From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Wavelets, coherent states and complex spacetimes.
Organization: Whole Earth 'Lectronic Link
Date: Thu, 7 Jan 1993 09:29:38 GMT


Wavelets, coherent states and complex spacetimes 1
quotes from Gerald Kaiser's Quantum Physics, Relativity and Complex
Spacetime, North Holland 1990

"...complex spacetime as a unification of spacetime and classical phase
space ... relativistic coherent states ... tube domains in axiomatic
quantum field theory ... result from analytic continuation of .. vacuum
expectation values ... to complex spacetime .... these tube domains ....
have a direct physical interpretation as extended classical phase spaces
.... relativistic coherent states are similar to wavelets .... coherent
states and time-frequency localization ... windowed Fourier transforms and
wavelet transforms ... frames ... discrete and continuous resolutions of
unity ... reproducing kernel .. orthonormal bases of wavelets ...complex
structure explains symmetry between the low and the high frequency filters
in wavelet theory ... complex wavelet decompositions ... are analogous to
analytic coherent state representations ... generalized coherent states
based on Lie groups and their homogeneous spaces ... holomorphic (analytic)
coherent-state representations .. result from .. Lie group
complexification. The rotation group ... 'spin coherent states' ... the
group associated with the harmonic oscillator is a weak contraction limit
(as the spin s-> infinity) of the rotation group... the canonical coherent
states are limits of the spin coherent states .... relativistic coherent
state representations admit a covariant probabilistic interpretation ..
absent in usual spacetime theories. In the non-relativistic limit they
contract smoothly to representations of the Galilean group .. The Gaussian
weight functions of the canonical coherent state representations emerge
from the geometry of the mass hyperbolid ... Analytic-Signal Transform ..
applied to an arbitrary function on R^n to give a function on C^n .. is
'analyticity-friendly'...even the most general fields satisfying the
Wightman axioms generate a complexification of spacetime which may be
interpreted as an extended classical phase space for certain special states
... The analytic Wightman two-point functions play the role of reproducing
kernels,, Complex spacetime densities of observables such as energy,
momentum, angular momentum and charge current are ... regularizations of
their counterparts in real spacetime.. Dirac particles do not undergoe
their usual zitterbewegung. The extension to complex spacetime separates ..
the positive and negative frequency parts of free fields so Wick ordering
becomes unnecessary. A functional integral representation ... for quantized
fields ..combines the coherent-state representations for particle (finite
number of degrees of freedom) with that for fields ..(infinite number ...)
Whereas in real spacetime it is not possible to derive gauge potentials and
gaughe fields from a fiber metric .. this can be done in complexspacetime
.. the analogy between general relativity and Yang Mills gauge theory
becomes much closer in complex spacetime than it is in real spacetime ...
the relation between the non-abelian Yang-Mills field and it potential
becomes linear due to the cancellation of the non-linear part ...
generalizing the Analytic-Signal Transform ... is a higher dimensional
version of the wavelet transform which is closely related to .. Hilbert,
Fourier-Laplace and Radon transforms."

to be continued


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