From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Wavelets 21: Group Theory, Fiber Bundles, Squeezed States.
Organization: Whole Earth 'Lectronic Link
Date: Fri, 22 Jan 1993 09:51:07 GMT


21. Group theory of wavelets.
(My comments between *..*)Given a unitary representation U of any Lie group
group G acting on a Hilbert space H. Choose arbitrary "fiducial (window)
vector" h in H. When G is the affine group of the plane R^2, then h is "the
basic wavelet". For every g in G

hg = U(g)h     (192)

U unitary implies "rigidity"

||hg|| = ||h||   (193)

*Recall that this rigidity (i.e., invariance of inner product) plus
orthogonality of single-particle kets in the entangled pair state,
together, prevent quantum connection communication outside the light cone
and backwards-in-time on or inside the light cone within standard quantum
mechanics.*

"Covariance" of the hg's means

U(g')hg = U(g')U(g)h = U(g'g)h = hg'g  (194)

i.e., covariance, in Kaiser's sense, means that U is at least a many -> one
homomorphic image of G.

Consider now the set K of all elements k in G for which the action U(k) on
h reduces to a multiplication by the phase factor e^i@(k). In the case of a
"slice" of W1 for the Weyl-Heisenberg group of a 1D oscillator, K consists
of all k = (0,0,@#(0,0)). In general U(k) is non-trivial acting trivially
as a phase factor only with K which depends on initial choice h, i.e.,
K(h). K is a subgroup of G.

The map

k -> e^i@(k) (195)

is a homomorphism of K to unit circle i.e. a "character" of K which is
defined as a unitary representation of K on the one-dimensional Hilbert
space C (the complex plane). K(h) is the "stability subgroup" of "basic
wavelet" (now for any G not just affine group). "Stability" can be replaced
by "normal" or "invariant" in this context. The Lie algebra of K(h) is the
"stability subalgebra". K is a "gauge symmetry" of the quantum mechanical
state |h> corresponding to he basic wavelet. If ||h|| = 1, the quantum pure
state is the projection operator

P(h) = |h><h|  (196)

P(hg) = |hg><hg| = U(g)P(h)U(g)* (197)

If g = k in K(h), then

P(k) = P(h)   (198)

P is stable or invariant under K.

Kaiser uses "states" in the sense of diagonal outer-product ket-bra
projection operators not as vectors (or projective rays) in the Hilbert
space. The use of projection operators "eliminates the fictitious degree of
freedom represented by the over-all phase (global gauge invariance). In
communication theory, an overall phase shift has no effect on the
informational content of the signal. In contrast, the Hilbert space vectors
paly an essential role for "holomorphy". For example, the coherent states
have an analytic representation f(z), whereas |f(z)|^2 is not analytic.

If g in G but k in K, then

hgk = U(gk)h = e^i@(k) hg    (199)

P(gk) = P(g)  (200)

Thus, Pg is the same for all members of the left coset gK. The set of all
translates is therefore parametrized by the left-coset space

M = G/K = {gK, g in G}   (201)

m is in M.

*How classical phase space relates to quantum Hilbert space?
For the Weyl-Heisenberg W1 (see 20) M is simply the classical phase space
(p,x) in R^2. The classical phase space is thus the quotient space of the
group G acting on the quantum Hilbert space H modulo the normal subgroup
K(h) generated from the basic wavelet h. The individual coset m = (p,x)K in
M = G/K which is a "point" in classical phase space, is a "line" in Weyl-
Heisenberg group space W1 (with the topology of R^3).

W1 is a degenerate non-relativistic limit of phase spacextime and the coset
(p,x)K is the trajectory of a free classical particle - says Kaiser on
p.109.

To build a "G-frame", pick a slice of G by choosinf a single representative
from each coset gK(h). (For the above example, recall from 20 that g =
(p,x,@#(p,x) in a slice.) This is a non-trivial map

s:G/K -> G   (202)

Note that G -> G/K defines a "fiber bundle" and the map s in eq. (202) is a
"section" or "cross-section" of the bundle (which will be a classical field
in physics if M is spacetime rather than phase space). s is smooth only
locally in a neigborhood of each point m. The state P(m) is independent of
the choice of s.

To finish building the frame we need the action of G on the quotient space
(i.e. base space of the fiber bundle) and a measure on M invariant under
the action of G.  The G-action is easy. It is

g'(gK) = (g'g)K  (203)

So, if m = gK in M, (g'g)K is g'm. M is also called a "homogeneneous
space". If G is unimodular (i.e. if left and right invariant measures are
proportional by a constant at every point in m, then Helgasson has a
theorem for the existence of the invariant measure - what about its
construction? Call the measure du(M).

J = Int(M)[du(M)|h(s)m><h(s)m|    (204)

If J commutes with every U(g') then irreducibility and Schur's lemma forces
J = cI. But like the Weyl-Hesienberg case in 20 there will be a phase
factor problem in the composition. The h(s)m are covariant only mod a
residual phase factor e^-i@#(g'g)e^i@#(g). But the states are covariant

U(g')P(m)U(g')* = P(m)  (205)

It is easy to show then that J commutes with every U(g) from unitarity and
invariance of the measure.

Given a fixed g in G, the stability subgroup for basic wavelet h is K(h)
but the stability subgroup for U(g)h is gK(h)g^-1 which is a subgroup of G
conjugate to K(h). In general, the stability subgroups of two possible
basic wavelets h1 and h2 may not be conjugate. K(h1) may have an invariant
measure while K(h2) does not. This suggests an objective choice for the
basic wavelet so as to maximize the stability subgroup which minimizes the
homogenous space M = G/K improving the chance that the resolution integral
for J will converge as it must for a finite theory.

*Both the Glauber-Klauder coherent states of optical coherence theory and
the windowed Fourier transforms of classical signal detection theory
provide "phase space " representations. However, the optical coherence
states have analytic (holomorphic) properties. All phase-space
representations appear to be analytic in this sense. The coherent quantum
state of a radiation oscillator is a circle of minimum area h in the
dynamical (p,x) of the corresponding classical oscillator. The optical
squeezed state, enhancing signal/noise ratio, achieved in laser
interferometry is an ellipse rather than a circle in phase space. There are
relativistic coherent states from the Poincare group. Are there coherent
states for the diffeomorphism of general relativity?

to be continued.