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- Xref: sparky sci.physics:23272 alt.sci.physics.new-theories:2799
- Newsgroups: sci.physics,alt.sci.physics.new-theories
- Path: sparky!uunet!well!sarfatti
- From: sarfatti@well.sf.ca.us (Jack Sarfatti)
- Subject: Wavelets 21: Group Theory, Fiber Bundles, Squeezed States.
- Message-ID: <C19218.A1s@well.sf.ca.us>
- Sender: news@well.sf.ca.us
- Organization: Whole Earth 'Lectronic Link
- Date: Fri, 22 Jan 1993 09:51:07 GMT
- Lines: 150
-
-
- 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.
-
-
-