Thanks Gary - good stuff. Reformulating quantum field theory and string
theory in the wavelet transform generalization of the Fourier transform
is important. Note how complex spacetime comes in. Possibly hypercomplex
non-commutative spacetime beyond that.

On Wednesday, November 19, 2003, at 07:02 AM, Gary S. Bekkum wrote:


http://www.arxiv.org/abs/math-ph/0303027

Authors: Gerald Kaiser
Comments: 56 pages, 3 figures. Invited "Topical Review" article for Journal
of Physics A: Mathematical and General, this http URL
Subj-class: Mathematical Physics; Complex Variables; Analysis of PDEs
Journal-ref: J.Phys. A36 (2003) R291-R338

For the first time, complete source distributions for the emission and
absorption of acoustic and electromagnetic wavelets are defined and
computed, both in spacetime and Fourier space. The biggest surprise is the
great simplicity of the Fourier sources as compared to the rather convoluted
spacetime expressions obtained from the original wavelets. This suggests
that the associated pulsed-beam propagators may play a fundamental role in
emission and absorption processes including focus or "directivity." It also
opens the way to constructing FFT-based algorithms for pulsed-beam analyses
of acoustic and electromagnetic waves.


Electromagnetic Wavelets as Hertzian Pulsed Beams in Complex Spacetime

http://www.arxiv.org/abs/gr-qc/0209031


Authors: Gerald Kaiser

Comments: 16 pages, 2 figures, "Topics in Mathematical Physics, General
Relativity and Cosmology" conference (in honor of Jerzy Plebanski) this
http URL
Subj-class: General Relativity and Quantum Cosmology; Mathematical Physics;
Complex Variables

Electromagnetic wavelets are a family of 3x3 matrix fields W_z(x')
parameterized by complex spacetime points z=x+iy with y timelike. They are
translates of a \sl basic \rm wavelet W(z) holomorphic in the
future-oriented union T of the forward and backward tubes. Applied to a
complex polarization vector p (representing electric and magnetic dipole
moments), W(z) gives an anti-selfdual solution W(z)p of Maxwell's equations
derived from a selfdual Hertz potential Z(z)=-iS(z)p, where S is the \sl
Synge function \rm acting as a Whittaker-like scalar Hertz potential.
Resolutions of unity exist giving representations of sourceless
electromagnetic fields as superpositions of wavelets. With the choice of a
branch cut, S(z) splits into a difference of retarded and advanced \sl
pulsed beams \rm whose limits as y\to 0 give the propagators of the wave
equation. This yields a similar splitting of the wavelets and leads to their
complete physical interpretation as EM pulsed beams absorbed and emitted by
a \sl disk source \rm D(y) representing the branch cut. The choice of y
determines the beam's orientation, collimation and duration, giving beams as
sharp and pulses as short as desired. The sources are computed as spacetime
distributions of electric and magnetic dipoles supported on D(y). The
wavelet representation of sourceless electromagnetic fields now splits into
representations with advanced and retarded sources. These representations
are the electromagnetic counterpart of relativistic coherent-state
representations previously derived for massive Klein-Gordon and Dirac
particles.

Non-linear Vacuum Phenomena in Non-commutative QED

http://www.arxiv.org/abs/hep-th/0006209


Authors: L. Alvarez-Gaume, J.L.F. Barbon
Comments: LaTeX, 23 pp
Report-no: CERN-TH/2000-181
Journal-ref: Int.J.Mod.Phys. A16 (2001) 1123-1146

We show that the classic results of Schwinger on the exact propagation of
particles in the background of constant field-strengths and plane waves can
be readily extended to the case of non-commutative QED. It is shown that
non-perturbative effects on constant backgrounds are the same as their
commutative counterparts, provided the on-shell gauge invariant dynamics is
referred to a non-perturbatively related space-time frame. For the case of
the plane wave background, we find evidence of the effective extended nature
of non-commutative particles, producing retarded and advanced effects in
scattering. Besides the known `dipolar' character of non-commutative neutral
particles, we find that charged particles are also effectively extended, but
they behave instead as `half-dipoles'.