From Osher Doctorow
It isn't that hard to get gravity from what we know, but equally
curiously it isn't that hard to get a dimension of mass beyond 3+1
spacetime. Kaluza and Klein "gave us" a 5th dimension but thought it
was compactified, and Randall and Sundrum have in effect given us a
non-compactified 5th mass-like dimension with gravitation somehow
involved and either one or two branes and an "outside" or "inter-brane"
bulk involved with curvature. Gravity may be lower on our brane
because it is attracted to a second possible brane. The curvature may
or may not be involved in vibration of brane(s).
There are two mysteries here: A. Where did Hilbert Space go, and B.
Where did vibration of strings/branes go?
From the looks of it, Hilbert Space went to Banach Space, a more
general space that includes HS but also doesn't necessarily have inner
products. Banach Space (for short BS) does have a norm. In fact, the
set of bounded linear transformations T from one vector space A to
another vector space B is itself a BS if B is with the operator norm:
1) //T// = sup{//T(x)//, x in A, //x// < = 1}
which is actually rather nice from a probabilistic viewpoint where
everything so to speak is on [0, 1] at least for probabilities and
probability density functions and cumulative distribution functions;
and for the uniform distribution on [0, 1] also the values of random
variables are in [0, 1].
How in the world can we get anywhere without inner products? A clue is
"sup" in (1). Since we're essentially left only with addition,
subtraction, and scalar multiplication kT for scalar k, how about
maximizing //Ti - Tj// for all linear transformations Ti, Tj in some
set(s)? The idea is that "objects" in our non-brane bulk and in the
second brane if any are at maximal "norm distance" or near it (norm of
differences with our brane's transformations exceeding some constant).
We would in effect be defining the 5th dimension by a "very far away"
concept (further than anything in our 3+1 dimensions).
Osher Doctorow
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