Unification At A Point or Little String 2
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 19 Jun 2005 07:52:02 -0700
>>From Osher Doctorow mdoctorow@xxxxxxxxxxx
Most of the previous postings in this thread were in verbal language,
so it might be asked whether there is any quantitative version. Let's
see.
1) (A-->A) = A' U A = universe
Equation (1) is direct from the definition of (A-->B) = (AB')' and the
fact from set theory that (CD)' = C' U D' for any sets C, D, and the
fact that (B')' = B from set theory.
Equation (1) applies to any set A, so it applies also to the physical
universe at every stage of its existence including the Big Bang if the
Big Bang actually existed. In any case, arguably the universe was once
a particle or little string or a singularity, or if the Bounce theories
are correct it came very close to it (a small "local" neighborhood).
Equation (1) also embodies the "push" (arrow) that the universe had and
has internally, and which is Causation or Influence. The probability
of the event (A-->A) is:
2) P(A-->A) = P(A' U A) = P(A') + P(A) = 1 - P(A) + P(A) = 1
since A' and A are disjoint and P(A U B) = P(A) + P(B) for any two
events A, B which are disjoint (have no intersections). So the
physical universe always has probability 1 of influencing or causing
itself, and did in its very early period too.
Now we come to the "extension of the tensor field beyond one point",
and with A still the universe and B any set/event (or process which can
be represented as a (generalized) set):
3) (A-->B) = (AB')' = A' U B
where the first equality is by definition and the second is from set
theory similarly to (1) above.
You might ask whether, in the "original" universe, B would not simply
be the null or empty set N. It could have been, but the null set is
not very much explored even today. It is the "set with no elements",
but nobody is quite sure whether that means anything, whether
abstractly or concretely. It needs to be there for the mathematics,
but its interpretation might be different from the conventionally
obscure one.
The point is that the "original" universe A did have a second "object"
or "set" incorporated into its causal/influence structure, which was
either the null set N or the complement of the universe A'. In rough
language, the universe was able to "know" about its opposite or
complement A', even though "physical universe" is usually thought by
researchers to mean "everything physical that exists or that is
observable". If this seems odd, think of existence or observability at
a different time or place, which in some sense doesn't exist in our
universe right now (although in a different sense it might).
This tendency to "push" outward is the wave or field tendency of the
universe, whether regarded as a matter/energy or geometric structure,
and is incorporated in (3).
Of course, the probability of (3) is:
4) P(A-->B) = P(A' U B) = P(A') + P(B) - P(A'B) = P(A') + P(AB) = 1 -
P(A) + P(AB) = 1 + P(AB) - P(A) = 1 + y - x
where x = P(A) and y = P(AB). Compare this with conditional
probability which replaces 1 + y - x by y/x if x is not 0 and labels
the result P(B|A), the "probabiliy of B given A".
If the universe "began" with a bounce, which really means that it keeps
regenerating itself with almost contraction to a point or little string
but not quite and then expansion, then things are quite simple from
equation (4) onward. In Nathan Seiberg's terminology (of the Princeton
Institute), A could be regarded as the singularity through which the
universe keeps passing, or in "non-singularity" language as whatever
point or little string the universe approaches before bouncing. The
wave or field can be distinguished from that limiting point as the
remainder of the very small neighborhood of a point, and A can be taken
as the point and B as the field "from the start". The existence of
other events or processes besides the particle or point A and its
associated field B follows from the fact that a field is spatially
extended and so passes through other points than A. If this seems
tautological, then we can specify that the "original" universe A (and
perhaps the later universe too) had a property of generating other
points that is implicitly contained in the word field (now
generalized).
Osher Doctorow
.
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