# Einstein and Standing Waves

Since Einstein's formulation of special relativity in 1905, the most important development in physics was the formulation of quantum mechanics in 1927.

• In quantum mechanics, free particles are running waves, and extended objects consist of standing waves. The hydrogen atom is a standing wave, so is the proton in the quark model. Click here for a story of the moving hydrogen atom.

• fig.(a). There are standing wave and running waves. Bound states are standing waves with discrete energy spectra. Is this difference derivable in Einstein's Lorentz-covariant world?

• fig.(b). In 1939, Wigner introduced his little groups which are for the internal space-time symmetry of the particles. The electron has its spin with rotational symmetries. The proton has the same symmetry, but it also has its space-time extension.

For running waves, with its scattering matrix, quantum field theory provides a satisfactory answer to the problem. The best was to approach this problem is to use Feynman diagrams.

For bound-states, the S-matrix approach lead to disasters, such as the Dashen-Frautchi fiasco.

 Feynman giving his talk in 1970. The room was dark. I did not use my flash when I took this photo. Chew-Frautshi plot indicating the hadronic mass spectra are like that of the energy levels of the three-dimensional harmonic oscillator.

• fig.(c). In 1970, at the Spring meeting of the American Physical Society, Feynman gave a talk which surprised many people. He remarked that quantum field theory is a giant step to make quantum mechanics consistent with Einstein's special relativity. However, this field theory is not able to produce hadronic mass spectra which are similar to the degeneracy of the three-dimensional harmonic oscillator.

He then worked about the two and three particles bound together by harmonic oscillator potential. In 1971, together with his students, published a a paper in the Physical Review D.

The point of this paper is to use harmonic oscillators for bound states, instead of Feynman diagrams. They then write Lorentz-invariant differential equations for the bound-state particles. They used harmonic oscillator potential for the bound state. However, their wave functions become infinite when the time-separation between the constituent become large. Their wave functions are wrong.

• fig.(d). In 1979, with my younger colleagues, I showed that their wave functions can be fixed up according to Wigner's little groups, and attempted to publish a paper in the Journal of Mathematical Physics. The referee said what we say is correct, but we should not make reference to the paper of Feynman et al. because their paper does not make any sense from the mathematical point of view. Here is my paper about the 1971 paper of Feynman et al. without making the reference to that paper. Interesting world! You may click here directly from my computer.

In any case, this paper has its own merit, as the referee pointed out. This paper gives a lorentz-covariant bound-state wave function for bound states, satisfying the symmetry of Wigner's little group for massive particles. Since Wigner's little groups are subgroups of the Lorentz group, Einstein prevails both insider and outside the particle.

• Thus, after appropriate fix-ups, Feynman's oscillator wave functions should satisfy the Lorentz-covariant boundary conditions dictated by the symmetry of Wigner's little group.

• There is at least one set of wave functions which satisfies the Lorentz-covariant boundary conditions, and which can be placed in Step 1 of the Comet/Planet table given at right. It is indeed gratifying to be fill in this step using only the existing principles of quantum mechanics and relativity. We did not have to invent any new physics. The next step is to construct one mathematical device which will take care of both scattering and bound states.

The most cruel question is whether this covariant bound-state wave function can explain manythings in the real world. Click here for an interesting story.

• It is not a trivial problem to construct Lorentz-covariant bound-state picture in Lorentz-covariant world. Many distinguished physicists made important contributions to this subject. I could mention Dirac, Wigner, and Feynman.