Hitoshi Kitada (hitoshi@kitada.com)
Mon, 22 Mar 1999 23:41:03 +0900
Dear Stephen,
Thanks for your information on n-body Dirac equation. I visited all pages, but
all seemed to be concerned with some NON-relativistic approximations.
I know Volker (Volker Enss, with whom I stayed at Caltech for almost 6 months
in 1985 or so and met also in Denmark and some other places). I have his
papers on inverse scattering on multi-dimensional scattering. His Hamiltonian
is also an approximation. One possibility is to choose Klein-Gordon equation,
but also in this case the invariance with respect to Lorentz or Poincare
transformation breaks down when one considers three or more body case. Also
there is an equation that seemed to have been abandoned at the discovery of
Dirac equation. The Hamiltonian of the equation is
H= \sqrt{p^2+m^2} + V(x),
where V(x) is the sum of pair potentials V_{ij}(x) over all pairs i, j of
particles. As V(x) is a potential describing action-at-a-distance, H is not
Lorentz invariant again. (I derived this type of equation as a Hamiltonian
describing actual observations in some of my papers (e.g. time_IV.tex).)
Volker's results cover this type of Hamiltonians.
To describe an exact N-body situation, it seems that we have to return to
Euclidean geometry if we want to retain quantum mechanics.
Best wishes,
Hitoshi
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