[time 47] Re: n-body dirac equation

Stephen P. King (stephenk1@home.com)
Mon, 22 Mar 1999 10:46:13 -0500

Dear Hitoshi,

Hitoshi Kitada wrote:
> 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

        I agree with the necessity of Euclidean geometry for QM, using
non-relativist equations. I am waiting for my friend Paul Hanna to
finish up his study of Clifford algrebras to see if they can provide a
way of working out the "correction". He is also familiar with the
Klein-Gordon equations and is interested in the Dirac equations. He has
figured out an equation for relativity in 6 dim. which I thought might
be useful in dealing with how to approximate the canonical Hamiltonian
variables of position and momentum for observers. The reduction of R^6
to R^4 may contain some symmetries that may be helpful. I will see if he
will give me permission to post it here... :)
        A while back I have thought about how strong gravitation acts to
"reduce" possible spatial motions 3 ->1, as in the case when test
particles cross the event horizon of a Schwarchild black hole. There
seems to be a 1 -> 3 "unfolding" (I don't have a good word for this) of
the possible temporal directions at the same time.
        One conceptual difficulty that I think we need to overcome is how sets
of LSs observing each other construct a local cosmos. I do not find the
usual explanation of "why all observers percieve the same arrow of time"
satisfied by Hawking & Ellis's argument, they seem to merely assume it.
There is more to this! The 'selection' of actual observations from the
ensemble of possibles demands are more careful consideration. Bohm
mentions a "contact matrix" C_ij in The Undivided Universe pg.377 that
might give us some clues. :) I have mentioned this before and had no
response. This relates directly to my posts about Weyl's gauge invariant
        While the emission and absorption of photons (and any other particle
for that mater) is well modeled by QM within LSs, the "propagation" and
"dispersion" 'between' LS is not. This related to the Robertson-Walker
metric question... How relativistic "corrections" are made upon
observations of EMF is in need of careful study.
        There is also a need understand the difference between the mass terms
in the internal LS Hamiltonian, such as that you gave above, and the
mass terms used in the "center of Mass" relativistic corrections. We
have a difference between internal "mass" and external "mass." The V(x)
term seems to 'tie' together the particles; could we describe/model this
internal/external relation with some fucntion of it?
        Since QM particles inside the LS can have infinite velocity, how do we
account, if at all, for inertia, e.g. resistence to a change in state of
motion. One of the goals of QG is to account for mass and inertia,
which, up to now, are "penciled in." Also, do we have a way of
predicting the Unruh effect within LS theory?



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