantum constants instead of the usual one. We compare the classical (CLHO), quantum (QLHO), and finite (FLHO) linear harmonic oscillators and their canonical or unitary groups. The FLHO is isomorphic to a dipole rotator with N=l(l+1) states where l is very large for physically interesting case. The position and momentum variables are quantized with uniform finite spectra. For fixed quantum constants and large N there are three broad classes of FLHO: soft, medium, and hard corresponding respectively to cases where ratio of the of potential energy to kinetic energy in the Hamiltonian is very small, almost equal to one, or very large The field oscillators responsible for infra-red and ultraviolet divergences are soft and hard respectively. Medium oscillators approximate the QLHO. Their low-lying states have nearly the same zero-point energy and level spacing as the QLHO, and nearly obeying the Heisenberg uncertainty principle and the equipartition principle. The corresponding rotators are nearly polarized along the z-axis. The soft and hard FLHO's have infinitesimal 0-point energy and grossly violate equipartition and the Heisenberg uncertainty principle. They do not resemble the QLHO at all. Their low-lying energy states correspond to rotators polaroizd along x-axis or y-axis respectively. Soft oscillators have frozen momentum, because their maximum potential energy is too small to produce one quantum of momentum. Hard oscillators have frozen position, because their maximum kinetic energy is too small to produce one quantum of momentum. Hard oscillators have frozen position, because their maximum kinetic energy is too small to excite one quantum of position.