# Harmonic Oscillators as Bridges between Theories

Let us see how these great physicists used harmonic oscillators to establish
beachheads to new physics.

- Albert Einstein used harmonic oscillators to understand specific heats of solids
and found that energy levels are quantized. This formed one of the
key bridges between classical and quantum mechanics.
- Werner Heisenberg and Erwin Schrödinger formulated quantum mechanics.
The role of harmonic oscillators in this process is well known.
- Paul A. M. Dirac was quite fond of harmonic oscillators. He used oscillator
states to construct Fock space. He was the first one to consider harmonic oscillator
wave functions normalizable in the time variable. In 1963, Dirac used coupled
harmonic oscillators to construct a representation of the O(3,2) de Sitter group
which is the basic scientific language for two-mode squeezed states.
- Hediki Yukawa was the first one to consider a Lorentz-invariant differential
equation, with momentum-dependent solutions which are Lorentz-covariant but
not Lorentz-invariant. He proposed harmonic oscillators for relativistic
extended particles five years before Hofstadter observed that protons are
not point particles in 1955. Some people say he invented a string-model approach to
particle physics.
- Richard Feynman was also fond of harmonic oscillators. When he gave a
**talk at the 1970 Washington meeting**of the American Physical Society, he stunned the audience by telling us not to use Feynman diagrams, but harmonic oscillators for quantum bound states. This figure illustrates what he said in 1970.We are still allowed to use Feynman diagrams for running waves. Feynman diagrams applicable to running waves in Einstein's Lorentz-covariant world. Are Feynman's oscillators Lorentz-covariant? Yes in spirit, but there are many technical problems. Then can those problems be fixed. This is the question. You may be interested in reading about this subject: Lorentz group in Feynman's world.

The photos of Einstein, Heisenberg, Schroedinger, Dirac, Yukawa, and Feynman are from the Niels Bohr Photo Library of the American Physics Institute. License fees paid.