From: Subject: The Wavelet Digest :: View topic - Wavelets in Quantum Field Theory? Date: Wed, 10 Jan 2007 12:12:45 -0800 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0007_01C734B0.A30F0EF0" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3028 This is a multi-part message in MIME format. ------=_NextPart_000_0007_01C734B0.A30F0EF0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.wavelet.org/phpBB2/viewtopic.php?t=5584 The Wavelet Digest :: View topic - Wavelets = in Quantum Field Theory?
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Wavelets in Quantum Field = Theory?
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Jack Sarfatti=20 (sarfatti@pacbell.net)
Guest
Posted: Sun Oct=20 05, 2003 10:47 pm   =  
Subject: Wavelets in Quantum Field = Theory?
Is there anything = published=20 on reformulating quantum field theory Green's function = propagaors in terms of wavelet transforms rather than = rigid=20 windowed Fourier transforms? Are the renormalization = group=20 theorists using a kind of wavelet transform without = knowing=20 it?
3D""=20
Laurent Demanet (ldemanet at hotmail dot=20 com)
Guest
Posted: Thu Oct=20 09, 2003 12:07 am   =  
Subject: Re: Wavelets in Quantum Field=20 Theory?
I am no = specialist but here=20 are some elements I could gather.

The context = is that=20 of canonical coherent states : they are orbits of = admissible=20 vectors along a representation of the Weyl-Heisenberg = group on=20 the Hilbert space of states. So they can be thought of = as a=20 windowed Fourier transform, I assume that's what you = are=20 referring to. Apparently these coherent states provide = a=20 representation in which UV divergences can be removed, = this=20 has been known since 1965.

Affine coherent = states can=20 be built from square-integrable representations of the = affine=20 group (ax+b) in a similar fashion. This is exactly the = continuous wavelet transform, and was proposed by = Aslaksen and=20 Klauder as early as 1969 (JMP vol. 10 p.2267). This=20 representation provides another formulation of usual = quantum=20 mechanics, see also T. Paul, 1984, JMP vol. 25 p. = 3252. So=20 yes, in that sense wavelets have been explicitly = constructed=20 by physicists long before they bloomed in other = fields.=20

Now whether this type of representation is = actually=20 more than a curiosity in quantum mechanics or QFT is = beyond=20 me. All I know is that the commutation relation for = quantum=20 observables in the affine coherent states = representation is=20 not the canonical one [Q,P]=3Di, but the 'affine' one = [Q,P]=3DiQ=20 or something like that. Klauder (the same as above) is = following up on this apparently interesting fact to=20 reformulate quantum gravity, as I could gather from = his recent=20 preprints (you can google his name).

Your = input on=20 these questions is most welcome !

- = L
3D""=20
Jack Sarfatti=20 (sarfatti@pacbell.net)
Guest
Posted: Thu Oct=20 09, 2003 12:32 am   =  
Subject: Re: Wavelets in Quantum Field=20 Theory?
"I am no = specialist but=20 here are some elements I could gather.

The = context is=20 that of canonical coherent states : they are orbits of = admissible vectors along a representation of the=20 Weyl-Heisenberg group on the Hilbert space of states. = So they=20 can be thought of as a windowed Fourier transform, I = assume=20 that's what you are referring to. Apparently these = coherent=20 states provide a representation in which UV = divergences can be=20 removed, this has been known since 1965. "

OK = in that=20 case squeezed states are like "wavelet transforms" in = the=20 number phase representation of phase space. <N> = is the=20 displacement of the "window" and the shape of the = displaced=20 ellipse is the distortion of the window away from the=20 "Gaussian circle" I suppose. Yes, I should have = thought of=20 that. Thanks.

"Affine coherent states can be = built=20 from square-integrable representations of the affine = group=20 (ax+b) in a similar fashion. This is exactly the = continuous=20 wavelet transform, and was proposed by Aslaksen and = Klauder as=20 early as 1969 (JMP vol. 10 p.2267). This = representation=20 provides another formulation of usual quantum = mechanics, see=20 also T. Paul, 1984, JMP vol. 25 p. 3252. So yes, in = that sense=20 wavelets have been explicitly constructed by = physicists long=20 before they bloomed in other fields.

Now = whether this=20 type of representation is actually more than a = curiosity in=20 quantum mechanics or QFT is beyond me. All I know is = that the=20 commutation relation for quantum observables in the = affine=20 coherent states representation is not the canonical = one=20 [Q,P]=3Di, but the 'affine' one [Q,P]=3DiQ or = something like that.=20 Klauder (the same as above) is following up on this = apparently=20 interesting fact to reformulate quantum gravity, as I = could=20 gather from his recent preprints (you can google his = name).=20

Your input on these questions is most welcome = !=20

- L"

Thanks for the tip. I actually = was aware=20 of the old Klauder work many years ago before there = were=20 "wavelets" in the modern sense of the word. =


Jack=20 Sarfatti (sarfatti@pacbell.net)


Posted: = Sun Oct=20 05, 2003 10:47 pm    Post subject: = Wavelets in Quantum Field Theory?


Is there = anything published on reformulating quantum field = theory=20 Green's function propagaors in terms of wavelet = transforms=20 rather than rigid windowed Fourier transforms? Are the = renormalization group theorists using a kind of = wavelet=20 transform without knowing it?
3D""=20
Pu Liu=20 (puliu@chem.columbia.edu)
Guest
Posted: Fri Oct=20 17, 2003 12:07 am   =  
Subject: Re: Wavelets in Quantum Field=20 Theory?
" WAVELETS AND=20 RENORMALIZATION"
by G Battle (Texas A&M = University)=20
ISBN 981-02-2624-1
3D""=20
Laurent Demanet (ldemanet at hotmail dot=20 com)
Guest
Posted: Mon Oct=20 20, 2003 6:07 pm  =20  
Subject: Re: Wavelets in Quantum Field=20 Theory?
One good = reference is the=20 book 'Coherent states' by Klauder and Skagerstam, = 1985. It=20 contains the seminal papers + a nice tutorial at the=20 beginning.

You can check out Ten Lectures p. = 33,=20 Daubechies has a word about Klauder whom she actually = worked=20 with before going into wavelets. That's where she = comes from -=20 theoretical physics. Actually the continuous wavelet = transform=20 was first introduced by Alberto Calderon in 1964 (I = think) ,=20 as a tool in order to solve some obscure problem in = harmonic=20 analysis...

And yes, there is Battle's book. = For some=20 reason I have a hard time getting into it...

- = L
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image/gif Content-Transfer-Encoding: base64 Content-Location: http://www.wavelet.org/phpBB2/templates/subSilver/images/spacer.gif R0lGODlhAQABAIAAAAAAAAAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw== ------=_NextPart_000_0007_01C734B0.A30F0EF0 Content-Type: text/css; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.wavelet.org/menubar/tooltip.css .tip { BORDER-RIGHT: #666666 1px solid; PADDING-RIGHT: 1px; BORDER-TOP: = #666666 1px solid; PADDING-LEFT: 1px; Z-INDEX: 100; LEFT: 90px; = VISIBILITY: hidden; PADDING-BOTTOM: 1px; FONT: 12px = Arial,Helvetica,sans-serif; BORDER-LEFT: #666666 1px solid; WIDTH: = 250px; COLOR: #333333; PADDING-TOP: 1px; BORDER-BOTTOM: #666666 1px = solid; POSITION: absolute; TOP: 20px; BACKGROUND-COLOR: #ffffcc; = layer-background-color: #ffffcc } ------=_NextPart_000_0007_01C734B0.A30F0EF0 Content-Type: text/css; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.wavelet.org/phpBB2/templates/subSilver/formIE.css INPUT { BORDER-TOP-WIDTH: 1px; BORDER-LEFT-WIDTH: 1px; BORDER-BOTTOM-WIDTH: = 1px; BORDER-RIGHT-WIDTH: 1px } TEXTAREA { BORDER-TOP-WIDTH: 1px; BORDER-LEFT-WIDTH: 1px; BORDER-BOTTOM-WIDTH: = 1px; BORDER-RIGHT-WIDTH: 1px } SELECT { BORDER-TOP-WIDTH: 1px; BORDER-LEFT-WIDTH: 1px; BORDER-BOTTOM-WIDTH: = 1px; BORDER-RIGHT-WIDTH: 1px } INPUT { TEXT-INDENT: 2px } INPUT.button { BORDER-TOP-WIDTH: 1px; BORDER-LEFT-WIDTH: 1px; BORDER-BOTTOM-WIDTH: = 1px; BORDER-RIGHT-WIDTH: 1px } .postbody { LINE-HEIGHT: 18px } ------=_NextPart_000_0007_01C734B0.A30F0EF0 Content-Type: application/octet-stream Content-Transfer-Encoding: quoted-printable Content-Location: http://www.wavelet.org/menubar/tooltip.js // Extended Tooltip Javascript // copyright 9th August 2002, by Stephen Chapman, Felgall Pty Ltd // permission is granted to use this javascript provided that the below = code is not altered var DH =3D 0;var an =3D 0;var al =3D 0;var ai =3D 0;if 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