[time 49] RE: n-body dirac equation

Hitoshi Kitada (hitoshi@kitada.com)
Tue, 23 Mar 1999 18:37:34 +0900

Dear Stephen,

In this note I comment on the direction of time (or the 'arrow' of time).

-----Original Message-----
From: Stephen P. King <stephenk1@home.com>
To: Hitoshi Kitada <hitoshi@kitada.com>
Cc: Time List <time@kitada.com>
Date: Tuesday, March 23, 1999 12:53 AM
Subject: [time 47] Re: n-body dirac equation

>Dear Hitoshi,
>Hitoshi Kitada wrote:
>> Dear Stephen,
>> Thanks for your information on n-body Dirac equation. I visited all pages,
>> all seemed to be concerned with some NON-relativistic approximations.
>> I know Volker (Volker Enss, with whom I stayed at Caltech for almost 6
>> 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
>> is also an approximation. One possibility is to choose Klein-Gordon
>> 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.

First a quotation from Unruh:

>From W.G. Unruh, "Time, Gravity, and Quantum Mechanics" (gr-qc/9312027, page

> III) Probabilities:
> One of the suggested resolutions of the problem mentioned above is to select
> one of the variables of the unconstrained theory is selected as the time.
> The physically intuitive reasoning is that time in reality is an
> unobservable feature of the world anyway. What actually passes for
> time is the reading on various and sundry pieces of physical apparatus
> called clocks. If you as a child are late for school, it is not because
> your arrival at the school is late in relation to any abstract notion of
> time. It is
> rather that the reading on the face of your teacher's watch is later than
> the reading
> at which school was supposed to start. Note that this approach is in direct
> contradiction to Newton's approach as stated in the quote which began this
> paper.
> Time, according to the proponents of this view is exactly the common view,
> and
> Newton's non-relationist view is wrong.
> The key problem with this approach is that it removes the foundation for
> the third
> aspect of time in quantum mechanics. At any one time, any variable has
> one and only
> one value. It is this which physically justifies the whole Hilbert space
> structure
> of quantum mechanics. But any real physical watches are imperfect. It can
> be proven that
> any realist watch not only has a finite probability to stop, it has a finite
> probability
> to run backwards. Now as long as the watch is simply the measure of some
> outside
> phenomenon, one could take these probabilities into account. If, however,
> time is {\bf defined} to be the reading on the face of the clock, the
> question
> as to whether or not the clock can stop or run backward is moot--- it cannot
> by definition.

The local time of each local system is defined as the reading of the local
clock of the LS in Local Time theory. Thus in the sense of Unruh above, the
direction of time is unique for each local system.

Let us consider two LSs, say L1 and L2, and let them observe the same local
system L other than them. The observed phenomena and values of L are
transformed from the data observed by L1 to the data observed by L2 in
accordance with the general relativistic transformation of coordinates (by
Axiom 6 of LS theory). Thus two observations by L1 and L2 give the same
direction concerning the observation of LSs (like L) other than L1 and L2.
(Here the use was made of the argument by Hawking and Ellis, p. 181 to show
the orientability of manifold.) This implies that the arrow of all local times
inside local systems coincide with each other.

Here we used: firstly that the direction of local time inside a local system
is unique _by definition_, and secondly that the manifold that satisfies the
_GR axioms_ is orientable. So it might be said that our argument is also based
on assumptions (i.e. GR axioms), but these assumptions seem to be natural

>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?

I would comment on other points when I can understand the questions or
problems you raised.

Best wishes,

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