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 Lorentzcovariant world?
 fig.(b). In 1939, Wigner introduced
his little groups which are
for the internal spacetime symmetry of the particles. The electron
has its spin with rotational symmetries. The proton has the same
symmetry, but it also has its spacetime 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 boundstates, the Smatrix approach lead to disasters, such as the DashenFrautchi fiasco.
 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 threedimensional 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 Lorentzinvariant differential equations for the boundstate particles. They used harmonic oscillator potential for the bound state. However, their wave functions become infinite when the timeseparation 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 lorentzcovariant boundstate 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 fixups, Feynman's oscillator wave functions
should satisfy the Lorentzcovariant boundary conditions dictated by
the symmetry of Wigner's little group.
 There is at least one set of wave functions which satisfies the Lorentzcovariant
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 boundstate wave function can explain manythings in the real world. Click here for an interesting story.
 It is not a trivial problem to construct Lorentzcovariant boundstate
picture in Lorentzcovariant world. Many distinguished physicists made
important contributions to this subject. I could mention Dirac, Wigner,
and Feynman.
Click here to see their contributions.  The history continues.
Crazy? You may look at my recent papers on this subject.
 copyright@2013 by Y. S. Kim, unless otherwise specified. Revised in
2020.
 Click here for his
home page.
