Tony Smith's Home Page

Segal's Conformal Theory


Conformal GraviPhotons

Here are some Obituaries describing aspects of Segal's work, and references to current work of Alexander Levichev.


Casimir Vacuum and Conformal GraviPhotons:

Some comments and questions:  

then would there be an induced current flow in the outer Conformal Hopf torus along Clifford-Hopf circles ?


Alastair Couper, in a web article, touches on relationships among

saying: "... Marko Rodin's ... vision ... in his book Aerodynamics ... was essentially a perception of a four dimensional sphere, which becomes a complicated toroidal structure when projected into three dimensions. He also perceived a mapping of an energy flow on the surface of this projected toroid. The Rodin coil is the best guess at this stage as to how this original insight could be made in physical form. ...

... The details of the flow of energy are encoded in his Sunflower Map,

which is an elaborate numerological scheme rendered on the surface of the toroidal form. He expects strong gravitational effects from a properly executed and powered winding. ...

... Marko begins with the sequence of powers of two: 1, 2, 4, 8, 16, 32, 64... .and turns this [numerologically by 64 = 6+4 = 10 = 1+0 = 1, etc] into a repeating pattern: 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5. The claim is that this doubling sequence is seen in all life processes and throughout nature. ...".

Another sequence uses Clifford Periodicity, equivalent to a sequence 1, 2, 4, 8, 16, 32, 64, 128, 256=1 with periodicity 8 instead of Marko Rudin's periodicity 7.

It might be possible to reconcile the two sequence-approaches by using some ideas of Alastair Couper, who points out that "... as has been pointed out by mathematician Jason Sharples ... cross addition of any number n, in base b, is equivalent to the result of n mod(b-1) ... the column of cross added numbers [of the Fibonacci sequence]... will ... repeat every 24 elements of the Fibonacci sequence, ad infinitum. In addition the elements of this repeated pattern show an additional bipolar symmetry, whereby the sum of any element plus the element 12 places away always results in 9. These relations are shown by plotting the 24 repeating elements on a wheel, bringing out the symmetries clearly:

... Marko Rodin has built a remarkable system based on the sequence which results from the cross addition of the powers of two; 1,2,4,8,16,32,64..... becomes 1,2,4,8,7,5, 1,2,4,8,7,5 ..... ad infintum. The numbers 3,6,9 have a special place in his system, which need not be detailed here. Basically, what he claims is that a very basic energy form in nature relies on three elements: one binary doubling sequence of 1,2,4,8,7,5..., another doubling sequence going in the opposite direction 5,7,8,4,2,1,...., and a sequence of 3,9,6,6,9,3,3,9,6,6,9,3..... ... When the previous wheel figure is inspected for Rodin's archetypal 'doubling circuits' and 3,6,9 'gap circuit', we find that they are indeed there:

... One can go on further, and do this same procedure over from the beginning, but starting with any two ascending numbers (instead of 1,1 in the standard Fibonacci sequence). The resulting symmetries and configurations will be essentially the same as shown here. In addition, one can use Theon's demonstration (which is similar to the Fibonacci sequence, but gives the square root of two) and obtain these same patterns. ...".

It seems to me that Alastair Couper's approach indicates that

Perhaps it is significant that the 8-dimensional vectors of 256-dimensional Cl(8) are isomorphic to its 8-dimensional +half-spinors and to its 8-dimensional -half-spinors, all three of which have total dimension 8+8+8 = 24, which correspond to the 24 vertices of the 24-cell root vector diagram of the 24+4=28-dimensional Cl(8) bivector Lie algebra Spin(8), whose outer triality automorphisms are the isomorphisms among vector and + and - half-spinors, and all of this is fundamental to the D4-D5-E6-E7-E8 VoDou Physics model, including the geometry of MacroSpace as related to 24+2=26-dimensional bosonic string theory with Monster symmetry and an extension to an M-theory related to the 24+3=27=dimensional Jordan algebra J3(O).


A version of a Rodin coil was used by J. L. Naudin in his B-Field Torsion Generator.


Spheres, Rodin Coils, Kron, and Negative Permittivity:

There are 3 parallelizable spheres: S1, S3, and S7. Penrose and Rindler, in Spinors and Spacetime, volume 1, (Cambridge 1986) vol. 1, p. 199, say: "... The (local) condidtion for a basis da to be holonomic is [da, d1] = 0, [,] being the Lie bracket operation ...", so I want to look at the Lie bracket product on those spheres.

On the 1-sphere S1, which is isomorphic to the abelian Lie group U(1), we have [x,y] = 0 for all x and y in S1, so S1 is holonomic and not a good example of anholonomy/nonholonomy.

On the 3-sphere S3, which is isomorphic to the abelian Lie group SU(2) = Spin(3), we have [x,y] =/= 0 for some x and y in S3, so look at S3. As Penrose and Rindler, in Spinors and Spacetime, volume 1, (Cambridge 1986) vol. 1, p. 199, say: "... Many more manifolds admit globally defined non-holonomic bases than admit globally defined holonomic ones. (An example is S3 ...) ...". So, let's take S3 as a concrete example.

What is the structure of S3, and for which x and y is [x,y] =/= 0 ? Since S3 = SU(2) = Spin(3) is the double-cover of 3-dim rotations, you can see that any element x of S3 has two parts:

The set of all axes of rotation corresponds to the points of a 2-sphere in 3-space.

The set of all magnitudes of rotations about a given axis corresponds to the points of a 1-sphere S1, which is the group of rotations in 2-dim space, or U(1).

Therefore, S3 is made up of S2 and S1. Topologically, S3 is constructed in accord with the Hopf fibration S3 / S1 = S2.

I stopped with S3 as an example, and did not go on to the 7-sphere S7. For S7, we also have [x,y] =/= 0 for some x and y in S7. However, the bracket product on S7 does NOT satisfy the Jacobi identity, so S7 (although it is parallelizable) is NOT a Lie algebra. If you want to make a Lie algebra out of the bracket product structure of S7, you have to realize that the base point of the tangent space DOES matter. You then have to expand S7 to 28-dim Spin(8) to take account of base point dependence and form a Lie algebra. One way to see why S3 is a Lie group and S7 must be expanded is that S3 is the unit sphere of quaternion space, and quaternions are associative, while S7 is the unit sphere of octonion space, and octonions are non-associative (but they are alternative).

I think that the Hopf fibration S3 / S1 = S2 is not just a fancy math example, but is a geometric picture of the type of physical arrangement you need to get engineering results such as those discussed by Jack Sarfatti, who said "... My primary objective here is to ... engineer with rotating superconducting machines, perhaps using ideas from Gabriel Kron for example. ...".

Since I don't have any of Kron's papers, I looked up some stuff about him on a web page that says:

"... To see how really complex are the actions in motors and generators, the reader is referred to the work of Gabriel Kron, possibly the greatest U.S. nonlinear electrical scientist of all time. Even full general relativity _ which Kron rigorously applied to rotating electrical machines _ still falls short of what is needed. E.g., the technical reader might wish to peruse his "Four abstract reference frames of an electric network." IEEE Transactions on Power Apparatus and Systems, PAS-87(3), Mar. 1968, p. 815-823. Electrical engineers often treat their stationary networks, rotating machines, and microwave electronic devices as a collection of impedance elements Z without decomposing Z into its RLC components. Kron shows that a lumped or distributed impedance network, surrounded by its own electromagnetic field, is actually the sum of four different types of multidimensional networks:
  • (1) the well-studied 1-network of branches in which the currents flow,
  • (2) a 0-network formed by all the point generators,
  • (3) a 2-network of equipotential surfaces that pass through the generators perpendicularly to the branches, and
  • (4) a 3-network composed of three-dimensional impedance blocks surrounding the branches.

Thus the topological structure of a stationary or rotating, electric or electronic network is neither a graph nor a polyhedron, but a so-called fiber bundle over a non-Riemannian manifold. ...".

I think that the basic example of the fiber bundle structure used by Kron may be the Hopf fibration S3 / S1 = S2 where:


Jack Sarfatti said "... I am thinking of engineering applications, when the ... [non-holonomic torsion gaps] ... are maybe meters in size which is OK if the radii of elastic curvature are millions of kilometers, ...".

However, I think that maybe you don't need huge radii to get meter-size non-holonomic torsion gaps, because if you think of the Lie bracket of rotations, which is what you have in S3 because of the S2 in the Hopf fibratioon, you can get a bracket product, or gap, that is on the same order of magnitude size as the two rotations whose product you are taking, so, with an S3 machine such as a Rodin coil, the gap can be roughly the same size as the Rodin coil machine itself, and you may be able to get a meter-size effect with a meter-size machine.


As to fabrication of materials that might be useful in S3 machines, a UCSD web page describes "... a new class of composite materials with unusual physical properties ... If these effects turn out to be possible at optical frequencies, this material would have the crazy property that a flashlight shining on a slab can focus the light at a point on the other side ... Underlying the reversal of the Doppler effect, Snell's law, and Cerenkov radiation ... is that this new material exhibits a reversal of one of the "right-hand rules" of physics which describe a relationship between the electric and magnetic fields and the direction of their wave velocity. ... The composite constructed by the UCSD team ... was produced from a series of thin copper rings and ordinary copper wire strung parallel to the rings. ... The idea for the new composite ... building on the work of John Pendry of Imperial College, London. In 1996, Pendry described a way of using ordinary copper wires to create a material with the property physicists call "negative electric permittivity." ... Pendry also recently suggested a way of using copper rings to make a material with negative magnetic permeability at microwave frequencies. ...". One paper by Pendry (and co-authors), specifically describing arrays of metallic cylinders, is cond-mat/9804195.

In cond-mat/9807022, Arriaga, Ward, and Pendry say:

"... In recent years, there has been much interest in artificially structured dielectrics, otherwise known as photonic crystals ... [with structure] ... on the scale of the wavelength of light. ... [These] materials promise new and exciting optical properties based on the novel dispersion relationships ... induced by the periodic structure. There is a strong analogy here with semiconductor physics, where band gaps in the dispersion relationships for electrons play a key role in determining the electronic properties. ... Conceptually, the problem reduces to solving Maxwell's equations ... where the photonic structure may be introduced either through the electric permittivity ... or, less commonly, the magnetic permeability ... As in the electronic case computation times for traditional schemes, such as plane wave expansions, scale as N^3 , where N is proportional to the size of the system. ... The optimum scaling possible is clearly O(N) - the time taken to define the problem or to look at the answer! Chan etal. showed it is possible to realize this optimal scaling by working in real space and in the time domain, provided that the resulting equations are local in space and time. ... Chan's method was not a new idea but had been known for some time to the electrical engineering community as the finite difference time domain (FDTD) method ... there is a problem with ... FDTD. In the time domain, it is not obvious how to deal with a frequency dependent dielectric permittivity ... Earlier calculations using the transfer matrix method show huge enhancements of local fields in nanostructured arrays of metal cylinders or spheres ... The electrical engineering community have known about this problem for some time and progress has been made towards solving it ... we treat metallic systems exhibiting a simple plasmon pole ... and we argue that our methods can be extended to treat more complex forms of dispersion. ... The poles have a simple interpretation of internal electromagnetically active modes which can absorb energy. Many inverse dielectric functions are well approximated by a sum over poles, particularly where the poles lie off the real axis and therefore give rise to rather broad structure ... For the two dimensional case we calculate the photonic band structures for a system consisting of an infinite array of parallel, metal rods of square cross-section, embedded in vacuum. ... The key features of these results are the low frequency cut-off in the E-polarized bands and the very flat bands at around the plasma frequency in the H-polarization. The cut-off is caused by the collective motion of electrons screening the electric field parallel to the rods, below some effective plasma frequency determined by the filling fraction. The at bands are caused by the resonant modes of the individual rods, excited by the electric field perpendicular to the rods. ...".

In cond-mat/9802259, Antonoyiannakis and Pendry say:  

"... We are particularly interested in calculating forces on nanostructures. Our findings confirm that a body reacts to the EM field by minimising its energy, i.e. it is attracted (repelled) by regions of lower (higher) EM energy. When travelling waves (of real wavevector) are involved, forces can be additionally understood in terms of momentum exchange between the body and its environment. However when evanescent waves (of complex wavevector) dominate, the forces are complicated, often become attractive and cannot be explained by means of real momentum being exchanged. We have studied the EM forces induced by a laser beam ona crystal of dielectric spheres of GaP. We observe effects due to the lattice structure, as well as due to the single scattering from each sphere. In the former case the two main features are a maximum momentum exchange (and largest forces) when the frequency lies within a band gap; and a multitude of force orientations when the Bragg conditions for multiple outgoing waves are met. In the latter case the radiation couples to the EM eigenmodes of isolated spheres (Mie resonances) and very sharp attractive and repulsive forces occur. Depending on the intensity of the incident radiation these forces can overcome all other interactions present (gravitational, thermal and Van der Waals) and may provide the main mechanism for formation of stable structures in colloidal systems. ...".

In cond-mat/9805375, Garcia-Vidal, Pitarke, and Pendry say:

"... We analyse from a theoretical point of view the optical properties of arrays of carbon nanotubes filled with silver. Dependence of these properties on the different parameters involved is studied using a Transfer Matrix formalism able to work with tensor-like dielectric functions and including the full electromagnetic coupling between the nanotubes. ... Enhancements of up to 10^6 in the Raman signal of molecules absorbed on these arrays could be obtained. ... Very localised surface plasmons, created by the electromagnetic interaction between the capped silver cylinders, are responsible of this enhancing ability. ... Very recently it has been possible to fill carbon nanotubes with a coinage metal like silver, using capillary forces. On the other hand, bulk alignment of nanotubes has been also reported using different techniques. In these ordered arrays the carbon nanotubes form very close-packed structures. ...".


Jack Sarfatti comments: "... Will the negative electric permittivity and permeability persist in the superconducting state. What does negative electric permittivity do to the Coulomb barrier against fusion of nuclei for example. Coulomb energy barrier ~ qQ/permittivity r So will like charges attract in a medium with negative permittivity dielectric screening? This will remove the repulsive Coulomb barrier completely, in fact, the colliding positively charged deuterium ions will attract forming helium with the release of fusion energy at room temperature. Is this correct? ... Also a plasma should explode if its permittivity makes a phase transition from positive to negative. Metals should explode as well? ...".


I comment that:

A useful configuration of Rodin coils might be

4 Rodin Coils arranged in the geometry of a Fuller Vector Eqilibrium Cuboctahedron as in an Unconventional Unispace Earth Experiment.



Wavelets and the Conformal Group

Gerald Kaiser, in A Friendly Guide to Wavelets (Birkhauser 1994), says:

"... Maxwell's equations for electromagnetic waves and the wave equation for acoustic waves ... are invariant under the group of conformal transformations of space-time, which includes dilations and translations - the basic operations of wavelet analysis - as a subgroup. ... the conformal group ... SU(2,2) ... here ... denoted by C. ... [is] ... a 15-dimensional Lie group ... A study of the action of SU(2,2) on solutions of Maxwell's equations has been made by ... Ruhl ... Distributions on Minkowski space and their connection with analytic representations of the conformal group, Commun. Math. Phys. 27 (1972), 53-86. ... The wavelet analysis of electromagnetics is based entirely on the homogeneous Maxwell equations. Acoustic waves ... obey the scalar wave equation, which has ... its symmetry group the same conformal group C. ... the two analyses are just different representations of C. ...

... All the physical wavelets [of electromagnetism or acoustic waves] in any representation can be obtained from a single "reference wavelet" by conformal space-time transformations, just as one-dimensional wavelets can all be obtained as dilations and translations of a single "mother wavelet". ...

... the physical wavelets PSIz, which are solutions of the homogeneous Maxwell and wave equations, naturally split into two parts:

This splitting is far from trivial, because of the analyticity constraints. Neither PSI-z nor PSI+z are global solutions of the homogeneous equation. Rather, they each solve the corresponding inhomogeneous equation, and the source terms are "currents" given by one-dimensional wavelets ... That establishes a deep connection between the physical wavelets and a certain special class of one-dimensional wavelets, and therefore between the wavelet analysis of physical waves and that of communication signals. ...

... It is possible to begin with C and construct ... a Hilbert space H consisting of solutions of Maxwell's equations. ... as a representation space for C, where conformal transformations act as unitary operators ...

... We define a Hilbert space H of ... solutions by integration over the light cone in the Fourier domain. ... the analytic-signal transform (AST) ... extends any function or vector field from Rn to Cn, and we call the extended function the analytic signal of the original fundtion. the analytic signal of any solution in H is analytic in a certain domain T in C4, the causal tube. ...

... If ... g in C ...[is in] ... E ... the ... subgroup of C ... of space translations and scalings acting on real space-time ... E is invariant under space rotations but not time translations, Lorentz transformations or special conformal transformations ... nothing new results ...

... If g is a time translation, then the wavelets parameterized by gE are all localized by at some time t =/= 0 rather than t = 0. ...

... If g is a Lorentz transformation, then gE is "tilted" and all the wavelets with z in gE have centers that move with a uniform nonzero velocity rather than being stationary. ...

... if g is a special conformal transformation, then gE is a curved submanifold of T and the wavelets parameterized by gE have centers with varying velocities. ... the .. special conformal transformations, which form a four-parameter subgroup. ... begin with the space-time inversion J : x -> x / x.x ... If translations are denoted by Tb x = x + b, then special conformal transformations are compositions Cb = J Tb J ... Since Cb is nonlinear, it maps flat subsets in R4 into curved ones. ... Cb maps flat surfaces in space at time zero to curved surfaces in the same space. ... for some different choices of b, Cb can be interpreted as boosting to an accelerating reference frame ... all conformal transformations are nonsingular on T. ...".

There exist Electromagentic and Acoustic SuperLuminal X Waves.

At different levels in the D4-D5-E6-E7 physics model there are at least 5 types of conformal transformations:

At each level of Conformal Structure, Physical Wavelets provide a connection between the World of Physics and the World of Information.

The Geometry of those connections is that of Bounded Complex Domains. A good introductory paper is Conformal Theories, Curved Phase Spaces Relativistic Wavelets and the Geometry of Complex Domains, by R. Coquereaux and A. Jadczyk, Reviews in Mathematical Physics, Volume 2, No 1 (1990) 1-44, which can be downloaded from the web as a 1.98 MB pdf file. See also Lie Balls and Relativistic Quantum Fields, by R. Coquereaux, Nuc. Phys. B (Proc. Suppl.) 18B (1990) 48-52 and Born's Reciprocity in the Conformal Domain, by A. Jadczyk, in Z. Oziewicz et al. (eds), Spinors, Twistors, Clifford Algebras and Quantum Deformations, 129-140 (Kluwer 1993).

The relevant Complex Structure can be seen in such physical concepts as Momentum Space; Position-Momentum Complementarity, Type IV(2) Domains, Hyperspace, Black Holes, and Conformal SpaceTime.



Since the Conformal Transformations include Dilatation Scale Transformation, physics models based on Conformal Transformations, such as

the D4-D5-E6-E7 model, have natural Renormalization Structure.

Not only does the D5-D4 structure with Spin(8) gauge group have conformal structure over 8-dim spacetime,


gravity has conformal structure over 4-dim spacetime, because you get gravity by starting with a gauge theory of the 15-dimensional conformal group whose Euclidean version is Spin(6) and then gauge-fix the 5 conformal degrees of freedom to restrict it to the 10-dim anti-deSitter group and then use the MacDowell-Mansouri mechanism to get gravity,


the standard model groups of SU(3 )x SU(2) x U(1) are compatible with conformal structure over 4-dim spacetime, because each of them is a subgroup of SU(4) = Spin(6).


Wavelet-induced renormalization group for the Landau-Ginzburg model has been described by C. Best in hep-lat/9909151, saying: "... The scale hierarchy of wavelets provides a natural frame for renormalization. ... A wavelet expansion can be used to derive the properties of the Landau-Ginzburg model and its nontrivial renormalization flow even in a rather simple approximation. The crucial features we have made use of are scaling and self-similarity of the basis and locality of the basis functions. They enabled us to focus on the fluctuation strengths at different scales as the quantities of interest that govern the phase transition. The effective free energy of the system exhibits in a minimal way the coupling between different scales. ...".

A book entitled Wavelets and Renormalization (World, 1999) has been written by G. Battle.


Momentum Space

is useful for calculations, but I consider that

J. J. Sakurai, in Modern Quantum Mechanics (Benjamin/Cummings 1985), says: "... There is ... a complete symmetry between x and p ... which we can infer from the canonical commutation relations. Let us now work in the p-basis, that is, in the momentum representation. ... the position-space wave function is related to the momentum-space wave function ... This ... is just what one expects from Fourier's inversion theorem. ... the mathematics ... somehow "knows" Fourier's ... transforms. ... an extremely well localized (in the x-space) state is to be regarded as a superposition of momentum eigenstates with all possible values of momenta. Even those mometum eigenstates whose momenta are comparable to or exceed mc must be included in the superpositon ... It turns out that the concept of a localized state in relativistic quantum mechanics is far more intricate because of the possibility of "negative energy states", or pair creation ...". Sakurai also says, in Advanced Quantum Mechanics (Benjamin/Cummings 1967):"... the wave function for a well-localized particle contains, in general, plane-wave components of negative energies. ...".

In the D4-D5-E6-E7-E8 VoDou Physics model, Position-Momentum Complementarity is related to Clifford Algebra Bit Duality.

The relevant Complex Structure can be seen in such physical concepts as Type IV(2) Domains, Hyperspace, Black Holes, Wavelets, and Conformal SpaceTime.


If you use Wavelet Transforms

instead of ordinary Fourier Transforms, you might avoid some of the difficulties of the conventional Momentum Space construction and also have a natural Dilatation Scale Transformation Renormalization Group structure.

In conventional Lorentz physics, translations are distinct from rotations, so that Linear Momentum and Angular Momentum are both separately conserved, and Linear Momentum and Angular Momentum cannot be converted into each other.

With Conformal Transformations,

Linear Momentum and Angular Momentum can be converted into each other.

In nlin.PS/0006047, Garcia-Ripol, Perez-Garcia, Krolikowski, and Kivshar say: "... We study the scattering properties of optical dipole-mode vector solitons recently predicted theoretically and generated in a laboratory. We demonstrate that such a radially asymmetric composite self-trapped state resembles "a molecule of light" which is extremely robust, survives a wide range of collisions, and displays new phenomena such as the transformation of a linear momentum into an angular momentum, etc. We present also experimental verifications of some of our predictions. ...".



Radon Transforms are generalizations of Fourier Transforms.

Sigurdur Helgason, in Geometric Analysis on Symmetric Spaces (AMS 1994), describes Radon Transforms, saying:

"... let

Given x in X, b in B there exists a unique horocycle passing through x with normal b. ...

Given a horocycle h and a point x in h there exist exactly |W| distinct horocycles passing through x with tangent space at x equal to h_x. ...

Each horocycle is a closed submanifold of X ... The group G acts transitively on the set of horocycles in X. The subgroup of G which maps ... [a] horocycle ... into itself equals M N. ...

The set of horocycles in X with the differentiable structure of G / M N ... is ... the dual space of X and denoted by Z ... Horocycles are classical objects for hyperbolic spaces Hn and are considered ... [by] Gelfand and Graev ... as orbits of the conjugates of the nilpotent group N. ...

[ a useful nilpotent group in physics is the Heisenberg group

while a useful compact group in physics is the Gauge group ]

X and Z have the same dimension and show many analogies reminiscent of the duality between points and hyperplanes in Rn. ...

the coset space G / M N is not symmetric in general ... it does possess curves resembling geodesics. ...

the case of bounded symmetric domains D. A harmonic function on D which is continuous on [the closure of] D has a Poisson integral representation involving the Shilov boundary S(D) ... [let * denote dual space] ... For a polydisk G* / K* = D* the Shilov boundary is the product B* of the (circle) boundaries. ...".


In his review (Bull. A.M.S. 32 (1995) 441-446)) of Helgason's book Geometric Analysis of Symmetric Spaces (AMS 1994), Francois Rouviere asks a question: "...

What are ... natural substitutes for the Euclidean lines in the hyperbolic unit disk X = H2 = SU(1,1) / SO(2) ?

first answer: take as Z the set of all geodesics of X (circles orthogonal to the unit circle). The corresponding Radon transform may be called a generalized X-ray transform, recalling ... the mathematical theory of tomography ...

A second answer is: take Z as the set of all horocycles of X, that is, the "wave surfaces" orthogonal to a "parallel beam of rays" (geodesics meeting at infinity on the unit circle). The horocycles are thus all circles inwardly tangent to the unit circle.

Both settings are considered in the book ... Helgason deals with the following Radon transforms:

... when dim X < dim Z and the range of R can be characterized as the kernel of a certain differential operator on Z ... This extends the Poisson integral example ... where X is the circle, Z is the disk, and the range of R is known to be the kernel of the Laplace operator on the disk. ...

... The Fourier transform ... on a Riemannian cymmetric space of the noncompact type X = G / K ... is ...

f^(l,b) = INTEGRAL(X) f(x) exp( - i l + r , A(x,b) dx .

It is inverted by

f(x) = INTEGRAL(a*xB) f(x) exp( - i l + r , A(x,b) ) | c(l) |^(-2) dl db .

... as a function of x, the exponential is an eigenfunction of all G-invariant differential operators on X ...

Horocycles are ... level surfaces of A(x,b) for fixed b ...

the Plancherel measure | c(l) |^(-2) dl db involves Harish-Chandra's celebrated, and explicitly known, c-function. ...

the Poisson kernel for the unit disk is an exponential of the function A(x,b) ...".



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