Tony Smith's Home Page

Segal's ConformalTheory



Here are some Obituariesdescribing aspects of Segal's work, and references to current work ofAlexander Levichev.


Casimir Vacuum andConformal 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: "... MarkoRodin's ... vision ... in his book Aerodynamics ... wasessentially a perception of a four dimensional sphere, which becomesa complicated toroidal structure when projected into threedimensions. He also perceived a mapping of an energy flow on thesurface of this projected toroid. The Rodin coil is the best guess atthis stage as to how this original insight could be made in physicalform. ...

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

which is an elaborate numerological scheme rendered on the surfaceof the toroidal form. He expects strong gravitational effects from aproperly 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 alllife processes and throughout nature. ...".

Another sequence uses CliffordPeriodicity, equivalent to a sequence 1, 2, 4, 8, 16, 32, 64,128, 256=1 with periodicity 8 instead of Marko Rudin's periodicity7.

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

... Marko Rodin has built a remarkable system based on thesequence 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 ..... adinfintum. The numbers 3,6,9 have a special place in his system, whichneed not be detailed here. Basically, what he claims is that a verybasic energy form in nature relies on three elements: one binarydoubling sequence of 1,2,4,8,7,5..., another doubling sequence goingin the opposite direction 5,7,8,4,2,1,...., and a sequence of3,9,6,6,9,3,3,9,6,6,9,3..... ... When the previous wheel figure isinspected 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 fromthe beginning, but starting with any two ascending numbers (insteadof 1,1 in the standard Fibonacci sequence). The resulting symmetriesand configurations will be essentially the same as shown here. Inaddition, one can use Theon's demonstration (which is similar to theFibonacci sequence, but gives the square root of two) and obtainthese same patterns. ...".

It seems to me that AlastairCouper's approach indicates that

Perhaps it is significant that the 8-dimensional vectors of256-dimensional Cl(8) are isomorphic to its 8-dimensional+half-spinors and to its 8-dimensional -half-spinors, all three ofwhich have total dimension 8+8+8 = 24, which correspond to the 24vertices of the 24-cell root vectordiagram of the 24+4=28-dimensional Cl(8) bivector Liealgebra Spin(8), whose outer triality automorphisms are theisomorphisms among vector and + and - half-spinors, and all of thisis fundamental to the D4-D5-E6-E7-E8 VoDouPhysics model, including the geometry of MacroSpace as related to24+2=26-dimensional bosonic stringtheory with Monster symmetry andan extension to an M-theoryrelated to the 24+3=27=dimensional Jordanalgebra J3(O).


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


Spheres, RodinCoils, Kron, and NegativePermittivity:

There are 3 parallelizablespheres: S1, S3, and S7. Penrose and Rindler, inSpinors and Spacetime, volume 1, (Cambridge 1986) vol. 1, p. 199,say: "... The (local) condidtion for a basis da to beholonomic is [da, d1] = 0, [,] being the Lie bracketoperation ...", so I want to look at the Lie bracket product on thosespheres.

On the 1-sphere S1, which is isomorphic to the abelian Liegroup U(1), we have [x,y] = 0 for all x and y in S1, soS1 is holonomic and not a good example of anholonomy/nonholonomy.

On the 3-sphere S3, which is isomorphic tothe abelian Lie group SU(2) = Spin(3), we have[x,y] =/= 0 for some x and y in S3, so look at S3. As Penroseand Rindler, in Spinors and Spacetime, volume 1,(Cambridge 1986) vol. 1, p. 199, say: "... Many more manifoldsadmit globally defined non-holonomic bases than admit globallydefined holonomic ones. (An example is S3 ...) ...". So, let's takeS3 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-coverof 3-dim rotations, you can see that any element x of S3 has twoparts:

The set of all axes of rotation corresponds to the points of a2-sphere in 3-space.

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

Therefore, S3 is made up of S2 and S1. Topologically, S3 isconstructed in accord with the Hopffibration 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 Hopffibration S3 / S1 = S2 is not just a fancy math example, but is ageometric picture of the type of physical arrangement you need to getengineering results such as those discussed by JackSarfatti, who said "... My primary objective here is to with rotating superconducting machines, perhaps usingideas from Gabriel Kron for example. ...".

Since I don't have any of Kron's papers, I looked up some stuffabout him on a webpage 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 usedby Kron may be the Hopffibration S3 / S1 = S2 where:


Jack Sarfattisaid "... I am thinking of engineering applications, when the ...[non-holonomic torsion gaps] ... are maybe meters in sizewhich is OK if the radii of elastic curvature are millions ofkilometers, ...".

However, I think that maybe you don't need huge radii to getmeter-size non-holonomic torsion gaps, because if you think of theLie bracket of rotations, which is what you have in S3 because of theS2 in the Hopf fibratioon, you can get a bracket product, or gap,that is on the same order of magnitude size as the two rotationswhose product you are taking, so, with an S3 machine such as aRodin coil, the gap can be roughly the samesize as the Rodin coil machine itself, and you may be able to get ameter-size effect with a meter-size machine.


As to fabrication of materials that mightbe useful in S3 machines, a UCSDweb page describes "... a new class of composite materials withunusual physical properties ... If these effects turn out to bepossible at optical frequencies, this material would have the crazyproperty that a flashlight shining on a slab can focus the light at apoint on the other side ... Underlying the reversal of the Dopplereffect, Snell's law, and Cerenkov radiation ... is that this newmaterial exhibits a reversal of one of the "right-hand rules" ofphysics which describe a relationship between the electric andmagnetic fields and the direction of their wave velocity. ... Thecomposite constructed by the UCSD team ... was produced from a seriesof thin copper rings and ordinary copper wire strung parallel to therings. ... The idea for the new composite ... building on the work ofJohn Pendry of Imperial College, London. In 1996, Pendry described away of using ordinary copper wires to create a material with theproperty physicists call "negativeelectric permittivity." ... Pendry also recently suggested a wayof using copper rings to make a material with negative magneticpermeability at microwave frequencies. ...". One paper by Pendry (andco-authors), specifically describing arrays of metallic cylinders, iscond-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 Sarfatticomments: "... Will the negative electric permittivity andpermeability persist in the superconducting state. What does negativeelectric permittivity do to the Coulomb barrier against fusion ofnuclei for example. Coulomb energy barrier ~ qQ/permittivity r Sowill like charges attract in a medium with negative permittivitydielectric screening? This will remove the repulsive Coulomb barriercompletely, in fact, the collidingpositively charged deuterium ions will attract forming helium withthe release of fusion energy at room temperature. Is thiscorrect? ... Also a plasma shouldexplode if its permittivity makes a phase transition from positive tonegative. Metals should explode aswell? ...".


I comment that:

A useful configuration of Rodin coilsmight be

4 Rodin Coils arranged inthe geometry of a Fuller Vector Eqilibrium Cuboctahedron as in anUnconventional Unispace Earth Experiment.



Waveletsand the Conformal Group

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

"... Maxwell'sequations for electromagneticwaves and the wave equation for acousticwaves ... are invariant under the group of conformaltransformations of space-time, which includes dilations andtranslations - the basic operations of wavelet analysis - as asubgroup. ... the conformal group ... SU(2,2) ... here ... denoted byC. ... [is] ... a 15-dimensional Lie group ... A study ofthe action of SU(2,2) on solutions of Maxwell's equations has beenmade by ... Ruhl ... Distributions on Minkowski space and theirconnection with analytic representations of the conformal group,Commun. Math. Phys. 27 (1972), 53-86. ... The wavelet analysis ofelectromagnetics is based entirely on the homogeneous Maxwellequations. Acoustic waves ... obey the scalar wave equation, whichhas ... its symmetry group the same conformal group C. ... the twoanalyses are just different representations of C. ...

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

... the physical wavelets PSIz, which are solutions of thehomogeneous Maxwell and wave equations, naturally split into twoparts:

This splitting is far from trivial, because of the analyticityconstraints. Neither PSI-z nor PSI+z are global solutions of thehomogeneous equation. Rather, they each solve the correspondinginhomogeneous equation, and the source terms are "currents" given byone-dimensional wavelets ... That establishes a deep connectionbetween the physical wavelets and a certain special class ofone-dimensional wavelets, and therefore between the waveletanalysis of physical waves and that of communication signals....

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

... We define a Hilbert space H of ... solutions by integrationover the light cone in the Fourier domain. ... the analytic-signaltransform (AST) ... extends any function or vector field from Rn toCn, and we call the extended function the analytic signal of theoriginal fundtion. the analytic signal of any solution in H isanalytic in a certain domain T in C4, the causal tube. ...

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

... If g is a time translation, then the waveletsparameterized by gE are all localized by at some time t =/= 0rather 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 auniform nonzero velocity rather than being stationary. ...

... if g is a specialconformal transformation, then gE is a curved submanifoldof T and the wavelets parameterized by gE have centers withvarying velocities. ... the .. special conformal transformations,which form a four-parameter subgroup. ... begin with the space-timeinversion 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 intocurved ones. ... Cb maps flat surfaces in space at time zero tocurved surfaces in the same space. ... for some different choices ofb, Cb can be interpreted as boosting to an accelerating referenceframe ... all conformal transformations are nonsingular on T....".

There exist Electromagenticand Acoustic SuperLuminal X Waves.

At different levels in the D4-D5-E6-E7physics model there are atleast 5 types of conformal transformations:

At each level of ConformalStructure, Physical Wavelets provide a connectionbetween the World of Physics and theWorld of Information.

The Geometry of those connections is that of BoundedComplex Domains. A good introductory paper is ConformalTheories, Curved Phase Spaces Relativistic Waveletsand the Geometry of Complex Domains, byR. Coquereaux andA. Jadczyk, Reviews inMathematical Physics, Volume 2, No 1 (1990) 1-44, which can bedownloadedfrom the web as a 1.98 MB pdf file. See also Lie Balls andRelativistic Quantum Fields, by R.Coquereaux, Nuc. Phys. B (Proc. Suppl.) 18B (1990) 48-52 andBorn's Reciprocity in the Conformal Domain, by A.Jadczyk, in Z. Oziewicz et al. (eds), Spinors, Twistors, CliffordAlgebras and Quantum Deformations, 129-140 (Kluwer 1993).

The relevant Complex Structure can be seen in such physicalconcepts as Momentum Space; Position-MomentumComplementarity, Type IV(2)Domains, Hyperspace,Black Holes, andConformalSpaceTime.



Since the Conformal Transformationsinclude Dilatation Scale Transformation, physics models based onConformal Transformations, such as

the D4-D5-E6-E7 model, havenatural RenormalizationStructure.

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


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


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


Wavelet-induced renormalization group for the Landau-Ginzburgmodel has been described by C. Best in hep-lat/9909151,saying: "... The scale hierarchy of waveletsprovides a natural frame for renormalization. ... A waveletexpansion can be used to derive the properties of the Landau-Ginzburgmodel and its nontrivial renormalization flow even in a rather simpleapproximation. The crucial features we have made use of are scalingand self-similarity of the basis and locality of the basis functions.They enabled us to focus on the fluctuation strengths at differentscales as the quantities of interest that govern the phasetransition. The effective free energy of the system exhibits in aminimal 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/Cummings1985), says: "... There is ... a complete symmetry between x and p... which we can infer from the canonical commutation relations. Letus now work in the p-basis, that is, in the momentum representation.... the position-space wave function is related to themomentum-space wave function ... This ... is just what one expectsfrom 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 ofmomentum eigenstates with all possible values of momenta. Even thosemometum eigenstates whose momenta are comparable to or exceed mc mustbe included in the superpositon ... It turns out that the conceptof a localized state in relativistic quantum mechanics is far moreintricate because of the possibility of "negative energy states", orpair creation ...". Sakurai also says, in Advanced Quantum Mechanics(Benjamin/Cummings 1967):"... the wave function for a well-localizedparticle contains, in general, plane-wave components of negativeenergies. ...".

In the D4-D5-E6-E7-E8 VoDou Physics model, Position-MomentumComplementarity is related to CliffordAlgebra Bit Duality.

The relevant Complex Structure can be seen in such physicalconcepts as Type IV(2) Domains,Hyperspace, BlackHoles, Wavelets, and ConformalSpaceTime.


If you use WaveletTransforms

instead of ordinary Fourier Transforms, you might avoid someof the difficulties of the conventional Momentum Space constructionand also have a natural Dilatation Scale Transformation RenormalizationGroup structure.

In conventional Lorentz physics, translations are distinct fromrotations, so that Linear Momentum and Angular Momentum are bothseparately conserved, and Linear Momentum and Angular Momentumcannot be converted into each other.

With Conformal Transformations,

Linear Momentum and Angular Momentum can be converted intoeach other.

In nlin.PS/0006047,Garcia-Ripol, Perez-Garcia, Krolikowski, and Kivshar say: "... Westudy the scattering properties of optical dipole-mode vectorsolitons recently predicted theoretically and generated in alaboratory. We demonstrate that such a radially asymmetric compositeself-trapped state resembles "a molecule of light" which isextremely robust, survives a wide range of collisions, and displaysnew phenomena such as the transformation of a linear momentum intoan angular momentum, etc. We present also experimentalverifications of some of our predictions. ...".



Radon Transforms are generalizations of FourierTransforms.

Sigurdur Helgason, in Geometric Analysis on Symmetric Spaces (AMS1994), describes Radon Transforms, saying:

"... let

Given x in X, b in B there exists a unique horocyclepassing 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 equalto h_x. ...

Each horocycle is a closed submanifold of X ... The group G actstransitively on the set of horocycles in X. The subgroup of G whichmaps ... [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 ... Horocyclesare classical objects for hyperbolic spaces Hn and are considered ...[by] Gelfand and Graev ... as orbits of the conjugates of thenilpotent group N. ...

[ a useful nilpotentgroup in physics is the Heisenberg group

while a useful compactgroup in physics is the Gauge group ]

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

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

the case of bounded symmetric domains D. A harmonicfunction on D which is continuous on [the closure of] Dhas a Poisson integralrepresentation involving the Shilovboundary S(D) ... [let * denote dual space] ... For apolydisk 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 bookGeometric Analysis of Symmetric Spaces (AMS 1994), Francois Rouviereasks a question: "...

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

first answer: take as Z the set of all geodesics of X (circlesorthogonal to the unit circle). The corresponding Radon transform maybe called a generalized X-ray transform, recalling ... themathematical theory of tomography ...

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

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

... when dim X < dim Z and the range of R can be characterizedas the kernel of a certain differential operator on Z ... Thisextends the Poisson integral example ... where X is the circle, Z isthe disk, and the range of R is known to be the kernel of the Laplaceoperator on the disk. ...

... The Fourier transform ... on a Riemannian cymmetric space ofthe 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 allG-invariant differential operators on X ...

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

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

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



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