Coupled oscillators, entangled oscillators, and Lorentz-covariant harmonic oscillators

Abstract

Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called $SL(2,c)$. Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.

Typos corrected

References in corpus (1)

• Entanglement of formation for symmetric Gaussian states

Cited by in corpus (10)

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• Coupled oscillators and Feynman's three papers
• Minimum Uncertainty and Entanglement
• Classical analogs of generalized purities, entropies, and logarithmic negativity
• Old Game, New Rules: Rethinking The Form of Physics
• Integration of Dirac's Efforts to construct Lorentz-covariant Quantum Mechanics
• Shannon Entropic Entanglement Criterion in the Simple Harmonic Oscillator