On Dec 30, 8:57=A0am, Daryl McCullough <stevendaryl3...@yahoo.com
> No, what I've said has nothing to do with topology.
But it most certainly does, whether you realize it or
not. You are implicitly assuming a wrong topology of
the physical space in your argument. Worse still, you
are confusing the topology of the space S^3 with the
global topology of the universe. Let me try to explain
this to you one more time. You wrote:
> Having the topology of S^3
> changes *global* properties of space--for example, it is compact,
> whereas R^3 is not--but in a small enough region, it doesn't
> change any properties that are locally measurable.
This is correct, but this is simply reiterating what
I have explained in my papers. Your argument
relies on comparing two measurement events, one
observed by Alice and the other observed by Bob,
in two remote, space-like separated regions. You,
or Bell, are therefore not justified in ignoring the
global properties of the physical space. It is evident
from my variables A(a, mu) and B(b, mu), which
are defined in small enough local regions, that they
are not affected by the global topology of S^3. Both
of them produce completely random binary numbers,
+1 or -1. But when these numbers are compared at
the end of a large number of runs, one finds that
they are strongly correlated. These correlations are
the result of the global properties of S^3, which
have nothing to do with the global topology of the
universe as a whole. And Just as Dr. Bertlmann's
socks have nothing to do with non-locality, they
have nothing to do with non-locality either.
> Assuming that an EPR type experiment takes place in a region
> that is small compared with the size of the universe, the topology
> would not make a difference to results.
Wrong. As I just explained, my argument has nothing
to do with the topology of the universe as a whole.
Have you ever tried to do the Dirac's belt trick? That
is a topological effect, exhibited in a small region of
space. Does that involve the size of the Universe?
Your assertions here are another indication that
you have not understood my argument at all.
My argument has nothing to do with the global
topology of the universe. It has to do with the
topology of S^3, for joint measurement events.
> No, I haven't said anything about topology, and
> I don't think it is relevant.
But you indeed have, and it is absolutely relevant.
You are assuming wrong topology of space -- R^3
instead of S^3. For example, in your argument you
are assuming two vectors, a and b. How are these
vectors defined? In my model all vectors are defined
as Clifford-algebraic elements. They are defined by
the trivector mu itself, since that is how vectors are
defined in Clifford algebra -- by equations mu /\ a =3D 0
and mu /\ b =3D 0. There is no analogue of these
equations in vector algebra (which is not even an
algebra). This is very important in my model, because
it is based on the even sub-algebra of Cl(3, 0), which
represents the 3-sphere. This is just one example of
how you are making implicit assumptions without
even realizing. You are assuming vectors a and b
which have nothing to do with the vectors of my
model. As a result, your argument has nothing to do
with my model, let alone the actual EPR statistics.
> I did not assume anything about topology.
You most certainly did, but without realizing it.
> ... in a small enough region, topology by itself would
> not play a role in an EPR-type experiment.
You are confusing the topology of space with the
topology of the universe. I am not concerned about
the topology of the universe. I am concerned about
the topology of S^3, as exhibited in Dirac=92s belt trick.
> What would play a role is curvature, or more generally,
> non-trivial parallel transport.
Curvature is completely irrelevant. It is zero. S^3 is as
flat as a sheet of paper. What is relevant is the torsion
within S^3. Once again, you have not understood my
argument at all, or even the basic physics of the EPR
correlations. And you have not read my papers. As I
have urged you many times before, read my papers
first and try to understand my argument. Read this paper,
for example, to understand how torsion is relevant, but
not curvature: http://arxiv.org/abs/1101.1958
> > Consequently, as I have explained in great detail in this paper:
, the four different outcomes, (+1,+1),
> > (+1,-1), (-1,+1), and (-1,-1), necessarily and deterministically occur
> > within my model even for the fixed detector directions a and b (but of
> > course with different probabilites depending on the angle between
> > a and b).
> That paper leaves the most important questions about your model
> unanswered. In particular, the interpretation of equation (46) as a
> *probabilistic* prediction is not supported by anything leading up to it.
No it does not. As I have explained to you many times before,
you must read the *whole* paper before making such false
claims. The argument leading up to equation (46) is developed
in equations (1) to (45) that come before equation (46). But you
have not paid any attention to these earlier equations.
> > If, however, one chooses to ignore the premises of my model (as you
> > have been doing) by replacing it with an old fashioned contextual
> > hidden variable model set up within a non-compact flat space with a
> > trivial topology (and that is indeed what you are doing whether you
> > realize it or not),
> You haven't shown how topology is even relevant to Bell's theorem.
> As I said, topology comes into play when the scale of the experiment
> becomes significant, compared to the size of the universe. But that's
> not the case in EPR experiments on Earth.
Yes I have. As I have explained to you before, the topology
of S^3, and more generally that of S^7, are crucially important
for the existence and strength of quantum correlations. The
global topology of the universe has nothing to do with this.
This is explained in great detail in my papers, which can be