NetNews Usenet Archive 1993 #1/NN_1993_1.iso/spool/sci/physics/22359

Internet Message Format | 1993-01-07 | 3.3 KB

Xref: sparky sci.physics:22359 alt.sci.physics.new-theories:2716
Newsgroups: sci.physics,alt.sci.physics.new-theories
Path: sparky!uunet!well!sarfatti
From: [e-mail] (Jack Sarfatti)
Subject: Wavelets, coherent states etc. 5
Message-ID: <[e-mail]>
Sender: [e-mail]
Organization: Whole Earth 'Lectronic Link
Date: Sat, 9 Jan 1993 02:53:28 GMT
Lines: 81


5.Kaiser presents "generalizations of the concepts of basis (HM),
reciprocal basis (HM*), Kronecker delta (K(m,m') and metric tensor
(g(m,m')) to the infinite-dimensional case where, in addition, the
requirement of linear independence is dropped. The point is that the all-
important reconstruction formula, which allows us to express any vector as
a linear combination of the frame vectors, survives under the additional
restriction that the consistency condition be obeyed. The useful concepts
of orthogonal and orthonormal frames generalize to tight frames and frames
with A = B = 1 (i.e. G = AI and B^-1I <=G^-1 <=A^-1I ) respectively. We
will call frames with A=B =1 'normal'. Thus normal frames are resolutions
of unity." p.25

* Quantum connection communication may require non-normal frames.
** This formalism may allow us to think of variably-curved Hilbert space in
a non-standard quantum mechanics. Standard quantum mechanics has a "flat"
Hilbert space. Curvature in the Hilbert space probably violates the
superposition principle - it is non-linear related to measurement -
entropy?  Curvature in spacetime is gravitation and mass. What is curvature
in Hilbert space?  Perhaps curvature is not the right term. In fiber bundle
theory the em-weak& strong forces are "curvatures" in the sense of parallel
transports around closed loops in the fibers. But this us a little
different. I visualize that gmk depends on location of point on fiber so
that "nonlinear" may be better term than "curvature" in this context. This
is half-baked.

g(m,m') = <m|G|m'>  (22)

note

|k> = Integral[du(m)|m>K(m,k)]     (23)

where the "frame vectors" |m> are in general not linearly independent.
There is a general "consistency condition"

g(m) = Integral[du(m')K(m,m')g(m')   (24)

recall

K(m,m') = <m|G^-1|m'>    (20)

(which also corrects a misprint in the orginal eq 20 in lecture 4)

Recall also that T is map from Hilbert space to pre-"complex spacetime" M
and

G = T*T = Integral[du(m)|m><m|]                  (25)

If the frame is "tight" G is proportional to the unit matrix.

If

g = Tf   (26)

f = G^-1Gf = G^-1T*(Tf) = G^-1T*g       (27)

S = G^-1T*        (28)

is a left inverse of T

There is also a distinction between upper and lower indices ("reciprocal
frames") like contravariant and covariant indices in relativity connected
by G^-1 . I'll modify notation now since this gets important. It is
important in the "unitary resolutions of unity in terms of pairs of
reciprocal frames with the bra covariant and the ket is contravariant in
the ket-bra outer product (or vice-versa) integrated over the measure du(m)
to get unit operator I when the resolution exists.

I'll call contravariant index ^m and covariant m so that

Integral[du(m)|^m><m| = Integral[du(m)|m><^m| = I    (29)

|^m> = G^-1|m>   (30)

If M is countable   Integral[du(m)...] replaced by sum S


|k> = Sm[|^m><m|k>   (31)

gmk = <m|k> = <^m|G|k>   (32)
to be continued