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- Xref: sparky sci.physics:22286 alt.sci.physics.new-theories:2713
- Newsgroups: sci.physics,alt.sci.physics.new-theories
- Path: sparky!uunet!well!sarfatti
- From: sarfatti@well.sf.ca.us (Jack Sarfatti)
- Subject: Wavelets, coherent states and complex spacetimes 4
- Message-ID: <C0IzDp.8M5@well.sf.ca.us>
- Sender: news@well.sf.ca.us
- Organization: Whole Earth 'Lectronic Link
- Date: Fri, 8 Jan 1993 07:56:13 GMT
- Lines: 47
-
-
- 4. Gerald Kaiser defines a "generalized frame" in abstract math that I will
- not even pretend to understand or try to describe - but we will see if we
- can decode a useful picture for simple-minded physicists. The math could
- all be a big con-game to hood-wink us.
-
- The frame is a set of vectors |m> and the canonical coherent states are a
- "tight frame". T is a map from vectors of the Hilbert space to functions on
- M. The resolution of unity is replaced by
-
- G(N) = T*T = Integral over N[du(m)|m><m|] (19)
-
- where N is a subset of the set M whose elements are m. Does G converge? is
- an issue.
-
- If the frame is tight then we get a resolution of unity.
-
- *It appears that if we want quantum connection communication we do not want
- entangled states with "tight frames" that strangle the communication!*
-
- A lot of obscure math that John Baez can slop around in follows but it
- leads up to the enigmatic image of a "reproducing Kernel Hilbert space".
- Along the way we have
-
- K(m,m') = <m|G^-1|m> (20)
-
- which "has a property similar to the Dirac delta function with respect to
- the measure du in that it reproduces functions in the space.." But K is
- bounded unlike the Dirac function and the "test functions" that K
- reproduces form a Hilbert space. K is an integral operator not merely a
- distribution.
-
- *M will become complex spacetime!*
-
- g(m) = Integral over M [du(m)K(m,m')g(m')] (21)
-
- So we can begin to see that the imaginary part of space-time will be a
- little like the momentum p of the harmonic oscillator. So these tight
- frames may play some role in Hawking's version of quantum gravity needing
- imaginary time?
-
- to be continued
-
-
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-