Finite Quantum Dynamics

Abstract We general-quantize the dynamics of the quantum harmonic oscillator to obtain a covariant finite quantum dynamics in a finite quantum time. The usual central ("superselected") time results from a self-organization. Unitarity necessarily fails, imperceptibly for middle times and grossly near the beginning and end of time. Time and energy interconvert during space-time decondensation or meltdown, at a rate governed by a constant like the Planck power

Simplicial quantum dynamics

Present-day quantum field theory can be regularized by a decomposition into quantum simplices. This replaces the infinite-dimensional Hilbert space by a high-dimensional spinor space and singular canonical Lie groups by regular spin groups. It radically changes the uncertainty principle for small distances. Gaugeons, including the gravitational, are represented as bound fermion-pairs, and space-time curvature as a singular organized limit of quantum non-commutativity. Keywords: Quantum logic, quantum set theory, quantum gravity, quantum topology, simplicial quantization.Comment: 25 pages. 1 table. Conference of the International Association for Relativistic Dynamics, Taiwan, 201

Finite Quantum Theory of the Harmonic Oscillator

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.Ph.D.Committee Chair: Finkelstein, David; Committee Member: Biritz, Helmut; Committee Member: Fox, Ron; Committee Member: Marks, Dennis; Committee Member: Wood, Joh

Finite Quantum Harmonic Oscillator ∗

The harmonic oscillator has infinities that presage those of field theory arising from its singular groups. Group regularization transforms it to a finite quantum theory based on a quantum time. The Heisenberg group, which is unstable with respect to small changes in its structure tensor, becomes the rotation group in three dimensions, which is stable. This results in pronounced violations of equipartition and of the usual uncertainty relations, and to interactions between the previously uncoupled excitation quanta of the oscillator. It freezes out the zero-point energy of extreme soft or hard oscillators, like those responsible for the infrared or ultraviolet divergencies of usual field theories, without much changing medium oscillators. It unifies and quantizes the time, energy, space, and momentum variables. In some extreme conditions this unification allows the interconversion of time and energy on a scale of 10 51 W, the Planck power. 1 The harmonic oscillator problem The three main evolutions of physics in the twentieth century have a suggestive family resemblance. Each introduced a small but new kind of non-commutativity, that of accelerations i