[time 50] Re: n-body dirac equation

Stephen P. King (stephenk1@home.com)
Tue, 23 Mar 1999 13:41:25 -0500

Dear Hitoshi and Friends,

        My comments are within the body of the text...
Hitoshi Kitada wrote:
> 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, 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... :)

        Another friend here in Greenville has also expressed interest in
Clifford algebras (and Grassmannian exterior algebras); we'll see where
it leads us.

> > 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 39)
> > 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.

        This is a very important thought! :) But we need to note that clocks
have no memory, thus do not define histories... Such are defined by
relations between clocks, but we need a way of defining memory in a
generic way...

> > 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.

        Ok, I understand that; but it seems that Unruh is assuming a
basis" or frame of reference for the clocks to synchronize to. The fact
that clocks can run at different speeds or have different probabilities
of stopping is not relevant here... it is the 'direction" of the
times... I am thinking of a time as a vector not as a scalar quantity!
> 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.

        Right, but this arguments about manifolds is assuming that all
clocks (here LSs) are "on one and the same manifold." This is the one
aspect of the Local Times theory that I have some difficulty with. While
I agree that there are an infinite number of LSs, simply mapping them to
a single Riemannian manifold X is problematic. Since, as we have
discussed before, we postulate no connection between the LSs, we are
free do define an infinite number of different Xs depending of an
arbitrary choice of connection. Such connections and their accompanying
metrics define a <<physics>>, since the transformations *allowed* by the
geometry are the <<physics>>! This follows from the relationship between
the group theoretical properties of the <<physics>> and the
corresponding geometry, which is "spacetime" in the usual consideration!
        This speaks to the idea of <<physics>> as constructed, not as a
imposed apartheids. Please read Pratt's ratmech.ps!
        Another way of thinking of my question is to consider the phase
of a n-body system of particles S. We can partition S up into subspaces
S_i depending on the relative orientation of the flows in S. But note
that to do so we start by superposing some arbitrary basis with which to
define a coordinate system.
        I think that it would be consistent to posit that independent of a
particular configuration and propagator, just as we can identify LSs to
individual points in an arbitrary Riemannian manifold X, we can also
identify an arbitrary X to each point in an LS! I believe that this is a
duality that needs to be axamines carefully!

> 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
> requirements.

        The "direction" of the time of an arbitrary LS is much like that of an
arrow in an empty space, devoid of any features, it is not *observable*
in itself; we always require a basis to orient the arrows. The
orientability of GR manifolds, I think, is refering to a topological
property such as we find when comparing Moebius loops to simple loops:

 | |
 | | Identifications: Moebius: A <-> D, C <-> B, Normal: A <-> B,
C <-> D
 | |

But note (!) without "parallel transport" and a way of "connecting the
points", e.g. a connection, this property is unknowable! A set of
disconnected points has very limited properties!
        The axioms of GR are only one of many possible assumptions. We must not
assume that our experinces are the only one's possible! I understand
that you wish to only deal with explaining that we can observe here and
now, but a good model of physics will enable us to extend our
understanding and thus our ability to observe/predict even more, and
that, I believe, is the main reason to do this work. :)

> >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
> >theory.
> > 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?
> >
> >Onward!
> >
> >Stephen
> >
> I would comment on other points when I can understand the questions or
> problems you raised.
> Best wishes,
> Hitoshi

        Could we discuss the internal and external definitions of mass? Some
people have been proposing that mass is defined in terms of zero point
energy. Do we have a way of thinking of such in our model?




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