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Wavelets in Quantum Field Theory?

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Jack Sarfatti
([email-masked])
Guest

Posted: Sun Oct
05, 2003 10:47 pm

Subject: Wavelets in Quantum Field Theory?

Is there anything published
on reformulating quantum field theory Green's function
propagaors in terms of wavelet transforms rather than rigid
windowed Fourier transforms? Are the renormalization group
theorists using a kind of wavelet transform without knowing
it?

Laurent Demanet (ldemanet at hotmail dot
com)
Guest

Posted: Thu Oct
09, 2003 12:07 am

Subject: Re: Wavelets in Quantum Field
Theory?

I am no specialist but here
are some elements I could gather.

The context is that
of canonical coherent states : they are orbits of admissible
vectors along a representation of the Weyl-Heisenberg group on
the Hilbert space of states. So they can be thought of as a
windowed Fourier transform, I assume that's what you are
referring to. Apparently these coherent states provide a
representation in which UV divergences can be removed, this
has been known since 1965.

Affine coherent states can
be built from square-integrable representations of the affine
group (ax+b) in a similar fashion. This is exactly the
continuous wavelet transform, and was proposed by Aslaksen and
Klauder as early as 1969 (JMP vol. 10 p.2267). This
representation provides another formulation of usual quantum
mechanics, see also T. Paul, 1984, JMP vol. 25 p. 3252. So
yes, in that sense wavelets have been explicitly constructed
by physicists long before they bloomed in other fields.

Now whether this type of representation is actually
more than a curiosity in quantum mechanics or QFT is beyond
me. All I know is that the commutation relation for quantum
observables in the affine coherent states representation is
not the canonical one [Q,P]=i, but the 'affine' one [Q,P]=iQ
or something like that. Klauder (the same as above) is
following up on this apparently interesting fact to
reformulate quantum gravity, as I could gather from his recent
preprints (you can google his name).

Your input on
these questions is most welcome !

- L

Jack Sarfatti
([email-masked])
Guest

Posted: Thu Oct
09, 2003 12:32 am

Subject: Re: Wavelets in Quantum Field
Theory?

"I am no specialist but
here are some elements I could gather.

The context is
that of canonical coherent states : they are orbits of
admissible vectors along a representation of the
Weyl-Heisenberg group on the Hilbert space of states. So they
can be thought of as a windowed Fourier transform, I assume
that's what you are referring to. Apparently these coherent
states provide a representation in which UV divergences can be
removed, this has been known since 1965. "

OK in that
case squeezed states are like "wavelet transforms" in the
number phase representation of phase space. <N> is the
displacement of the "window" and the shape of the displaced
ellipse is the distortion of the window away from the
"Gaussian circle" I suppose. Yes, I should have thought of
that. Thanks.

"Affine coherent states can be built
from square-integrable representations of the affine group
(ax+b) in a similar fashion. This is exactly the continuous
wavelet transform, and was proposed by Aslaksen and Klauder as
early as 1969 (JMP vol. 10 p.2267). This representation
provides another formulation of usual quantum mechanics, see
also T. Paul, 1984, JMP vol. 25 p. 3252. So yes, in that sense
wavelets have been explicitly constructed by physicists long
before they bloomed in other fields.

Now whether this
type of representation is actually more than a curiosity in
quantum mechanics or QFT is beyond me. All I know is that the
commutation relation for quantum observables in the affine
coherent states representation is not the canonical one
[Q,P]=i, but the 'affine' one [Q,P]=iQ or something like that.
Klauder (the same as above) is following up on this apparently
interesting fact to reformulate quantum gravity, as I could
gather from his recent preprints (you can google his name).

Your input on these questions is most welcome !

- L"

Thanks for the tip. I actually was aware
of the old Klauder work many years ago before there were
"wavelets" in the modern sense of the word.

Jack
Sarfatti ([email-masked])

Posted: Sun Oct
05, 2003 10:47 pm    Post subject:
Wavelets in Quantum Field Theory?

Is there
anything published on reformulating quantum field theory
Green's function propagaors in terms of wavelet transforms
rather than rigid windowed Fourier transforms? Are the
renormalization group theorists using a kind of wavelet
transform without knowing it?

Pu Liu
([email-masked])
Guest

Posted: Fri Oct
17, 2003 12:07 am

Subject: Re: Wavelets in Quantum Field
Theory?

" WAVELETS AND
RENORMALIZATION"
by G Battle (Texas A&M University)

ISBN 981-02-2624-1

Laurent Demanet (ldemanet at hotmail dot
com)
Guest

Posted: Mon Oct
20, 2003 6:07 pm

Subject: Re: Wavelets in Quantum Field
Theory?

One good reference is the
book 'Coherent states' by Klauder and Skagerstam, 1985. It
contains the seminal papers + a nice tutorial at the
beginning.

You can check out Ten Lectures p. 33,
Daubechies has a word about Klauder whom she actually worked
with before going into wavelets. That's where she comes from -
theoretical physics. Actually the continuous wavelet transform
was first introduced by Alberto Calderon in 1964 (I think) ,
as a tool in order to solve some obscure problem in harmonic
analysis...

And yes, there is Battle's book. For some
reason I have a hard time getting into it...

-
L

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