[time 228] Re: Weyl's theory

Stephen P. King (stephenk1@home.com)
Tue, 13 Apr 1999 11:32:24 -0400

Dear Hitoshi,

Hitoshi Kitada wrote:
> Dear Stephen,
> Thank you for your sending the copies.
> I read the copy of a part of Eddington's book and your comments on the pages.
> I noticed that the theory described there is an old version of Weyl's gauge
> theory, which Weyl himself withdrew as soon as QM appeared as a clear theory.
> Then he soon proposed another type of gauge theory, which is the origin of
> modern gauge theory, and was developed to the weak-electro unification theory.

        I believe that it was the idea of spectral smearing that stoped work on
Weyl's origional theory. I do not think that this is a fatal flaw at
all, the smearing effect might be connected to the way that we construct
the geometry of a set of interacting LSs, e.g. since each LS can only
observe a discrete spectra, any part of the spectra that is not observed
would appear to be red or blue shifted, e.g. the Hubble expasion
illusion. I think of this as like trying to receive an analogue signal
with a finite number of discrete antennas, or like catching water in
small buckets, the distributions of signal or water are dependent on the
number of buckets available and their size. Robert's discussions are
relevant here. I will look for his comments on this...
> As far as you remain at classical level, the old type gauge theory may be
> useful in that it gives a mathematical formulation that includes gravity and
> electromagnetic forces in the classical regime (with some problems still
> remaining about the ambiguity of gauge). The problem may be how we can make
> use of it. I am not sure about this point. I myself want to try to study weak
> and strong forces to see if these forces could be incorporated into my theory.
> But before that, I need to wait Ben's response to see if he can understand the
> point on observation that he raised in [time 226].

        I am thinking that the LSs act as the local gauge at each point in X,
where X is Weyl's generalization of Riemannian space, but there are
details that remain to be discussed. What ambiguity do you see in the
local gauge idea? I think that weak and strong forces are internal to an
LS in the sense that perhaps they are manifestations of symmetries of
the LS's subsystems. But this is just a rough idea. I too need to
restudy that the implications would be given the changes that LS theory
brings to particle physics. Perhaps Prugovecki's ideas would help. :) (I
whish he would explain why he uses a Lorentz geometry for the Hilber
space fiber!)
> Also on your [time 225]:
> [SPK]
> > > > > Again I am thinking in external terms. How is the "speed c" defined for
> > > > > a group of LSs communicating to each other? Do we assume an absolute
> > > > > interval for all? How do we get around the need of a invariant interval
> > > > > such as is assumed in GR? An LS defines a local clock, do they also
> > > > > define a local "length"?
> [HK]
> > > > Light speed c in vacuum and the time of an LS are used to measure the length
> > > > of the path (outside the LS) through which the light passes from the begining
> > > > to the end, under the situation that the path is stationary with respect to
> > > > the LS. Thus c (speed of ligh in vacuum) is assumed as an absolute constant.
> > > > The length of a path moving with respect to the LS is defined by relativistic
> > > > change of coordinates (e.g. Lorentz transformation).
> [SPK]
> > > I don't think of this "stationary path" as a priori existing, it is
> > > dependent on the decomposition of the LSs, among other things...
> [HK]
> > Since the path is "stationary" with respect to the observer's LS, the length
> > of the path is the same as that measured by the internal scale of the LS. My
> > definition above thus is concerned with the paths moving relative to the LS.
> > This transformation of length was used in time_IV.tex, I.3.2 (page 19, where
> > Lorentz transformation suffices) to measure the length between two centers of
> > mass of sublocal systems of the observed LS.
> [SPK]
> Could we elaborate more on this thinking of the Lorentz transformation?
> There is something strange about the geometry of these paths that I
> don't have words for right now. :( I need to talk to my friend Paul
> again, he helps me clear up my thinking. It has something to to with how
> these "stationary" paths related to simultaneity planes...
> I am not sure what you try to clarify by "elaborate more on this thinking of
> the Lorentz transformation." Could you explain it some more? I tried to see
> Paul Hanna's draft, but I could not understand what he wants to do.

        I don't know when Paul will have something for me, he tends to be
meticulous and shy. :) Perhaps, I need to better understand what the
Lorentz transformation implies to you. :) I need to understand how you
think about this in order to communicate my ideas better.
> > I am thinking here of the Feynman summation of the paths.
> I seem not know "Feynman summation of the paths" if it is not about Feynman
> path integrals. Would you explain the meaning?

        I am sorry, we are speaking of the same thing, I misunderstood you. :)
It is how, in thinking of morphing paths into each other in homology
theory, that sometimes paths will not be redusible. This implies the
existence of "holes", the algebraic property is that of
non-commutativity, topological it is multiply connected. When Feynman
path integration is performed, do we think of the surface on which the
paths exists as always simply connected, e.g. no holes? When we are
looking at the causal connection of events to each other in computer
science, we do not assume that the "surface" of events is simply
connected, there are many cases where certain events entail conflict, in
the sense that not all events can occur simultaneously. I think of this
as trying to compute the best schedule for constructing a house; we have
many different workers that can do their jobs simultaneously, but
certain jobs have to be scheduled in certain orders in order for
construction to move forward. Pratt discusses this in
http://boole.stanford.edu/chuguide.html#P3 and
http://boole.stanford.edu/chuguide.html#ph94 and



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