aa r X i v : . [ m a t h - ph ] D ec Trkalian fields: ray transforms and mini-twistors
K. Saygili ∗ Department of Mathematics, Istanbul University, Beyazit Campus, 34134 Vezneciler Istanbul, Turkey
We study X-ray and Divergent beam transforms of Trkalian fields and their relation with Radontransform. We make use of four basic mathematical methods of tomography due to Grangeat,Smith, Tuy and Gelfand-Goncharov for an integral geometric view on them. We also make use ofdirect approaches which provide a faster but restricted view of the geometry of these transforms.These reduce to well known geometric integral transforms on a sphere of the Radon or the sphericalCurl transform in Moses eigenbasis, which are members of an analytic family of integral operators.We also discuss their inversion. The X-ray (also Divergent beam) transform of a Trkalian field isTrkalian. Also the Trkalian subclass of X-ray transforms yields Trkalian fields in the physical space.The Riesz potential of a Trkalian field is proportional to the field. Hence, the spherical mean ofthe X-ray (also Divergent beam) transform of a Trkalian field over all lines passing through a pointyields the field at this point. The pivotal point is the simplification of an intricate quantity: Hilberttransform of the derivative of Radon transform for a Trkalian field in the Moses basis. We also definethe X-ray transform of the Riesz potential (of order 2) and Biot-Savart integrals. Then, we discussa mini-twistor respresentation, presenting a mini-twistor solution for the Trkalian fields equation.This is based on a time-harmonic reduction of wave equation to Helmholtz equation. A Trkalianfield is given in terms of a null vector in C with an arbitrary function and an exponential factorresulting from this reduction. ∗ [email protected] rkalian fields: ray transforms and mini-twistors I. INTRODUCTION
This is the second manuscript of a series aimed at studying the mathematical structure of Trkalian class ofBeltrami fields (eigenvectors of Curl operator with constant eigenvalue) in integral geometric and twistor terms.Integral geometry and Twistor theory provide new mathematical methods for studying the geometry of Trkalianfields. These lead to a deeper understanding of their physical aspects. The Trkalian fields arise in different areas ranging from fluid dynamics and plasma physics to field theories. Thefield theoretic examples of Trkalian vectors are the force-free magnetic field and the Euclidean topologically massiveAbelian gauge field.In plasma physics, a Trkalian field simply corresponds to a force-free equilibrium state of a plasma. The math-ematical tomography is based on applications of integral geometric methods in tomography. It makes use of bothpure and applied techniques. It provides the mathematical basis for modern tomography in the realistic sense. Thetomographical study of an equilibrium state of a plasma is an active field of research in plasma tomography.The topologically massive gauge theories are qualitatively different from Yang-Mills type gauge theories besidestheir mathematical elegance and consistency. These are introduced as an alternative to the mechanism of spontaneoussymmetry breaking for generation of mass. The study of their physical and mathematical aspects is an active andexciting field of reseach today providing new insight into the relation of gauge theories and gravity in low dimensions.In this context, Trkalian type solutions on 3-sphere S , anti-de Sitter space H and other spaces in connection withcontact geometry are discussed in Refs. 6, 10–12.The Trkalian (Beltrami) fields equation also arises in connection with harmonic morphisms. See Refs. 14 and15 and the refence therein for the solution on S (consisting of right/left-invariant 1-forms) and its uniqueness (uptopermutation).A purpose of this manuscript is to develop physical insight for Trkalian fields in integral geometric and twistorterms. Intuitively speaking, from a higher point of view, we expect these to be related to the representations of thegroup underlying Trkalian fields. Basically, we are trying to investigate the aspects of functions invariant under theCurl operator generating rotations. These integral transforms naturally arise in geometric analysis (in Fourier sense)of physical systems.As a physical example of integral geometry in tomography, we shall frequently benefit the example of Lundquist field which is used to model solar magnetic clouds, (see Refs. 4, 5 and the references therein). We shall present amathematical R¨ontgen of these clouds.Mathematically, we aim to expose the interrelations of the most basic transforms in integral geometry, for Trkalianfields. We are motivated by the geometric picture that is provided by the Radon transform which is at a centralplace in integral geometry. This endows us with an intuition for Trkalian fields. These transforms which arisenaturally in the study of integral geometric and tomographical aspects of Trkalian fields are intimately connected.The mathematical methods of tomography provide delicate ways for exposing their interrelations. This leads to anintuitive, integral understanding, besides its potential physical applications.We also aim to discuss a mini-twistor representation for Trkalian fields presenting a mini-twistor solution for them.This leads to new mathematical challenges beside providing new physical insight into the Trkalian fields.In the first part of this manuscript we shall study X-ray (John ) and Divergent beam transforms of Trkalian fields.Then we shall discuss the mini-twistor representation for Trkalian fields in the second part.The Radon transformation provides a geometric formulation of Trkalian fields. Especially, Moses eigenfun-ctions of the Curl operator, which form a complete orthonormal basis, leads to a helicity decomposition of theRadon transform of Trkalian fields. The spherical Curl transform is a Radon probe transformation in this basis.The Radon transform of a Trkalian field is tangent to a sphere in the transform space. It satisfies a correspondingeigenvalue equation on this sphere.Furthermore, we can associate an eigenvalue equation for Biot-Savart integral operator with Trkalian fields ifthey vanish at infinity. Meanwhile, the Riesz potential and Biot-Savart integrals naturally arise in Radon transform. The Radon-Biot-Savart (
RBS ) integral is defined as the Radon transform of the Biot-Savart operator. We can studyTrkalian fields using the RBS operator in transform space.First we shall study X-ray and Divergent beam transforms of Trkalian fields. The field is to be taken from Schwartzspace S [ R ] of rapidly decreasing functions on R . The X-ray and Divergent beam transforms are closely connected with the Radon transform. The mathematicalmethods of tomography (respectively Smith’ s and Tuy’ s methods discussed below) show that these transforms arebasically in the form of a Minkowski-Funk and a closely related (well known, but no specific name in the literatureknown to the author) integral transform of certain intricate quantities (Hilbert transform of the derivative of Radontransform). The Moses eigenbasis is especially efficient in exhibiting this connection for Trkalian fields. Then, thesenaturally reduce to well known geometric integral transforms on a sphere (mentioned above) of the Radon or thespherical Curl transforms. More precisely, the X-ray transform reduces to Minkowski-Funk transform of the Radon rkalian fields: ray transforms and mini-twistors ∇× ) and alsothe Divergence ( ∇ · ), Gradient ( ∇ ) and Laplacian ( ∇ ) operators. Thus the X-ray (Divergent beam) transform ofa Trkalian field is Trkalian. Also, the Trkalian subclass of X-ray transforms X F (satisfying John’ s differentialequation) yields Trkalian fields in the physical space. We shall also write John’ s differential equation for Trkalianfields in an equivalent form. Thus, we can study Trkalian fields either in physical space or in the transform space.Another crucial quantity in integral geometry is the Riesz potential. The Riesz potential, of order α where 0 < α <
3, of a Trkalian field is proportional to the field. Hence, the spherical mean of X-ray (or Divergent beam) transformof a Trkalian field over all lines passing through a point yields the field itself at this point. This endows us with a newsimple inversion formula for the X-ray (or Divergent beam) transform of Trkalian fields. This result is also logicallyimplied by Gelfand-Goncharov’ s mathematical approach to tomography below.Then we shall return back to the mathematical methods of tomography. The first purpose of this section is toprovide a unified geometric view and motivation for the interrelations of integral transforms arising in our discussion,as mentioned above. The second purpose is to present a discussion of these mathematical methods with a viewtowards tomographical studies of Trkalian field models in nature. For this purpose, we shall study these mathematicalmethods using Trkalian fields. Especially for the sake of the second purpose and also for a clean presentation, thisdiscussion will be postponed until the direct (but unmotivated) discussion of the geometry of X-ray and Divergentbeam transforms finish.We shall make use of four basic mathematical approaches of tomography due to Grangeat, Smith, Tuy andGelfand-Goncharov for studying the X-ray and Divergent beam transforms of Trkalian fields and expressing theirrelations with the other transforms. These relations are outflow of a formula essentially obtained in Ref. 32. These methods basically make use of the Radon inversion (for tomographical reconstruction). They lead us to newinversion formulas for the X-ray and Divergent beam transforms of Trkalian fields with a view towards tomographicalapplications. They also provide the geometric motivation underlying the interrelations of the transforms mentioned.We shall adopt a mathematical approach rather than a tomographical implementation.The Grangeat approach leads to another simple, direct inversion formula for the Divergent beam transform ofTrkalian fields.The Smith method reveals that the X-ray transform is in the form of a Minkowski-Funk transform of an intricatequantity related to the Radon transform. This simply reduces to Minkowski-Funk transform of the Radon transformon a sphere, yielding the result mentioned above. In this approach, the inversion formula can be expressed in termsof the Radon transform of the field.The Tuy method enables us to investigate the Divergent beam transform in detail. It reveals that the Divergentbeam transform is in the form of another closely related integral transform of a quantity related to the Radontransform. This reduces to the above mentioned integral transform of the spherical Curl transform. In this case, theinversion formula can be expressed in terms of the spherical Curl transform of the field.We calculate the Divergent beam transform of the Lundquist field using the Tuy method. This yields a mathematical R¨ontgen of solar magnetic clouds.Meanwhile, Gelfand-Goncharov’ s approach leads to a direct inversion through the spherical mean that is men-tioned above. This naturally makes use of the inverse transform that belongs to the above family of integrals operators.The basic simplification in these approaches are due to the same intricate quantity: Hilbert transform of thederivative of Radon transform in the Moses basis.The direct inversion formulas arising in Grangeat’ s and Gelfand-Gonchorav’ s approaches mathematically seemmore feasible than the inversions in Smith’ s and Tuy’ s methods.These approaches provide different inversion formulas which may serve useful for designing reconstruction methodsin tomographical studies of Trkalian field models in nature, depending on real physical situation. We shall not discusstomographical implementations of these inversion formulas. rkalian fields: ray transforms and mini-twistors It has led to a deeper understanding ofnature. In simplest terms, this is based on writing contour integral solution for wave equation in (3 + 1) dimensionalMinkowski space, using a holomorphic function. Similar formulas date back to Whittaker and Bateman.
The X-ray transform is a real analogue of the Penrose transform and a predecessor of Twistor theory. Seefor example Refs. 38–44 for relation of the X-ray transform and Twistor theory.Mini-twistor space as an intrinsic structure has been introduced by Hitchin. The (mini-)twistor space of R isthe space TS of oriented lines in R . This can be identified with TCP , the holomorphic tangent bundle of Riemannsphere CP . This is also given by the quotient of twistor space ( CP \ CP ) of the Minkowski space by the action oftime translation. The X-ray transform and the mini-twistors are both defined on the space TS ∼ TCP of orientedlines in R .We shall discuss a mini-twistor representation, presenting a mini-twistor solution for the Trkalian fields equation.We shall make use of the solution of (vector) Helmholtz equation which is based on a time-harmonic reduction of thewave equation. A Trkalian field is given in terms of a null vector in C with an arbitrary function and an exponentialfactor that results from the reduction.The exponential factor contains the spatial part of an integrating factor for the time-harmonicity condition. Thesolution is of the same form containing the spatial part of any choosen integrating factor. We shall also use the generalsolution of this condition for writing the solution.This solution can also be derived as a time-harmonic reduction of the twistor solution for electromagnetic fieldsin (3 + 1) dimensions.We are led to a time-harmonic extension of Trkalian fields implicitly keeping this condition. This can be interpretedas a time-harmonic electromagnetic field.We shall also present examples of Debye potentials for Chandrasekhar-Kendall (CK) type solutions using thetwistor solution of (scalar) Helmholtz equation.The relation of (mini-)twistors and ray transforms for Trkalian fields is beyond the limitations of this manuscript. II. X-RAY AND DIVERGENT BEAM TRANSFORMSA. Trkalian fields: Radon transform
Trkalian fields are eigenvectors of the curl operator ∇ × F ( x ) − ν F ( x ) = 0 , (1)with constant eigenvalue ν . The Radon transform F R ( p, κ ) = R [ F ( x )]( p, κ ) = Z F ( x ) δ ( p − κ · x ) d x, (2)of a field F ( x ) (that belongs to Schwartz class) on R is defined as the integral of the field over a hyperplane at(orthogonal) distance p to the origin, with unit normal vector κ . The Radon transform F R ( p, κ ) of a Trkalian fieldsatisfies Γ × F R ( p, κ ) − ν F R ( p, κ ) = 0 , (3)where Γ = κ ∂/∂p . We also have: κ · F R ( p, κ ) = 0 which leads to Γ · F R ( p, κ ) = 0. Because the Radon transformintertwines the operator ∇ with Γ . We can write this equation as ∂∂p F R ( p, κ ) + νκ × F R ( p, κ ) = 0 . (4) rkalian fields: ray transforms and mini-twistors is based on decomposing a vector field into helical eigenfunctions χ λ ( x | k ) = (2 π ) − / e i k · x Q λ ( k )of the curl operator, which form an orthogonal and complete set, in the fashion of a Fourier transform refining theHelmholtz decomposition. This is a helicity ( λ = − , ,
1) decomposition in the basis { Q λ ( k ) } .A Trkalian field can be expressed as: F ( x ) = Σ ′ F λ ( x ) excluding the divergenceful component, where F λ ( x ) =(1 /g ) R χ λ ( x | k ) f λ ( k ) d k . Then we find f λ ( k ) = (cid:2) δ ( k − λν ) /k (cid:3) s λ ( k ) relating the Curl transform f λ ( k ) and thespherical Curl transform s λ ( k ) of the field F ( x ). Thus an arbitrary solution is given entirely in terms of its transformon a sphere of radius k = λν = | ν | in transform space. Further, only the eigenfunctions for which λ = sgn ( ν )contribute to the field. The Radon transform of a Trkalian field is tangent to this sphere F R λ ( p, κ ) = (2 π ) / g ν (cid:2) e iλνp Q λ ( κ ) s λ ( λν κ ) + e − iλνp Q λ ( − κ ) s λ ( − λν κ ) (cid:3) . (5)The factor 1 /g is introduced for the sake of a proper strength for the gauge potential in topologically massive gaugetheory. This can be taken as 1 for general Trkalian fields. The inverse transform is given as F λ ( x ) = 18 π ν Z S κ F R λ ( κ · x , κ ) d Ω κ (6)= 1(2 π ) / g Z S κ e iλν κ · x Q λ ( κ ) s λ ( λν κ ) d Ω κ , using the adjoint Radon transform R † , where S κ is the unit sphere in the transform space.The simplest example of Trkalian fields is F ( x ) = e i k · x F where k = k κ , k = λν > F = Q λ ( κ ): κ × F = − iλ F , κ · F = 0. Its Radon transform is F R ( p, κ ) = (2 π ) k (cid:2) e ik p δ ( κ − κ ) + e − ik p δ ( κ + κ ) (cid:3) F . (7)The Lundquist solution is given as F L ( x ) = F [ λJ ( λνr ) e φ + J ( λνr ) e z ] , (8)in cylindrical coordinates, where J n is the n th order Bessel function. Its Radon transform, for λ = 1 is F R L ( λ =1) ( p, κ ) = 2 πiF ν δ ( κ z )( e iνp L + e − iνp L ′ ) , (9)where L = sin ψ e x − cos ψ e y − i e z , L ′ = − sin ψ e x + cos ψ e y − i e z and k = k r cos ψ e x + k r sin ψ e y + k z e z . The Radon transform of the Riesz potential of order 2: I [ F ]( x ) = (1 / π ) R † R [ F ]( x ) is given as R { I [ F ] } ( p, κ ) = 18 π RR † R [ F ]( p, κ ) = F − { k F [ F R ( q, κ )]( k, κ ) } ( p, κ ) . (10)We can associate an eigenvalue equation for Biot-Savart integral operator: BS [ F ]( x ) = ∇ × I [ F ]( x ) with aTrkalian field, if its Radon transform exists (if the field is divergence-free and it vanishes at infinity): BS [ F ] =(1 /ν ) F . For example, the Lundquist field (8) is an eigenvector of the BS operator.The Radon-Biot-Savart integral RBS [ F R ( q, κ )]( p, κ ) = Γ p × R { I [ F ] } ( p, κ ) = i κ × F − { k F [ F R ( q, κ )]( k, κ ) } ( p, κ ) , (11)is defined as the Radon transform of the BS integral. The Radon transform (5) of a Trkalian field is an eigenvectorof the
RBS integral operator. The inverse spherical Curl transform expression (6) for a Trkalian field is in the form of Whittaker’ s solution tothe (scalar) Helmholtz equation (one for each Cartesian component or simply: f −→ f there and λν is normalized).If we substitute (the vectorial form of) Whittaker’ s solution of the Helmholtz equation in (1), we are led to a vectoridentity which is simply satisfied by the Moses basis vectors: κ × Q λ ( κ ) ∼ − i Q λ ( κ ) (up to helicity factor λ = ± rkalian fields: ray transforms and mini-twistors B. X-ray and Divergent beam transforms
The X-ray transform of a vector-valued function F ( x ) (that belongs to Schwartz class) on R is defined as X F ( θ , x ) = Z ∞−∞ F ( x + s θ ) ds, (12)the integral of the field over line L passing through point x in the direction determined by the unit vector θ . This is a componentwise generalization of the X-ray transform of scalar fields. The X-ray transform is defined on thespace of oriented lines in R . Note, X F ( θ , x ) is unchanged if x is translated in the direction of θ . Therefore we restrict x to θ ⊥ : x · θ = 0. Hence X F ( θ , x ) is a function defined on the tangent bundle TS = { ( θ , x ) , θ ∈ S , x ∈ θ ⊥ } ofthe sphere S . Also X F ( − θ , x ) = X F ( θ , x ).The Divergent beam or Cone beam transform is defined as D F ( θ , x ) = Z ∞ F ( x + s θ ) ds, (13)the integral of the field over the half-line. We have: X F ( θ , x ) = D F ( θ , x ) + D F ( − θ , x ).The X-ray transform (12) [also the Divergent beam transform (13)] satisfies John’ s equation below. The inversionproblem of the X-ray transform is overdetermined, that is the data of all line integrals are redundant. There arevarious inversion methods for the X-ray (also Divergent beam) transform. For example, one can reconstruct F ( x )knowing X F ( θ , x ) where θ ∈ S , x ∈ L and L is a suitable curve, (see Ref. 48, p. 52, p. 276 and Ref. 51).The X-ray transform of the Trkalian field F ( x ) = e i k · x F where k = k κ , k > X F ( θ , x ) = 2 π k e ik κ · x δ ( κ · θ ) F . (14)Its Divergent beam transform is D F ( θ , x ) = 2 π k e ik κ · x δ + ( κ · θ ) F , D F ( − θ , x ) = 2 π k e ik κ · x δ − ( κ · θ ) F , (15)where δ ± ( x ) = (1 /
2) [ δ ( x ) ∓ / ( iπx )] is the socalled Heisenberg distribution and 1 /x is to be understood in thesense of Cauchy principal-value: 1 /x = P (1 /x ). Here we have used Fourier transform of the Heaviside step function(al) H ( x ): F [ H ( ± x )]( k ) = √ πδ ∓ ( k ).The X-ray transform of the Lundquist field (8) is X F L ( θ , x ) = 2 F λν v r (cid:8) sin [ νr sin( θ − φ )] e r ( θ ) + cos [ νr sin( θ − φ )] e z (cid:9) , (16)where x = r e r ( φ ) + z e z , θ = v r e r ( θ ) + v z e z in cylindrical coordinates. See Appendix A 1. We shall calculate itsDivergent beam transform in Section IV C.The main results of this section are given in the following two propositions. Note that these results are logicallymotivated in a unified view by the mathematical methods in Sections IV B: Smith’ s method and IV C: Tuy’ s method.However, we have interchanged the logical order for a clean presentation, because the direct approach here is moreappropriate for exposing the relations of X-ray and Divergent beam transforms of Trkalian fields to the Radon andother transforms. Also, the geometry of these results deserves a separate discussion in its own right. We shall discussthe mathematical point of view originating from tomography later on. We shall first present direct proofs withoutmotivation below. Proposition 1:
The X-ray transform of a Trkalian field is given as X F λ ( θ , x ) = 1(2 π ) / g λν Z S κ e iλν κ · x Q λ ( κ ) s λ ( λν κ ) δ ( κ · θ ) d Ω κ (17)= 14 π λν Z S κ F R λ ( κ · x , κ ) δ ( κ · θ ) d Ω κ , the Minkowski-Funk transform (in the transform space) of its Radon transform. rkalian fields: ray transforms and mini-twistors Proof:
We substitute F λ ( x + s θ ) using the second line of (6) in (12). See Appendix A 2 a. Note that a result validin a basis should be valid in any basis. ✷ This proposition is based on the motivation in equation (59) below which makes use of Smith’ s formula (55) inSection IV B. See equation (61) for the underlying intricate geometric quantity and its simplification.The Minkowski-Funk transform M of a continuous (even) function on a sphere is given by the integral of thisfunction over great circles: geodesics in the sphere. This yields a function on the space of geodesics. This clearlyannihilates odd functions.We symbolically write X = [ λν/ (4 π )] MR respectively relating the X-ray (John ), Minkowski-Funk andRadon transforms for Trkalian fields. In R , this corresponds to integral of the Radon transform of the field over apencil of planes intersecting at a line (passing through the point x in the direction of θ ) which are parametrized bya circle. Previously, Gonzales called this plane-to-line transform. See Ref. 43 for a twistor view of this transform.There are various inversion methods for the Minkowski-Funk transform. We shall briefly discuss an inversionformula which is appropriate for our purpose below.If we use the example (7) in the second line of (17) we find the result (14). Also, we find (16), ( λ = 1) substituting(9) in the second line of (17), where κ r = sin α = p − κ z , κ z = cos α and α is the polar angle on S κ with sphericalangles ψ , α . We shall avoid the details of these straightforward calculations.We can arrive at the same result, as expressed in equation (A11), using the Curl expansion for Trkalian fields inFourier slice-projection theorem F [ X F ( θ , x )]( θ , ξ ) = (2 π ) / F [ F ( x )]( ξ ) , ξ ∈ θ ⊥ (18)for the X-ray transform. See Appendix A 2 b. On the left-hand side F stands for Fourier transform in θ ⊥ planewhereas on the right-hand side it is a Fourier transform in three dimensions. Proposition 2:
The Divergent beam transform of a Trkalian field is given as D F λ ( θ , x ) = 1(2 π ) / g λν Z S κ e iλν κ · x Q λ ( κ ) s λ ( λν κ ) δ + ( κ · θ ) d Ω κ , (19) in the Moses basis. Proof:
We substitute F λ ( x + s θ ) using the second line of (6) in (13) and use F [ H ( ± x )]( k ) = √ πδ ∓ ( k ). ✷ This proposition is based on the motivation in equation (69) below which makes use of (67) in Section IV C: Tuy’s method. See equation (74) for the underlying intricate geometric quantity and its simplification.This is another basic transform (with no specific name in the literature known by the author) in integral geometrycontaining the Heisenberg delta function δ ± . The first term in δ ± is associated with the Minkowski-Funk transformmentioned above. Hence, this is an extension of the Minkowski-Funk transform. The second term is to be understoodin the sense of Cauchy principal value (see above). Roughly speaking, it provides a description of the behaviour of afunction on the sphere except the geodesics, [see (21) below for the physical meaning].This leads to D F λ ( − θ , x ) = 1(2 π ) / g λν Z S κ e iλν κ · x Q λ ( κ ) s λ ( λν κ ) δ − ( κ · θ ) d Ω κ , (20)since δ + ( − κ · θ ) = δ − ( κ · θ ).For F ( x ) = e i k · x F , [ k = k κ , k = λν > F = Q λ ( κ )], we find the result (15) substituting s λ ( λν κ ) =(2 π ) / gδ ( κ − κ ) in (19, 20).In analogy with the X-ray transform (12), the difference Y F ( θ , x ) = D F ( θ , x ) − D F ( − θ , x ) = Z ∞−∞ F ( x + s θ ) sgn ( s ) ds, (21)where sgn ( s ) = H ( s ) − H ( − s ) is the signum function, reduces to Y F λ ( θ , x ) = i π ) / g λν Z S κ e iλν κ · x Q λ ( κ ) s λ ( λν κ ) 1 κ · θ d Ω κ , (22) rkalian fields: ray transforms and mini-twistors F [ sgn ( x )]( k ) = √ π (1 /iπ )(1 /k ).For the Trkalian field F ( x ) = e i k · x F , [ k = k κ , k = λν > F = Q λ ( κ )], we find Y F λ ( θ , x ) = 2 i k e ik κ · x κ · θ F . (23)We can write the equation (19) as D F λ ( θ , x ) = 12 / g λν A [ G ] = 12 / g λν (cid:0) U [ G ] + i V [ G ] (cid:1) , (24)where G ( κ , x ) = e iλν κ · x Q λ ( κ ) s λ ( λν κ ), A = U + i V and U [ G ] = 12 1 π / Z S κ G ( κ , x ) δ ( κ · θ ) d Ω κ V [ G ] = 12 1 π / Z S κ G ( κ , x ) 1 κ · θ d Ω κ , (25)= 12 1 π / Z κ · θ =0 G ( κ , x ) d θ κ , with [a slight change of notation: e r ( φ ) −→ κ in (A11)] the integration measure d θ κ on the great circle C determinedby θ . The integrals in U and V respectively correspond to the Minkowski-Funk transform in (17, A11) and thetransform in (22). The Minkowski-Funk transform U describes behaviour of the function on great circles in the2-sphere. Roughly, the transform V describes behaviour of the function on the 2-sphere except the great circles.The transform A is a member (via analytic continuation) of an analytic family of integral operators A α = U α + i V α which arise in the study of Fourier transforms of homogeneous functions. Recently, these have been studied byRubin. For a brief summary of these transforms see Appendix A 3.The equations (17, A11), (19) and (24, 25) respectively suggest their inversion methods for the X-ray and Divergentbeam transforms of Trkalian fields. For the sake of motivation, we shall briefly discuss the inversion of Minkowski-Funktransform U using a member of this family. This was established by Semyanistyi. Because a detailed study ofthese would be distracting.We have X F = [ λν/ (4 π )] M [ F R ], (17) and U = [1 / (2 π / )] M , (25), hence X F = [ λν/ (2 π / )] U [ F R ]. Theinverse of U is ( U ) − = U − (Appendix A 3), hence M − = [1 / (2 π / )] U − . Thus F R λ ( κ · x , κ ) = − π λν Z S θ X F λ ( θ , x ) 1 | κ · θ | d Ω θ , (26)which is to be understood in a regularized sense. See equation (81) and the following discussion in Section IV Dfor a derivation of this inversion formula from the mathematical methods of tomography.If we substitute (17) in (26) and interchange the order of integrations, we find F R λ ( κ · x , κ ) = − π ) / g ν Z S κ e iλν κ ′ · x Q λ ( κ ′ ) s λ ( λν κ ′ ) I ( κ , κ ′ ) d Ω κ ′ . (27)The integral I ( κ , κ ′ ) = Z S θ δ ( κ ′ · θ ) | κ · θ | d Ω θ = − π [ δ ( κ − κ ′ ) + δ ( κ + κ ′ )] , (28)can be evaluated using the planewave decomposition of Dirac delta function. See Appendix A 3 a. Then theequation (27) yields F R λ ( κ · x , κ ), (5). We can find F λ ( x ) using the inverse Radon transform (6). See Section IV Dfor this.For F ( x ) = e i k · x F , [ k = k κ , k = λν > F = Q λ ( κ )], if we substitute (14) in (26), we find F R λ ( κ · x , κ ),(7) with a similar reasoning as in (28), ( κ ′ −→ κ ). For the Lundquist solution (8), if we substitute (16), [that is (A9)]in (26) and use the planewave decomposition of Dirac delta function, then we find F R L ( λ =1) ( κ · x , κ ), (9) for λ = 1.See Appendix A 3 b. rkalian fields: ray transforms and mini-twistors F λ ( x ) is given by the spherical mean of X F λ ( θ , x ). We shall derive this in amore direct way below, [see (43), (B35)].As a different alternative, one can try expressing F R in terms of X F : F R ( p, κ ) = [ R X F ( θ , x )] ( p, κ ) using aRadon transform: R in the plane θ ⊥ . We can also write the Divergent beam transform (19, 24) as D F λ ( θ , x ) = 1(2 π ) / g λν η Z R k k G ( κ , x ) e i k · η d k, (29)the equation (A17) with Re ( α ) = 0, where k = k κ , k = | k | and η = η θ , η = | η | . It is straightforward to verify thisfor the field F ( x ) = e i k · x F , [ k = k κ , k = λν > F = Q λ ( κ ) and s λ ( λν κ ) = (2 π ) / gδ ( κ − κ )], using (15). III. JOHN’ S EQUATION
The X-ray transform (12) [also the Divergent beam transform (13)] satisfies John’ s equation ∂ ∂x i ∂θ j X F ( θ , x ) = ∂ ∂x j ∂θ i X F ( θ , x ) . (30)The John’ s equation is the necessary and sufficient condition for a function to be expressible as an X-ray transform. This can be written as an ultrahyperbolic wave equation.We can easily check (17) [also (19)] satisfies the equation (30). It is also easy to check (14) [also (15)].
Proposition 3:
The X-ray (also Divergent beam) transform intertwines the Curl operator ( ∇× ): X [ ∇ × F ]( θ , x ) = ∇ x × X F ( θ , x ) , (31) and also the Divergence ( ∇ · ), Gradient ( ∇ ) and Laplacian ( ∇ ) operators. Proof:
Let x ′ = x + s θ , then ∇ x ′ = ∇ x and we have ∇ x ′ × F ( x ′ ) = ∇ x × F ( x + s θ ). Then we find, the X-ray(Divergent beam) transform intertwines the Curl operator: X [ ∇ × F ]( θ , x ) = ∇ x × X F ( θ , x ), using the definition(12), [(13)]. The proofs for the Divergence: X [ ∇ · F ]( θ , x ) = ∇ x · X F ( θ , x ), Gradient: X [ ∇ f ]( θ , x ) = ∇ x X f ( θ , x )and hence the Laplacian: X [ ∇ f ]( θ , x ) = ∇ x X f ( θ , x ) operators work with a similar reasoning. ✷ Proposition 4:
The X-ray (also Divergent beam) transform intertwines the operator ( ∇× ) − ν = 0 for constant ν . Proof:
The X-ray (Divergent beam) transform is linear. ✷ Thus the X-ray (Divergent beam) transform X F ( θ , x ) of a Trkalian field F ( x ′ ), (1) is also Trkalian ∇ x × X F ( θ , x ) − ν X F ( θ , x ) = 0 , (32)[ ∇ x × D F ( θ , x ) − ν D F ( θ , x ) = 0 ] and ∇ x · X F ( θ , x ) = 0.We also find ∇ θ · X F ( θ , x ) = 0, ( ∇ θ = s ∇ x ′ = s ∇ x ) for a Trkalian field.The X-ray transform (17) [also the Divergent beam transform (19)] satisfies (32) and ∇ θ · X F ( θ , x ) = 0. Wecan easily check the X-ray transform X F ( θ , x ), (14) [also the Divergent beam transform D F ( θ , x ), (15)] of the field F ( x ) = e i k · x F , [ k = k κ , k = λν > F = Q λ ( κ )] is Trkalian and ∇ θ · X F ( θ , x ) = 0. It is also straightforwardto show that the X-ray transform X F L ( θ , x ), (16) of the Lundquist field is Trkalian and ∇ θ · X F L ( θ , x ) = 0.We immediately find ∇ θ · ( ∇ x × X F ) = ( ∇ θ × ∇ x ) · X F = − ∇ x · ( ∇ θ × X F ) = ν ∇ θ · X F = 0 , (33)using ∇ θ · X F ( θ , x ) = 0, for a Trkalian field.We find rkalian fields: ray transforms and mini-twistors ∇ x × ( ∇ θ × X F ) − ν ∇ θ × X F = 0 , (34)that is ∇ θ × X F is also Trkalian, using a straightforward reasoning similar to that above. This yields ∇ x · ( ∇ θ × X F ) = 0 as we expect, (33). We can also prove the equation (34) using John’ s equation (30).It is straightforward to verify (34) respectively for the X-ray transforms (14) of the field F ( x ) = e i k · x F , [ k = k κ , k = λν > F = Q λ ( κ )] and (16) of the Lundquist field. Proposition 5:
For a Trkalian field the John’ s equation (30) is equivalent (both necessary and sufficient) to ∂ x m ǫ ijk ∂ θ i ( X F ) j = ν∂ θ m ( X F ) k that is ∂ x m ( ∇ θ × X F ) = ν∂ θ m X F . Proof:
We can write (30) as ∇ θ ×∇ x X A = 0, ∇ θ ×∇ x X B = 0, ∇ θ ×∇ x X C = 0 where X F = ( X A, X B, X C ).The proof is straightforward writing these and (32) and ∇ θ · X F ( θ , x ) = 0 in components. ✷ The difference of symmetric ( m ↔ k ) equations in this yields (34). The sum of diagonal equations ( m = k ) yields ∇ x · ( ∇ θ × X F ) = ν ∇ θ · X F which identically vanishes, (33).We also have ∇ θ × ( ∇ x × X F ) − ν ∇ θ × X F = 0, (34). Then ∇ θ · ( ∇ θ × X F ) = 0 trivially.The X-ray transform and its formal adjoint X † [ G ]( x ) = Z S θ G ( θ , E θ x ) d Ω θ , E θ x = x − ( x · θ ) θ (35)where [ G ( θ , x ) is to be taken in the range of X-ray transform:] G ( θ , x ) = X [ F ]( θ , x ), are related to the Rieszpotential of order α through I α [ F ]( x ) = [1 / (2 π ) ] X † I α − X [ F ]( x ). Here I α − is the Riesz potential on TS acting on the second variable x of X [ F ]( θ , x ) and 0 < α < n = 3. For α = 1 this yields I [ F ]( x ) = 1(2 π ) X † X [ F ]( x ) , (36)where I [ F ]( x ) = 12 π Z F ( y ) | x − y | d y. (37)It is straightforward to prove Z S θ D F ( θ , x ) d Ω θ = 2 π I [ F ]( x ) , (38)see Ref. 58, p. 283. This leads to I [ F ]( x ) = 14 π Z S θ X F ( θ , x ) d Ω θ , (39)using θ −→ − θ in (38). We could infer this from (36) using invariance of X F ( θ , x ) under translation of x in thedirection θ : X F ( θ , E θ x ) = X F ( θ , x ). We can use equation (39) for inverting the X-ray transform. Proposition 6:
The inversion of X-ray transform intertwines the Curl ( ∇× ) and also the Divergence ( ∇ · ) ,Gradient ( ∇ ) and Laplacian ( ∇ ) operators. Proof:
We can easily show: ∇ x × I [ F ]( x ) = I [ ∇ × F ]( x ) for F ( x ) in the Schwartz class, using the Fouriertransform of Riesz potential F { I α [ F ] } ( ξ ) = | ξ | − α F [ F ]( ξ ), α < n = 3. We also have I − α I α = 1. Thus I − h π Z S θ ∇ x × X F ( θ , x ) d Ω θ i = ∇ × F , (40)(39). ∇ x ×X F ( θ , x ) satisfies (30), if X F ( θ , x ) does. The inversion of X-ray transform also intertwines the Divergence( ∇ · ), Gradient ( ∇ ) and hence the Laplacian ( ∇ ) operators with a similar reasoning. ✷ rkalian fields: ray transforms and mini-twistors Proposition 7:
The inversion of X-ray transform intertwines the operator ( ∇× ) − ν = 0 for constant ν . Proof:
The transforms I − and X † are linear. ✷ Thus the Trkalian subclass (32) of functions X F ( θ , x ) satisfying John’ s equation (30) yields Trkalian fields in thephysical space.The propositions 3, 4, 5, 6 and 7 enable us to study Trkalian fields either in physical space or in the transformspace. Proposition 8:
The Riesz potential for a Trkalian field is given by I α [ F λ ]( x ) = ( λν ) − α F λ ( x ) , (41) where α < . Proof:
We are led to F { I α [ F ] } ( ξ ) = 1 g δ ( ξ − λν ) ξ α +2 Q λ ( u ) s λ ( λν u ) , (42)substituting the inverse spherical Curl transform (6) into: F { I α [ F ] } ( ξ ) = | ξ | − α F [ F ]( ξ ), α < n = 3 and using δ ( λν κ − ξ ) = [ δ ( ξ − λν ) /ξ ] δ ( κ − u ), [ ξ = ξ u , ξ = | ξ | , | u | = 1]. The result follows by inverting this. ✷ The Riesz potential (37) for F ( x ) = e i k · x F , [ k = k κ , k = λν >
0] and the Lundquist field (8) respectivelylead to these fields themselves. We shall not present the details of these calculations here.
Proposition 9:
The spherical mean of the X-ray (or Divergent beam) transform of a Trkalian field over all linespassing through a point yields the field at this point Z S θ X F λ ( θ , x ) d Ω θ = 4 π λν F λ ( x ) . (43) Proof:
The result follows from (39) and Proposition 8. [Divergent beam transform: X F ( θ , x ) = D F ( θ , x ) + D F ( − θ , x ), the integrals of Divergent beam transforms in opposite directions are equal. See equation (82) below.] ✷ This provides us a simple inversion formula for Trkalian fields. This proposition is geometrically motivated byequation (79) below in Section IV D: Gelfand-Goncharov’ s method. The underlying geometry is again based on thesimplification in the same intricate geometric quantity: Hilbert transform of the derivative of Radon transform in theMoses basis.The spherical mean of the X-ray transform (14) yields the field F ( x ) = e i k · x F , decomposing θ into componentswhich are respectively parallel and orthogonal to κ . If we substitute (17) in (43), we find (6) with a similar reasoning.An analogous result for the Radon transform of Trkalian fields: I [ F λ ] = [1 / (8 π )] R † R [ F λ ] = (1 /ν ) F λ alsofollows from equations (34, 40) in Ref. 1. IV. MATHEMATICAL METHODS OF TOMOGRAPHY
We shall use four basic mathematical approaches of tomography due to Grangeat, Smith, Tuy and Gelfand -Goncharov. These are based on relations of the X-ray and Divergent beam transforms to Hilbert transform of thederivative of Radon transform. Mathematically, these relations are outflow of the formula Z S θ D F ( θ , x ) h ( θ · b ) d Ω ~θ = Z F R ( p, b ) h ( p − b · x ) dp, (44)essentially obtained in Ref. 32. Here we have introduced the normalization: β = β b , β = | β | , s = βp , F R ( s, β ) =(1 /β ) F R ( p, b ) for the sake of our conventions. The distribution h satisfies h ( ap ) = (1 /a ) h ( p ), a > rkalian fields: ray transforms and mini-twistors A. Grangeat’ s method
We obtain Grangeat’ s formula ∂∂p F R ( p, κ ) (cid:12)(cid:12)(cid:12) p = κ · x = − Z S θ D F ( θ , x ) δ ′ ( κ · θ ) d Ω θ , (45)using h ( p ) = δ ′ ( p ) in (44). This is a componentwise generalization of the formula for scalar fields. For a derivationof this formula see Appendix B 2 a. Thus we find ∂∂p F R ( p, κ ) (cid:12)(cid:12)(cid:12) p = κ · x = Z S θ ∩ κ ⊥ ∂∂ κ D F ( θ , x ) d θ , (46)using the identity (B3). Here ∂/∂ κ denotes directional derivative along κ .We find F R ( κ · x , κ ) = − ν κ × Z S θ D F ( θ , x ) δ ′ ( κ · θ ) d Ω θ , (47)for a Trkalian field using (3, 45). We also have κ · R S θ D F ( θ , x ) δ ′ ( κ · θ ) d Ω θ = 0, (4, 45). The equation (47) leads to F R ( κ · x , κ ) = −
12 1 ν κ × Z S θ Y F ( θ , x ) δ ′ ( κ · θ ) d Ω θ , (48)using θ −→ − θ and rearranging.We can check (45) for F ( x ) = e i k · x F , [ k = k κ , k > Y F λ ( θ , x ), (23)in (48), we are similarly led to F R ( κ · x , κ ), (7).If we substitute (47) [or (48)] into the first line of (6), we find F ( x ) = − π ν Z S κ κ × Z S θ D F ( θ , x ) δ ′ ( κ · θ ) d Ω θ d Ω κ . (49)This leads to F ( x ) = ± π ν Z S θ θ × D F ( ± θ , x ) d Ω θ , (50)interchanging the order of integrations and using the identity (B3), (also θ −→ − θ ).This is another simple, direct inversion (reconstruction) formula for the Divergent beam transform of Trkalianfields. Note that the simplification is basically due to the eigenvalue equation for the Radon transform of Trkalianfields. We shall not discuss the tomographical implementation of this inversion formula.We can verify the formula (50) using (19,20). See Appendix B 2 c. If we substitute D F ( ± θ , x ), (15) in (50), wesimilarly find F ( x ) = e i k · x F , ( κ −→ κ ). rkalian fields: ray transforms and mini-twistors B. Smith’ s method
Smith’ s formula can be written in various ways. We shall follow Ref. 33, p. 24 (correcting misprints there)and Ref. 62. We extend the X-ray transform (12) as a function g F ( α , x ) = Z ∞−∞ F ( x + t α ) dt = 1 α X F ( θ , x ) , (51)homogeneous of degree −
1, using a non-unit vector α = α θ , α = | α | , s = αt . Then its Fourier transform is G F ( β , x ) = F [ g F ( α , x )]( β , x ) = 1(2 π ) / β Z S θ D F ( θ , x ) h ( θ · b ) d Ω θ . (52)See Appendix B 3. Here h ( p ) = Z ∞ α = −∞ | α | e − iαp dα, ( p = θ · b ) (53)where h ( ap ) = (1 /a ) h ( p ), a > h ( − p ) = h ( p ).Thus, we find G F ( β , x ) = 1(2 π ) / β Z F R ( p, b ) h ( p − b · x ) dp, (54)using (44). This leads to Smith’ s formula G F ( β , x ) = 1(2 π ) / β [ H ∂ p F R ( p, b )]( b · x , b ) , (55)for vector fields, using the identity Z F R ( p, b ) h ( p − p ) dp = 2 π [ H ∂ p F R ( p, b )]( p , b ) , ( p = b · x ) . (56)Here H is the Hilbert transform defined (on R ) by the principal-value integral H [ r ( y )]( x ) = 1 π Z r ( y ) x − y dy. (57)We can prove this identity noting h ( p − p ) = 2 ∂ p [1 / ( p − p )]. We also have[ H ∂ y r ( y )]( x ) = 1 π Z ∂ y r ( y ) x − y dy = − π Z r ( y )( x − y ) dy. (58)If we invert (52) using (55) and (51), we find X F ( θ , x ) = α F − [ G F ( β , x )]( α , x ) = 14 π Z S b [ H ∂ p F R ( p, b )]( b · x , b ) δ ( b · θ ) d Ω b , (59)using β = β b ⇒ d β = β dβd Ω b , α = α θ and [ H ∂ p F R ( p, b )]( b · x , b )] is even under b −→ − b . Hence, the X-raytransform is in the form of a Minkowski-Funk transform: X F ( θ , x ) = [1 / (2 π / )] U { [ H ∂ p F R ( p, b )]( b · x , b ) } ( θ , x ),(25) of H ∂ p F R ( p, b ). rkalian fields: ray transforms and mini-twistors G F ( β , x ), (54) satisfies: ∇ x × G F ( β , x ) − ν G F ( β , x ) = 0, [equivalently from (52, 51) and(32)]. The equation (59) reduces to X F ( θ , x ) = − π ν Z S b b × [ H F R ( p, b )]( b · x , b ) δ ( b · θ ) d Ω b , (60)using (4). We find[ H ∂ p F R λ ( p, b )]( p , b ) = − ν b × [ H F R λ ( p, b )]( p , b ) = λν F R λ ( p , b ) , ( p = b · x ) (61)using the Moses basis, (5). Thus (59), (60) reduce to (17), ( b −→ κ ) which can be directly proven as in Proposition 1in Section II. This can be inverted, for example using (26). The simplification here is basically due to the eigenvalueequation and the Hilbert transform of the derivative for the Radon transform of Trkalian fields in the Moses basis.Smith’ s inversion method makes use of the intermediate function K ( ω, β ) = Z F [ F ( x )]( τ β ) | τ | e iωτ dτ = 1(2 π ) / Z F R ( s, β ) h ( s − ω ) ds, (62) G F ( β , x ) = K ( β · x , β ), (see also Ref. 59). Here we have used the Fourier slice theorem: F [ F R ( s, β )]( τ, β ) =2 π F [ F ( x )]( τ β ). This leads to K ( ω, β ) = [1 / (2 π ) / ](1 /β )[ H ∂ p F R ( p, b )]( ω/β, b ) using (56) and K λ ( ω, β ) =[1 / (2 π ) / ](1 /β ) λν F R λ ( ω/β, b ) using (61), [or appropriately using (6) in (62)] for a Trkalian field. Hence K λ ( β · x , β ) = [1 / (2 π ) / ](1 /β ) λν F R λ ( b · x , b ). Then the inversion formula F λ ( x ) = 1(2 π ) / Z Z K λ ( ω, β ) e − i ( ω − β · x ) dωd β, (63)can be expressed in terms of the Radon transform of the field. This reduces to (6). We shall not discuss itstomographical implementation. C. Tuy’ s method
We shall follow Ref. 59 for Tuy’ s approach. We extend the Divergent beam transform (13) as a function g F ( α , x ) = Z ∞ F ( x + t α ) dt = 1 α D F ( θ , x ) , (64)homogeneous of degree −
1, using a non-unit vector α = α θ , α = | α | , s = αt . Then its Fourier transform is G F ( β , x ) = F [ g F ( α , x )]( β , x ) = 1(2 π ) / β Z S θ D F ( θ , x ) f ( θ · b ) d Ω θ , (65)See Appendix B 4. Here f ( p ) = 2 πi∂ p δ − ( p ), ( p = θ · b ) where f ( ap ) = (1 /a ) f ( p ), a > f ( − p ) = − πi∂ p δ + ( p ).Thus we find G F ( β , x ) = 1(2 π ) / β Z F R ( p, b ) f ( p − b · x ) dp, (66)using (44). This leads to G F ( β , x ) = π (2 π ) / β n (cid:2) H ∂ p F R ( p, b ) (cid:3) ( b · x , b ) − i (cid:2) ∂ p F R ( p, b ) (cid:3) ( b · x , b ) o , (67)using the identity rkalian fields: ray transforms and mini-twistors Z F R ( p, b ) f ( p − p ) dp, = π n [( H − i ) ∂ p ] F R ( p, b ) o ( p , b ) , ( p = b · x ) . (68)We can prove this identity noting f ( p − p ) = 2 πi∂ p δ − ( p − p ), upon simple manipulations.If we invert (65) using (67) and (64), we find D F ( θ , x ) = α F − [ G F ( β , x )]( α , x ) = π (2 π ) Z S b n [( H − i ) ∂ p ] F R ( p, b ) o ( b · x , b ) δ + ( b · θ ) d Ω b , (69)using β = β b ⇒ d β = β dβd Ω b , α = α θ and F [ H ( − β ′ )]( p ) = √ πδ + ( p ), β ′ = αβ . Hence, the Divergentbeam transform is in the form of the integral transform: D F ( θ , x ) = [1 / (4 π / )] A (cid:8) { [( H − i ) ∂ p ] F R ( p, b ) } ( b · x , b ) (cid:9) ( θ , x ), (25) of [( H − i ) ∂ p ] F R ( p, b ). An easy check reveals (cid:2) ∂ p F R ( p, b ) (cid:3) ( b · x , b ) is odd under b −→ − b while (cid:2) H ∂ p F R ( p, b ) (cid:3) ( b · x , b ) is even. Therefore only the even terms survive the integration. Thus we find D F ( θ , x ) = 12 X F ( θ , x ) + 12(2 π ) Z S b (cid:2) ∂ p F R ( p, b ) (cid:3) ( b · x , b ) 1 b · θ d Ω b , (70)using (59). The second term is associated with the difference Y F ( θ , x ) = 1(2 π ) Z S b (cid:2) ∂ p F R ( p, b ) (cid:3) ( b · x , b ) 1 b · θ d Ω b , (71)(21).For a Trkalian field (3), G F ( β , x ), (66) satisfies: ∇ x × G F ( β , x ) − ν G F ( β , x ) = 0, [equivalently from (65, 64) and(32)]. The equation (70) reduces to D F ( θ , x ) = 12 X F ( θ , x ) − π ) ν Z S b b × F R ( b · x , b ) 1 b · θ d Ω b , (72)using (4). This leads to Y F ( θ , x ) = − π ) ν Z S b b × F R ( b · x , b ) 1 b · θ d Ω b . (73)The equations (70, 72) reduce to (19), ( b −→ κ ) using the Moses basis (5), [also (17)] which can be directly provenas in Proposition 2 in Section II. Similarly (71, 73) reduce to (22). Note { [( H − i ) ∂ p ] F R λ ( p, b ) } ( p , b ) = − ν b × [( H − i ) F R λ ( p, b )]( p , b ) (74)= 2(2 π ) / (1 /g )(1 /λν ) e iλνp Q λ ( b ) s λ ( λν b ) , ( p = b · x )in the Moses basis. The simplification here is also due to the eigenvalue equation and the Hilbert transform of thederivative for the Radon transform of Trkalian fields in the Moses basis.It is straightforward to find the difference for Lundquist field ( λ = 1) Y F L ( λ =1) ( θ , x ) = − F ν v r ( ∞ X k =1 sin[2 k ( θ − φ )] J k ( νr ) e r ( θ ) − J ( νr ) e θ (75)+ 2 ∞ X k =0 cos[(2 k + 1)( θ − φ )] J k +1 ( νr ) e z ) , substituting the Radon transform (9) into (71). See Appendix B 5. Then, we find the Divergent beam transform rkalian fields: ray transforms and mini-twistors D F L ( λ =1) ( θ , x ) = F ν v r ( − ∞ X n =1 ( − n sin[ n ( θ − φ )] J n ( νr ) e r ( θ ) + J ( νr ) e θ (76)+ " J ( νr ) + 2 ∞ X n =1 ( − n cos[ n ( θ − φ )] J n ( νr ) e z ) , of the Lundquist field using (B33) in (16) and then in (70). See p. 23 and p. 538 in Ref. 63 for these series.Tuy’ s inversion method makes use of the intermediate function K ( ω, β ) = 12 Z F [ F ( x )]( τ β )( | τ | + τ ) e iωτ dτ = 1(2 π ) / Z F R ( s, β ) f ( s − ω ) ds, (77) G F ( β , x ) = K ( β · x , β ). This yields K ( ω, β ) = [1 / (2 π ) / ](1 /β ) I ( ω/β, b ) where I ( p, b ) = π (cid:2) ( H − i ) ∂ q F R ( q, b ) (cid:3) ( p, b ), [see (67)]. For a Trkalian field in the Moses basis G F ( β , x ) = K ( β · x , β ) = [1 / (2 π ) / ](1 /β ) I ( b · x , b ) =[ π/ (2 π ) / ](1 /β ) { [( H − i ) ∂ p ] F R ( p, b ) } ( b · x , b ) = (1 /g )(1 /λν )(1 /β ) e iλν b · x Q λ ( b ) s λ ( λν b ). The inversion formula F ( x ) = − π ) i Z S b Z I ( p, b ) δ ′ ( p − b · x ) dpd Ω b , (78)can be expressed in terms of the spherical Curl transform of the field. This reduces to (6). We shall not discuss itstomographical implementation. D. Gelfand-Goncharov’ s method
If we use h ( p ) = 1 /p , p = θ · b , then (44) leads to (cid:2) H ∂ p F R ( p, b ) (cid:3) ( b · x , b ) = − π Z S θ D F ( θ , x )( θ · b ) d Ω θ , (79)see (58). This can also be inferred from (52, 55) and (B14).For a Trkalian field (4), this yields b × (cid:2) H F R ( p, b ) (cid:3) ( b · x , b ) = 1 π ν Z S θ D F ( θ , x )( θ · b ) d Ω θ . (80)This reduces to F R λ ( b · x , b ) = − π λν Z S θ D F λ ( θ , x )( θ · b ) d Ω θ , (81)using the Moses basis, (61). The simplification here is again due to the eigenvalue equation and the Hilbert transformof the derivative for the Radon transform of Trkalian fields in the Moses basis.We can write similar formulas with X F ( θ , x ) using the substitution θ −→ − θ through the equations (79, 80, 81).This leads to Semyanistyi’ s inversion formula (26) in Section II.If we substitute the equation (81), ( b −→ κ ) in (6), we find Z S θ D F λ ( θ , x ) d Ω θ = 2 π λν F λ ( x ) , (82)which can be directly proved using the Riesz potential as in Proposition 9 in Section III. See Appendix B 6. We shallnot discuss its tomographical implementation. rkalian fields: ray transforms and mini-twistors F R λ ( b · x , b ) = − ν e ik κ · x I ( b , κ ) F , (83)where the integral I ( b , κ ) is given in (28), ( κ −→ b , κ ′ −→ κ ). Then we find F R λ ( b · x , b ), (7). Similarly, we areled to (5) substituting (19) in (81).If we substitute (76) in (82) we are led to (8), ( λ = 1). V. RIESZ POTENTIAL AND BIOT-SAVART INTEGRALS
We can write the X-ray transform of Riesz potential (of order 2) and Biot-Savart ( BS ) integrals using (59) in termsof their Radon transform.If we replace F −→ I [ F ] = 1 / (8 π ) R † R [ F ] ⇒ F R −→ R { I [ F ] } in (59) using (10), we find X I [ F ]( θ , x ) = 18 π X R † R [ F ]( θ , x ) = 14 π i Z S κ (cid:8) HF − { k F [ F R ( q, κ )]( k, κ ) } ( p, κ ) (cid:9) ( κ · x , κ ) δ ( κ · θ ) d Ω κ . (84)For Trkalian fields I [ F ] = 1 / (8 π ) R † R [ F ] = (1 /ν ) F , (Proposition 8) and hence X I [ F ]( θ , x ) = 18 π X R † R [ F ]( θ , x ) = 1 ν X F ( θ , x ) . (85)We can easily verify this substituting (5) in (84) and comparing with (17).The equation (11) yields ∂ p RBS [ F R ( q, κ )]( p, κ ) = − κ × F R ( p, κ ). If we replace F −→ BS [ F ] = ∇ × I [ F ] ⇒ F R −→ RBS [ F R ] in (59) using this, we find X BS [ F ] ( θ , x ) = − π Z S κ κ × (cid:2) H F R ( p, κ ) (cid:3) ( κ · x , κ ) δ ( κ · θ ) d Ω κ . (86)Thus X BS [ F ] = (1 / π ) M (cid:8) H ∂ p RBS (cid:2) F R ( q, κ ) (cid:3) ( p, κ ) (cid:9) = − (1 / π ) M (cid:8) κ × (cid:2) H F R ( p, κ ) (cid:3) (cid:9) . This is again in theform of a Minkowski-Funk transform. We call this John-Biot-Savart integral. Further, one can express F R in termsof X F . The equation (86) also follows from (84):
X BS [ F ] = X { ∇ × I [ F ] } = ∇ × X I [ F ], (Proposition3).For Trkalian fields (4), we find X BS [ F ] ( θ , x ) = 14 π ν Z S κ (cid:2) H ∂ p F R ( p, κ ) (cid:3) ( κ · x , κ ) δ ( κ · θ ) d Ω κ = 1 ν X F ( θ , x ) , (87)(59) as we expect, since: BS [ F ] = (1 /ν ) F .We can write DBS [ F ]( θ , x ) and YBS [ F ]( θ , x ) integrals respectively using (69, 70) and (71), with a similarreasoning. For example, we find YBS [ F ] ( θ , x ) = − π ) Z S κ κ × F R ( κ · x , κ ) 1 κ · θ d Ω κ . (88)We have YBS [ F ] ( θ , x ) = (1 /ν ) Y F ( θ , x ), (73) for Trkalian fields.These, together with the Radon transform, lead to an integral geometric understanding of these integrals. However,a physical or tomographical discussion of these integrals is beyond the scope of this manuscript. rkalian fields: ray transforms and mini-twistors VI. MINI-TWISTOR REPRESENTATION
The mini-twistor space as an intrinsic structure have been introduced by Hitchin. The Twistor theory, in simplestterms, is based on writing contour integral solutions for the wave equation in (3 + 1) dimensional Minkowski space,using holomorphic functions.
We can write the solution of Helmholtz equation as a time-harmonic reductionof this, using mini-twistor space variables. See Appendix C 1. The quotient of twistor space ( CP \ CP ) of theMinkowski space by the action of time translation yields the mini-twistor space TCP . The (mini-)twistor space of R is the space TS = (cid:8) ( u , v ) ∈ S × R , u , v ∈ R , | u | = 1 , v · u = 0 (cid:9) ⊂ S × R oforiented lines l : p = v + t u where u is the direction vector, v is the position (shortest) vector of l and p denotes a pointof this line. This has a natural complex structure which can be identified with the holomorphic tangent bundle
TCP of projective line: the Riemann sphere CP , with local coordinates ( η, ω ), [on S − { N } , u = (0 , , Here u ∈ S ∼ CP , ω is the coordinate on the base CP and η denotes the fiber coordinate which decribes a holomorphicsection.We can regard a point p in R as the intersection of all oriented straight lines through it which are parametrisedby a 2-sphere S p in TS ∼ TCP . More precisely, each point p corresponds to a holomorphic section of TCP . These are fixed by an involutive map τ on TS , τ = 1 reversing the orientation of lines which is called the real structure. The incidence relation between a point p ( x, y, z ) and a twistor ( η, ω ) that defines this section is givenby η = (1 / (cid:2) ( x + iy ) + 2 zω − ( x − iy ) ω (cid:3) . The set of twistors incident with a given point (the set of lines passingthrough this point) form a copy of CP which lies as a real section of TCP . If we hold ( η, ω ) fixed, then( x, y, z ) satisfying the incidence relation defines a line in R . If we hold ( x, y, z ) fixed, then ( η, ω ) satisfying theincidence relation parametrises the set of all lines through the point p ( x, y, z ). The local coordinates on S − { S } ,[ u = (0 , , − ω ′ = 1 /ω , η ′ = − η/ω = (1 / (cid:2) ( x − iy ) − zω ′ − ( x + iy ) ω ′ (cid:3) . We shall ignorethe factor 1 / η .We shall restrict a function defined on a domain of the mini-twistor space to a (projective) line and then integratealong a closed contour contained in this CP . We shall use (mini-)twistor solution of the Helmholtz equation for finding Trkalian fields. A Trkalian field (1) alsosatisfies the vector Helmholtz equation ∇ F ( x ) = − k F ( x ) , k = ν (89)but the converse is not necessarily true. We can write a solution of this equation as F ( x ) = [ A ( x ) , B ( x ) , C ( x )] = Z C e − ikf ( L, M, N ) dω, (90)where L ( η x ( ω ) , ω ), M ( η x ( ω ) , ω ), N ( η x ( ω ) , ω ) are holomorphic functions of η x ( ω ) = x + iy + 2 zω − ( x − iy ) ω and ω .Here f = ω ( x − iy ) − z is, for example chosen as the spatial part of integrating factor (C5, C8) for the time-harmonicitycondition (C4).If we substitute (90) in (1), we find i (cid:0) ω (cid:1) N η − ωM η = k ( L + iM + ωN ) , ωL η − (cid:0) − ω (cid:1) N η = − ik ( L + iM + ωN ) , (91) (cid:0) − ω (cid:1) M η − i (cid:0) ω (cid:1) L η = − k [ ω ( L − iM ) − N ] . We shall avoid further considerations such as modifying this equation by introducing arbitrary holomorphic functionsor modifying the contour C which calls for sheaf cohomology here. The first and second equations in (91) yields − ω (cid:2)(cid:0) − ω (cid:1) M η − i (cid:0) ω (cid:1) L η (cid:3) = 2 k ( L + iM + ωN ). This leadsto (cid:0) − ω (cid:1) L + i (cid:0) ω (cid:1) M + 2 ωN = 0 , (92)that is L + iM + 2 N ω − ( L − iM ) ω = 0 using the third equation. If we substitute N , (92) in (91), we find − i (cid:0) ω (cid:1) L + (cid:0) − ω (cid:1) M = 0 . (93) rkalian fields: ray transforms and mini-twistors L ( η, ω ) = (cid:0) − w (cid:1) u ( η, ω ), M ( η, ω ) = i (cid:0) w (cid:1) u ( η, ω ), N ( η, ω ) = 2 wu ( η, ω ) where u ( η, ω ) is an arbitrary holomorphic function (except some poles) of η and ω .Thus a Trkalian field (1) is given by F ( x ) = [ A ( x ) , B ( x ) , C ( x )] = Z C (cid:2)(cid:0) − ω (cid:1) , i (cid:0) ω (cid:1) , ω (cid:3) e − ikf u ( η, ω ) dω. (94)Note (cid:2)(cid:0) − ω (cid:1) , i (cid:0) ω (cid:1) , ω (cid:3) is a null vector in C .This solution is in the form of twistor solution to Maxwell equations in (3 + 1) dimensions, as we expect (sinceTrkalian fields correspond to the spatial part of time-harmonic electromagnetic fields with no source). We can alsoderive this solution from the twistor solution of Maxwell equations (see Ref. 35, p. 33, pp. 206-207), using a similartime-harmonic reduction (with minor changes of conventions).For example, we choose u ( η, ω ) = g ( η x ( ω )) /h ( ω ) with h ( ω ) = ( ω − ω ) m . If g ( η x ( ω )) = η n x ( ω ) where n is positive, m = 1 and ω = 0, that is u ( η, ω ) = η n x ( ω ) /ω , we find F ( x ) = 2 πie iνz ζ n (1 , i, , ζ = x + iy. (95)If ω = 0, that is u ( η, ω ) = η n x ( ω ) / ( ω − ω ), we find F ( x ) = 2 πie − iν [ ω ( x − iy ) − z ] η n x ( ω ) (cid:0) (1 − ω ) , i (1 + ω ) , ω (cid:1) . (96)Hence for a holomorphic function g ( η ) = P ∞ n =0 a n η n and ω = 0 that is u ( η, ω ) = g ( η x ( ω )) /ω we find F ( x ) = 2 πie iνz g ( ζ )(1 , i, . (97)The orthogonality of real contact structures arising in case g ( ζ ) is given by a derivative: g −→ g ′ is discussed in Ref.6. If we choose u ( η, ω ) = g ( η x ( ω )) /ω , we find F ( x ) = 2 πie iνz { (cid:2) − iνζg ( ζ ) + 2 zg ′ ( ζ ) (cid:3) (1 , i,
0) + 2 g ( ζ )(0 , , } . (98)If we choose u ( η, ω ) = (1 /ω ) e − i ( ν/ ω − η , [ − (1 / ω − η = f − g , g = x ( ω + w − ) / − iy ( ω − w − ) /
2] where x = ( x, y, z ) = ( r cos ϕ, r sin ϕ, z ) in cylindrical coordinates and ω = e iθ , C is a circle of unit radius about the origin,then we find the Lundquist solution (8) with F = 4 πi and λ = 1, using the integrals (C14) in Appendix C 2. A. Arbitrary integrating factor
We can choose different integrating factors for the time-harmonicity condition (C4). In fact, we do not have tochoose an integrating factor initially. We can see this in a time-harmonic extension of the solution above. Theintegrating factor in (C5) yields a time-harmonic extension of Trkalian fields: F −→ e − ikt F . This satisfies (vector)wave equation which reduces to (89).If we use an arbitrary integrating factor g ( p, q, ω ) = e − ik ˜ f ( p,q,ω ) h ( p, q, ω ) = e − ik ˜ f ( p,q,ω ) H ( η x ( ω ) , ω ) , (99)in (C4), we find ω ∂ ˜ f∂p + ∂ ˜ f∂q = 1 , (100)that is ∂ ˜ f /∂t = 1 as a condition on the integrating factor. Then the field (90), (with f −→ ˜ f containing bothtemporal and spatial pieces) satisfies the wave equation which reduces to (89) upon imposing the condition (100). Ifwe substitute this field in (1), we find rkalian fields: ray transforms and mini-twistors i (cid:0) ω (cid:1) N η − ωM η = kL − k ( iωM + N ) ˜ f p + k ( iM + ωN ) ˜ f q , ωL η − (cid:0) − ω (cid:1) N η = kM + ik ( ωL − N ) ˜ f p − ik ( L + ωN ) ˜ f q , (101) (cid:0) − ω (cid:1) M η − i (cid:0) ω (cid:1) L η = kN + k ( L + iM ) ˜ f p − kω ( L − iM ) ˜ f q . This reduces to (91) for ˜ f = q . The equations (101) lead to the same equations (92, 93) using a similar reasoning.These yield the solution (94), [ f −→ ˜ f satisfying (100)] with a harmonic time dependence now.Thus the solution is of the same form containing the spatial part of the chosen integrating factor.Any solution of the time-harmonicity condition (C4) can be written using ˜ f = (1 / p/ω + q ), (C9), see (C11).This (excluding the temporal piece) leads to the solution (94) with f = (1 / ω ( x − iy ) + ( x + iy ) /ω ], see (C10).In this case, the Lundquist solution is simply given by u ( η, ω ) = 1 /ω , ( ω = e iθ , C : unit circle about the origin).If we use u = h ( ω ′ ) which has a Laurent series: h ( ω ′ ) = 1 /ω ′ ( n +1) with ω = iω ′ , ( k = ν ), we find F ( x ) = 4 πie − imϕ (cid:20) im νr J m ( νr ) e r + J ′ m ( νr ) e ϕ − J m ( νr ) e z (cid:21) , m = n − This is a circular cylindrical CK solution with no z dependence,upto conventions. B. Chandrasekhar-Kendall type solutions: Debye potentials
We can use the solution φ ( x, y, z ) = Z C e − iσf H ( η x ( ω ) , ω ) dω, f = 12 [ ω ( x − iy ) + 1 ω ( x + iy )] (103)of the scalar Helmholtz equation ∇ φ = − k φ , ( k = σ ) as Debye potential for CK type Trkalian fields F ( x ) = − [ σ ∇ × ( φ w ) + ∇ × ∇ × ( φ w )] , (104)where ω is a fixed vector and ∇ × F ( x ) − σ F ( x ) = 0. It is straightforward to write a time-harmonic extension ofthe CK solution.The potential for the circular cylindrical CK solution is effectively (apart from z coordinate) a 2 dimensionalsolution (satisfying the Helmholtz equation: ∇ φ + ν φ = 0 in 2 dimensions). We can express this as φ ( r, ϕ, z ) = e − ikz Z C e − iνf Hdω = 2 πi i m J m ( νr ) e imϕ − ikz , σ = ν + k (105)where H = ω m − , ω = e iθ , C : unit circle about the origin and w = e z . Here we use (C12, C13). This reduces to φ ( r, ϕ, z ) = Z C e − iνf Hdω = 2 πiJ ( νr ) , (106)the potential for the Lundquist solution, for m = 0, k = 0, ( σ = ν ), H = ω − .An interesting case is the class of axially symmetric potentials. If we assume axial symmetry about z -azis, then arotation in xy -plane is given by ω −→ e iψ ω (treated as a spinor coordinate) which induces the rotation x + iy −→ e iψ ( x + iy ) (see Refs. 46 and also 45, 65) and η x ( ω ) −→ e iψ η x ( ω ), dω −→ e iψ dω while f = (1 / ω ( x − iy ) + ( x + iy ) /ω ] −→ f . Hence we consider fields of the form φ ( x, y, z ) = Z C e − iσf G (cid:18) η x ( ω ) ω (cid:19) ω dω, (107) rkalian fields: ray transforms and mini-twistors G . We assume that G has a Laurent series about the origin: G = [ η x ( ω ) /ω ] n , then φ ( x, y, z ) = Z C e − iσf (cid:20) η x ( ω ) ω (cid:21) n ω dω. (108)For n = 0 ( σ = ν ), this reduces to the potential (106).We can use ω = − ie iu for parametrizing (108). This yields φ ( x, y, z ) = 2 n i Z C e − iσf ( z + ix cos u + iy sin u ) n du, (109)where f = x sin u − y cos u , with η/ω = 2( z + ix cos u + iy sin u ). For n = 1, we find φ = 2 iR Z u =2 πu =0 e − iσR sin θ sin( u − ϕ ) [cos θ + i sin θ cos( u − ϕ )] du (110)= 2 iR Z u =2 πu =0 e − iσR sin θ sin u (cos θ + i sin θ cos u ) du, using spherical coordinates: x = R sin θ cos ϕ , y = R sin θ sin ϕ , z = R cos θ . The integral of the second term vanishesand the first term yields φ = 4 πizJ ( σr ) , r = R sin θ, (111)in cylindrical coordinates, using (C15): ( β = σr ). The potential (111), with ω = e z , leads to F ( x ) = − πiσ n − σ J ( σr ) e r + z [ J ( σr ) e ϕ + J ( σr ) e z ] o , (112)a generalization of the Lundquist field (8), ( σ = ν ).In case n = −
1, we use f = ω ( x − iy ) − z for the sake of simplicity. The denominator in (108) can be factored as η x ( ω ) = − ( x − iy )( ω − ω )( ω − ω ) where ω = ( z −| x | ) / ( x − iy ) = − e iϕ tan( θ/ ω = ( z + | x | ) / ( x − iy ) = e iϕ cot( θ/ z . . If z > | ω | <
1, then the single residue inside the unit circle C yields φ = 12 e iσ | x | | x | . (113)This is the fundamental solution of the scalar Helmholtz equation: ∇ φ + σ φ = − πδ ( x ). It is beyond the scope ofthis manuscript to provide a complete treatment of this case. The classical spheromak equilibrium solution (see also Ref. 5 and the references therein) is given by F = F n j ( kR ) kR cos θ e R + 1 kR [ j ( kR ) − sin( kR )] sin θ e θ + j ( kR ) sin θ e φ o , (114)in spherical coordinates, where j ( kR ) is the spherical Bessel function, w = R , (with the conventions of Ref. 72) and σ = k , (104). We need to consider an expression of the form (103) which reduces to φ = − i F k Z π e − ikR cos θ cos α J ( kR sin θ sin α ) P (cos α ) sin αdα = − F k j ( kR ) P (cos θ ) . (115)An integral of this type was recently reconsidered by various authors, refering to Refs. 63, 78, and 79 andincluding alternative proofs. To the knowledge of the author, the simplest derivation of this integral expression isgiven in Ref. 73, p. 411 (with a misprint of coefficient). rkalian fields: ray transforms and mini-twistors VII. CONCLUSION
We have studied the X-ray and Divergent beam transforms of Trkalian fields in connection with their Radontransform. We remind that the Radon transform of a Trkalian field is defined on a sphere in the transform space andit satisfies a corresponding eigenvalue equation there. The mathematical methods of tomography (respectively Smith’s and Tuy’ s methods) show that these transforms are basically in the form of a Minkowski-Funk and a closely relatedintegral transform of certain intricate quantities (Hilbert transform of the derivative of Radon transform). The Moseseigenbasis is especially efficient in exhibiting this connection for Trkalian fields. These naturally reduce to well knowngeometric integral transforms on a sphere of the Radon or the spherical Curl transform.More precisely, the X-ray transform of a Trkalian field is given by the Minkowski-Funk transform of its Radontransform on this sphere. In R , this corresponds to the integral of the Radon transform of the field over a pencilof planes intersecting at a line. Previously, this transform was introduced by Gonzalez , called the plane-to-linetransform, as an elementary geometric transform in integral geometry. This transform naturally arises for Trkalianfields. We refer the interested reader to Ref. 43 for a twistor approach to this transform.Meanwhile the Divergent beam transform is given by another closely related (an extension of Minkowski-Funk)geometric integral transform of the spherical Curl transform of the field on the sphere. This also provides an extensionover the plane-to-line transform. This seems a natural extension from the point of view of generalized functions.We remark that these transforms are naturally defined on the sphere in the transform space. Geometrically, we areendowed with a simple picture showing the interrelations of these transforms for Trkalian fields on this sphere. Thisis made possible with the Moses basis.Intuitively speaking, the X-ray or Divergent beam transform of a Trkalian field respectively integrates the field ona whole line or a half-line which is determined by a direction vector. This direction vector determines a great circleon the sphere in the transform space. Then these transforms are given by integrals of the Radon transform on thisgreat circle. The X-ray: whole-line transform is given by a whole (in the distributional sense: Dirac delta) integral.This leads to the plane-to-line picture in R . Meanwhile, the Divergent beam: half-line transform is given by a half(in the distributional sense: Heisenberg delta) integral, depending on the orientation. The picture in R for this isleft to the imagination of the reader.We can logically derive these results starting from the fundamental relation (44) of mathematical tomography in aunified manner.However, we have postponed the mathematical discussion originating from tomography until the basic investigationof X-ray and Divergent beam transforms of Trkalian fields had finished. This gave us the opportunity to consider themathematical basis for tomographical studies of Trkalian field models in nature, for its own sake.These transforms are members (via analytic continuation) of a well known analytic family of integral operatorswhich arise in the study of Fourier transforms of homogeneous functions. Recently, these integral operators have beenstudied by Rubin. We have inverted the X-ray transform of a Trkalian field using Semyanistyi’ s formula which also belongs to this family, so as to yield its Radon transform. This leads to an inversion through the sphericalmean of the X-ray transform of Trkalian fields.The X-ray (also the Divergent beam) transform and its inversion intertwine the Curl operator and also the Diver-gence, Gradient and Laplacian operators. Thus the X-ray (Divergent beam) transform of a Trkalian field is Trkalian.Also, the Trkalian subclass of X-ray transforms X F yields Trkalian fields in the physical space. We have also writtenthe John’ s equation for Trkalian fields in an equivalent form. Thus, we can study Trkalian fields either in physicalspace or in the transform space.Another crucial quantity in integral geometry is the Riesz potential. The Riesz potential, of order α where 0 <α <
3, of a Trkalian field is proportional to the field. Hence, the spherical mean of the X-ray (or Divergent beam)transform of a Trkalian field over all lines passing through a point yields the field at this point. This provides a newsimple inversion formula for the X-ray (or Divergent beam) transform of Trkalian fields. This result is also logicallyimplied by Gelfand-Gonchorav’ s mathematical approach (making use of the same intricate quantity) to tomography.Then we have returned back to the mathematical methods of tomography. These methods provide us elegantmathematical tools for investigating the interrelations of these integral transforms in a unified view. First, theseendowed us with an integral geometric view and motivation for the discussions above. Second, these enabled us todiscuss these mathematical methods with a view towards tomographical studies of Trkalian field models in nature. Forthis purpose, we have studied these mathematical methods using Trkalian fields. We have adopted a mathematicalapproach rather than a tomographical implementation.We have made use of four basic mathematical approaches of tomography due to Grangeat, Smith, Tuy andGelfand-Goncharov . These methods are outflow of the fundamental relation of mathematical tomography. Theyare based on relations of the X-ray and Divergent beam transforms to the intricate quantities mentioned above: theHilbert transform of the derivative of Radon transform. rkalian fields: ray transforms and mini-twistors R¨ontgen of these clouds.Meanwhile, the Gelfand-Goncharov approach leads to a direct inversion through the spherical mean that is men-tioned above. This naturally makes use of the inverse transform that belongs to the above family of integral operators.The simplification is again based on the same intricate quantity: Hilbert transform of the derivative of Radon trans-form in the Moses basis.Briefly, the intricate quantity: Hilbert transform of the derivative of Radon transform which intrigued tomographysimplifies for a Trkalian field in the Moses basis.The direct inversion formulas arising in Grangeat’ s and Gelfand-Gonchorav’ s approaches mathematically seemmore feasible than the inversions in Smith’ s and Tuy’ s methods.These approaches provide different inversion formulas which may serve useful for designing reconstruction methodsin tomographical studies of Trkalian field models in nature, depending on real physical situation. The author expectsthat the Moses basis which has led to a drastical simplification in the crucial quantity may also be of practical use intomographical studies. We shall not discuss tomographical implementations of these inversion formulas.Furthermore, the Smith and Tuy methods mathematically enable us to define the X-ray and Divergent beamtransforms of the Riesz potential (of order 2) and Biot-Savart integrals. The X-ray transform of the Biot-Savartintegral of a Trkalian field reduces to the X-ray transform of the field. The Radon, X-ray and Divergent beamtransforms of the Riesz potential (of order 2) and Biot-Savart integrals lead to an integral geometric understandingof these integrals. However, a physical or tomographical discussion of these integrals is beyond the scope of thismanuscript.In the second part of this manuscript we have discussed Trkalian fields using (mini-)twistors. The X-ray transformis a real analogue and a predecessor of Twistor theory.
The X-ray transform and the mini-twistors are bothdefined on the space TS ∼ TCP of oriented lines in R .We have discussed a mini-twistor representation, presenting a mini-twistor solution for the Trkalian fields equation.This is based on twistor solution of the (vector) Helmholtz equation which makes use of a time-harmonic reductionof the wave equation. A Trkalian field is given in terms of a null vector in C with an arbitrary holomorphic functionof two variables and an exponential factor that results from the reduction.The exponential factor contains the spatial part of an integrating factor for the time-harmonicity condition. Thesolution is of the same form containing the spatial part of any choosen integrating factor. We have also used thegeneral solution of this condition for writing our solution.This solution can also be derived using a time-harmonic reduction of the twistor solution for electromagnetic fieldsin (3 + 1) dimensions.We are led to a time-harmonic extension of Trkalian fields, implicitly keeping this condition. This can be interpretedas a time-harmonic electromagnetic field.We have also presented examples of Debye potentials for CK type solutions using mini-twistors.This manuscript is aimed at studying the most basic integral geometric aspects and the mini-twistor representationof Trkalian fields. We have made use of the mathematical methods of tomography (but not the tomography). Thesemay serve useful for studying their physical properties in a realistic environment.The Trkalian class of fields may also provide a simple and interesting example for studying the relation of raytransforms with Twistor theory. However, twistor tomography is beyond the limitations of this manuscript. rkalian fields: ray transforms and mini-twistors APPENDIX A: RAY TRANSFORMS1. X-ray transform of Lundquist Field
We write x = r e r ( φ ) + z e z , r > θ = v r e r ( θ ) + v z e z , v r > x ′ = x + s θ = r ′ e r ( φ ′ ) + z ′ e z where r ′ cos φ ′ = r cos φ + sv r cos θ , r ′ sin φ ′ = r sin φ + sv r sin θ , r ′ = r + s v r + 2 rsv r cos( θ − φ ), z ′ = z + sv z and e r ( φ ′ ) = cos φ ′ e x + sin φ ′ e y , e φ ′ = − sin φ ′ e x + cos φ ′ e y (A1)= 1 r ′ [ r e r ( φ ) + sv r e r ( θ )] = 1 r ′ [ r e φ + sv r e θ ] . Hence F L ( x + s θ ) = F [ λJ ( λνr ′ ) e φ ′ + J ( λνr ′ ) e z ] = F (cid:20) λr r ′ J ( λνr ′ ) e φ + λv r sr ′ J ( λνr ′ ) e θ + J ( λνr ′ ) e z (cid:21) , (A2)and we have X F L ( θ , x ) = F (cid:20) λr Z ∞−∞ r ′ J ( λνr ′ ) ds e φ + λv r Z ∞−∞ sr ′ J ( λνr ′ ) ds e θ + Z ∞−∞ J ( λνr ′ ) ds e z (cid:21) , (A3)where r ′ ( s ). We define a new variable: t = v r s + r cos( θ − φ ) ⇒ ds = (1 /v r ) dt , r ′ = √ t + u , u = r sin( θ − φ ). Then X F L ( θ , x ) = F v r ( λr Z ∞−∞ √ t + u J ( λν p t + u ) dt [ e φ − cos( θ − φ ) e θ ] + λ Z ∞−∞ t √ t + u J ( λν p t + u ) dt e θ + Z ∞−∞ J ( λν p t + u ) dt e z ) (A4)= 2 F v r ( λr Z ∞ √ t + u J ( λν p t + u ) dt [ e φ − cos( θ − φ ) e θ ] + Z ∞ J ( λν p t + u ) dt e z ) . We introduce another variable: sinh y = t/ | u | ⇒ dt = | u | cosh ydy . This reduces to X F L ( θ , x ) = 2 F v r ( λu Z ∞ J (2 ω cosh y ) dy e r ( θ ) + | u | Z ∞ J (2 ω cosh y ) cosh ydy e z ) , (A5)using e φ − cos( θ − φ ) e θ = sin( θ − φ ) e r ( θ ), where ω = (1 / λν | u | >
0. We can evaluate these integrals using Z ∞ J m + n (2 z cosh x ) cosh[( m − n ) x ] dx = − π J m ( z ) N n ( z ) + J n ( z ) N m ( z )] , z > J n and N m are respectively Bessel functions of the first and second (Neumann)kind. We also need N − / ( z ) = J / ( z ) , N / ( z ) = − J − / ( z ) , (A7) J / ( z ) = (cid:18) πz (cid:19) / sin z, J − / ( z ) = (cid:18) πz (cid:19) / cos z. We respectively find rkalian fields: ray transforms and mini-twistors Z ∞ J (2 ω cosh y ) dy = 12 1 ω sin 2 ω, Z ∞ J (2 ω cosh y ) cosh ydy = 12 1 ω cos 2 ω, (A8)for m = 1 / n = 1 / m = 1 / n = − /
2. This leads to X F L ( θ , x ) = F v r ω [ λu sin 2 ω e r ( θ ) + | u | cos 2 ω e z ] , (A9)that is equation (16) upon rearranging.
2. X-ray transform of a Trkalian Field a. Reduction of the integral
We can simplify (17) decomposing κ into two components which are respectively parallel and orthogonal to θ : κ = κ k + κ ⊥ = u θ + v e r ( φ ). That is κ k = u θ , u = cos θ and κ ⊥ = v e r ( φ ), v = √ − u = sin θ where e r ( φ ) is theunit radial vector parametrized by angle φ in the plane θ ⊥ orthogonal to θ : e r ( φ ) · θ = 0. Then κ · x = [ u θ + v e r ( φ )] · x , δ ( κ · θ ) = δ ( u ) and d Ω κ = sin θdθdφ = − dudφ . Hence we can write (17) as X F λ ( θ , x ) = 14 π λν Z C Z u =1 u = − F Rλ (cid:0) [ u θ + p − u e r ( φ )] · x , u θ + p − u e r ( φ ) (cid:1) δ ( u ) dudφ, (A10)where C is the unit circle in the plane θ ⊥ which corresponds to a great circle on S κ (the intersection of the planewith the sphere). This reduces to X F λ ( θ , x ) = 14 π λν Z C F Rλ (cid:0) e r ( φ ) · x , e r ( φ ) (cid:1) dφ = 1(2 π ) / g λν Z C e iλν e r ( φ ) · x Q λ ( e r ( φ )) s λ ( λν e r ( φ )) dφ, (A11)the integral along the great circle C in S κ determined by θ in the transform space.One can further try writing (A11) in terms of rotation about the axis determined by θ through angle φ in the plane θ ⊥ since Q λ ( e r ( φ )) can be regarded as an eigenfunction of this rotation. b. Fourier slice-projection theorem If we use the Curl expansion for a Trkalian field, we are led to F [ F λ ( x )]( ξ ) = 1(2 π ) / Z e − i ξ · x F λ ( x ) d x = 1(2 π ) g Z e − i ξ · x Z e i k · x Q λ ( k ) f λ ( k ) d kd x (A12)= 1 g Q λ ( ξ ) f λ ( ξ )= 1 g Q λ ( ξ ) δ ( ξ − λν ) ξ s λ ( ξ ) , ξ = | ξ | . If we substitute this in Fourier slice-projection theorem (18), we immediately find X F λ ( θ , x ) = (2 π ) / F − (cid:8) F [ F λ ( x )]( ξ ) (cid:9) = 1(2 π ) / Z θ ⊥ e i ξ · x F [ F λ ( x )]( ξ ) d ξ (A13)= 1(2 π ) / g λν Z C e iλν e r ( φ ) · x Q λ ( e r ( φ )) s λ ( λν e r ( φ )) dφ, (A11), using ξ = ξ e r ( φ ) ∈ θ ⊥ , d ξ = ξdξdφ and Q λ ( λν e r ( φ )) = Q λ ( e r ( φ )) since λν = | ν | > rkalian fields: ray transforms and mini-twistors
3. The integral operators: A α = U α + i V α The family of integrals A α = U α + i V α , α ∈ C , for Re ( α ) >
0, ( n = 2) are given by U α [ G ( κ )]( θ ) = Γ((1 − α ) / π Γ( α/ Z S θ G ( κ ) 1 | θ · κ | − α d Ω κ , α = 1 , , , ... (A14) V α [ G ( κ )]( θ ) = Γ(1 − α/ π Γ((1 + α ) / Z S θ G ( κ ) 1 | θ · κ | − α sgn ( θ · κ ) d Ω κ , α = 2 , , , ... G ( κ ) ∈ C ∞ ( S ). See the Refs. 26 and 27 and the references therein. The transform U α ( V α ) represents the even(odd) part of A α and annihilates odd (even) functions. For Re ( α ) ≤
0, these are to be understood in the sense ofanalytic continuation. In the case α −→ U [ G ( κ )] = 12 π / M [ G ( κ )] , (A15)and V is related to hemispherical transform. The inverse transforms are given as( U α ) − = U − − α , ( V α ) − = V − − α , (A16)in the sense of analytic continuation for certain values of α . This inversion of U α was established by Semyanistyi whostudied the connection of U α with Fourier transform. We have the relation Z R G ( κ ) k α e i k · η d k = c α η α − A α [ G ( κ )]( θ , x ) , c α,n = 2 − α π / (A17)where k = k κ , k = | k | , η = η θ , η = | η | . This can also be extended to all α ∈ C by analytic continuation.Note Γ(1) = 1, Γ( − /
2) = − π / for our purpose. a. Evaluation of an integral We decompose θ and κ into two components which are respectively parallel and orthogonal to κ ′ : θ = θ k + θ ⊥ = u κ ′ + v e r ( φ ). That is θ k = u κ ′ , u = cos θ and θ ⊥ = v e r ( φ ), v = √ − u = sin θ where e r ( φ ) is the unit radialvector parametrized by angle φ , in the plane κ ′⊥ orthogonal to κ ′ : e r ( φ ) · κ ′ = 0. Also κ = κ k + κ ⊥ . Then κ · θ = u κ k · κ ′ + √ − u κ ⊥ · e r ( φ ), δ ( κ ′ · θ ) = δ ( u ) and d Ω θ = sin θdθdφ = − dudφ . The integral (28) becomes I ( κ , κ ′ ) = Z πφ =0 Z u =1 u = − δ ( u ) | u κ k · κ ′ + √ − u κ ⊥ · e r ( φ ) | dudφ. (A18)This reduces to I ( κ , κ ′ ) = Z πφ =0 κ ⊥ · e r ( φ )] dφ. (A19)If we use the planewave decomposition of Dirac delta function δ ( x ) = − π Z S x · v ( ψ )] dψ, | v | = 1 (A20)in the plane κ ′⊥ , we find rkalian fields: ray transforms and mini-twistors I ( κ , κ ′ ) = − π δ ( κ ⊥ ) . (A21)Thus we are led to I ( κ , κ ′ ) = − π [ δ ( κ − κ ′ ) + δ ( κ + κ ′ )] , (A22)using the fact that as κ ⊥ = κ − κ k = 0, we have κ k = ± κ ′ and hence the identity δ ( κ ⊥ ) = δ ( κ − κ k ) = δ ( κ − κ ′ ) + δ ( κ + κ ′ ) . (A23)[Simply: κ = cos α κ ′ + sin α e r in the plane spanned by κ ′ and e r , thus δ ( κ ⊥ ) = δ (sin α ) = δ ( α ) + δ ( α − π ) = δ ( κ − κ ′ ) + δ ( κ + κ ′ ).] b. Example: Lundquist field If we substitute the X-ray transform (16) [that is (A9)] of the Lundquist field in (26), we find F RLλ ( κ · x , κ ) = − π F ν Z πθ =0 (cid:8) λ cos θ sin [ λνr sin( θ − φ )] e x + λ sin θ sin [ λνr sin( θ − φ )] e y (A24)+ cos [ λνr sin( θ − φ )] e z (cid:9) I ( β, θ − ψ ) dθ, where I ( β, θ − ψ ) = Z πα =0 α sin β cos( θ − ψ ) + cos α cos β ] dα. (A25)Here θ = v r e r ( θ ) + v z e z , v r = sinα , v z = cos α ⇒ d Ω θ = sin αdαdθ and κ = κ r e r ( ψ ) + κ z e z , κ r = sinβ , κ z = cos β .Consider the integral I = Z πα =0 y · v ( α )] dα = I + I , (A26)where y = cos β e x + sin β cos( θ − ψ ) e y and v ( α ) = cos α e x + sin α e y , | v | = 1. Here I is the integral (A25) and I = Z πα = π y · v ( α )] dα. (A27)We immediately find I = I using: α ′ = α − π . This leads to I = 12 I = − π δ ( y ) = − π δ (cos β ) δ (sin β cos( θ − ψ )) , (A28)using (A20) and δ ( y ) = δ ( y e x + y e y ) = δ ( y ) δ ( y ). This reduces to I = − π δ (cos β ) δ (cos( θ − ψ )) = − π δ ( κ z ) [ δ ( θ − ψ − π/
2) + δ ( θ − ψ − π/ . (A29)If we substitute this in (A24), then we find F R L ( λ =1) ( κ · x , κ ), (9) for λ = 1 after a straightforward manipulation. rkalian fields: ray transforms and mini-twistors APPENDIX B: MATHEMATICAL METHODS OF TOMOGRAPHY1. The identity
The identity (44) follows as Z S θ D F ( θ , x ) h ( θ · b ) d Ω θ = Z S θ Z ∞ t =0 F ( x + t θ ) h ( t θ · b ) t dtd Ω θ , x ′ = t θ , x ′′ = x + x ′ (B1)= Z Z ∞−∞ F ( x ′′ ) δ ( p − b · x ′′ ) h ( p − b · x ) dpd x ′′ = Z ∞−∞ F R ( p, b ) h ( p − b · x ) dp, similar to the derivation for scalar fields in Ref. 48, p. 277, Ref. 59. We have only assumed h ( ap ) = (1 /a ) h ( p ), a >
2. Grangeat’ s method a. Derivation
The derivation ∂∂p F R ( p, κ ) (cid:12)(cid:12)(cid:12) p = κ · x = Z F ( r ) δ ′ ( p − κ · r ) d r (cid:12)(cid:12)(cid:12) p = κ · x (B2)= − Z F ( x + r ′ ) δ ′ ( κ · r ′ ) d r ′ , r ′ = r − x , δ ′ ( − t ) = − δ ′ ( t )= − Z S θ D F ( θ , x ) δ ′ ( κ · θ ) d Ω θ , r ′ = s θ , s = | r ′ | > ⇒ d r ′ = s dsd Ω θ , δ ′ ( st ) = 1 s δ ′ ( t )follows the same reasoning for scalar fields.The identities Z S θ F ( θ ) δ ′ ( κ · θ ) d Ω θ = − Z S θ ∩ κ ⊥ ∂∂ κ F ( θ ) d θ = − Z πψ =0 ∂∂q F ( θ = q κ + r v ( ψ )) (cid:12)(cid:12)(cid:12) q =0 dψ, (B3)and Z S θ F ( x · θ ) δ ′ ( κ · θ ) d Ω θ = − x · κ Z S θ ∩ κ ⊥ F ′ ( x · θ ) d θ , (B4)are useful in handling these type of integrals. Here ∂/∂ κ denotes directional derivative along κ [ | θ | = 1, | κ | = 1 and v ( ψ ) takes values on the unit circle: | v | = 1 ⇒ q + r = 1, parametrized by angle ψ in the plane κ ⊥ through theorigin, Ref. 50, p. 71.] One can derive (B4) with a reasoning similar to the derivation of (B3). See Ref. 48, p. 277. b. Example If we substitute D F ( θ , x ), (15) in (45), the term containing δδ ′ vanishes and we find ∂∂p F R ( p, κ ) (cid:12)(cid:12)(cid:12) p = κ · x = − i k e ik κ · x I ′ F , (B5) rkalian fields: ray transforms and mini-twistors I ′ = Z S θ κ · θ δ ′ ( κ · θ ) d Ω θ = κ · κ Z S θ ∩ κ ⊥ κ · v ) dψ = κ · κ I ( κ , κ ) , (B6)(A19). Here we have used the identity (B3) or (B4) and (1 / κ · θ ) ′ = − / ( κ · θ ) (in distributional sense). Wedecompose the vector κ = κ k + κ ⊥ into components which are respectively parallel and orthogonal to κ and v ( ψ ) ∈ κ ⊥ , | v | = 1. This yields I ′ = − π κ · κ [ δ ( κ − κ ) + δ ( κ + κ )] = − π [ δ ( κ − κ ) − δ ( κ + κ )] , (B7)using the planewave decomposition of Dirac delta function (A20, A22). If we substitute this in (B5), we find ∂∂p F R ( p, κ ) (cid:12)(cid:12)(cid:12) p = κ · x = 4 π i k e ik κ · x [ δ ( κ − κ ) − δ ( κ + κ )] F , (B8)(7). c. Inversion for Trkalian fields If we substitute (19,20) into the righthand side of (50) and interchange the order of integrations, we find Z S θ θ × D F λ ( ± θ , x ) d Ω θ = 1(2 π ) / g λν Z S κ e iλν κ · x I ( κ ) × Q λ ( κ ) s λ ( λν κ ) d Ω κ , (B9)where I ( κ ) = Z S θ θ δ ± ( κ · θ ) d Ω θ = ± π i I ′ . (B10)Here the first integral vanishes. We can evaluate the second integral I ′ decomposing θ into components which arerespectively parallel and orthogonal to κ . That is θ = u κ + v e r ( φ ), u = cos θ , v = sin θ where e r ( φ ) is the unit radialvector parametrized by angle φ in the plane κ ⊥ orthogonal to κ . Then κ · θ = u and d Ω θ = − dudφ . Hence I ′ = Z S θ θκ · θ d Ω θ = Z φ =2 πφ =0 Z u =1 u = − [ u κ + v e r ( φ )] 1 u dudφ = 4 π κ , (B11)since the second integral vanishes, ( e r = cos φ e + sin φ e in the plane κ ⊥ ). Thus we find Z S θ θ × D F λ ( ± θ , x ) d Ω θ = ± π ν F λ ( x ) , (B12)using κ × Q λ ( κ ) = − iλ Q λ ( κ ) and (6). rkalian fields: ray transforms and mini-twistors
3. Smith’ s method
The equation (52) follows as G F ( β , x ) = F [ g F ( α , x )]( β , x ) , α = α θ ⇒ d α = α dαd Ω θ (B13)= 12 1(2 π ) / Z S θ Z ∞ α = −∞ X F ( θ , x ) | α | e − iα θ · β dαd Ω θ = 1(2 π ) / Z S θ D F ( θ , x ) h ( θ · β ) d Ω θ , where h ( s ) = Z ∞ α = −∞ | α | e − iαs dα = (2 π ) / i∂ s F [ sgn ( α )]( s ) = 2 ∂ s s = − s , ( s = θ · β ) (B14)satisfies h ( as ) = (1 /a ) h ( s ), a > h ( − s ) = h ( s ). Hence we obtain (52), since β = β b , s = βp .
4. Tuy’ s method
The equation (65) follows as G F ( β , x ) = F [ g F ( α , x )]( β , x ) , α = α θ ⇒ d α = α dαd Ω θ (B15)= 12 1(2 π ) / Z S θ Z ∞ α = −∞ D F ( θ , x )( | α | + α ) e − iα θ · β dαd Ω θ = 1(2 π ) / Z S θ D F ( θ , x ) f ( θ · β ) d Ω θ , where f ( s ) = 12 Z ∞ α = −∞ ( | α | + α ) e − iαs dα = (2 π ) / F [ R ( α )]( s ) = 12 Z ∞ α = −∞ ( | α | − α ) e iαs dα, ( s = θ · β ) (B16)= 12 [ h ( s ) − i k ( s )] . Here R ( α ) = αH ( α ) is the Ramp function, h ( s ) = 2 ∂ s /s , (B14) and k ( s ) = ∂ s Z ∞ α = −∞ e iαs dα = 2 πδ ′ ( s ) . (B17)Hence f ( s ) = 2 πi∂ s δ − ( s ). This also satisfies f ( as ) = (1 /a ) f ( s ), a > f ( − s ) = − πi∂ s δ + ( s ). Thus we obtain(65), since β = β b , s = βp .
5. Divergent beam transform of Lundquist field
If we substitute the Radon transform (9) of the Lundquist field ( λ = 1) into the integral in (71) using x = r e r ( φ ) + z e z , θ = v r e r ( θ ) + v z e z , and b = b r e r ( ψ ) + b z e z where b r = p − b z , d Ω b = − db z dψ , the integration over b z leads to I ( θ , x ) = (2 π ) Y F ( θ , x ) (B18)= − πF ν v r Z πψ ′ =0 h e iνr cos( ψ ′ + θ − φ ) L ( ψ ′ + θ ) e − iνr cos( ψ ′ + θ − φ ) L ′ ( ψ ′ + θ ) i ψ ′ dψ ′ . rkalian fields: ray transforms and mini-twistors ψ ′ = ψ − θ (and the integral over ψ ′ is insensitive to shift of limits of integration). This yields I ( θ , x ) = − πF ν v r (cid:2) ( I +3 + I − ) e r ( θ ) − ( I +1 + I − ) e θ − i ( I +2 − I − ) e z (cid:3) , (B19)where I ± = Z πψ ′ =0 e ± iνr cos( ψ ′ + θ − φ ) dψ ′ , I ± = Z πψ ′ =0 e ± iνr cos( ψ ′ + θ − φ ) ψ ′ dψ ′ , (B20) I ± = Z πψ ′ =0 e ± iνr cos( ψ ′ + θ − φ ) tan ψ ′ dψ ′ . We shall use the variable ζ = iωz where z = e iψ ′ and ω = e i ( θ − φ ) . Hence i cos( ψ ′ + θ − φ ) = (1 / ζ − /ζ ),cos ψ ′ = (1 / /iω )(1 /ζ )( ζ − ω ), tan ψ ′ = − i ( ζ + ω ) / ( ζ − ω ), and dψ ′ = − i (1 /ζ ) dζ . These integrals become I ± = − i Z C ′ e u ± ( ζ − /ζ ) ζ dζ, I ± = 2 ω Z C ′ e u ± ( ζ − /ζ ) ζ − ω dζ, (B21) I ± = − Z C ′ e u ± ( ζ − /ζ ) ζ ζ + ω ζ − ω dζ, where u ± = ± νr and C ′ is a unit circle around the origin (the integrals are insensitive to shift of limits of integration).Note the notation: the Cauchy principal value is meant in these contour integrals.We shall make use of the generating function e u ± ( ζ − /ζ ) = P ∞ n = −∞ ζ n J n ( u ± ) = J ( u ± ) + P ∞ n =1 [ ζ n +( − n ζ − n ] J n ( u ± ) for Bessel functions.In I ± there are no poles on C ′ and only the term of order n = 0 in the generating function contributes to theintegral I ± = 2 πJ ( νr ) . (B22)If we substitute the generating function in I ± , we get I ± = 2 ω ( I ± a J ( u ± ) + ∞ X n =1 (cid:2) I ± b + ( − n I ± c (cid:3) J n ( u ± ) ) . (B23)The contributions to the principal value integral I ± a of the residues of simple poles at ζ = ± ω on C ′ cancel out I ± a = Z C ′ ζ − ω dζ = πiRes [ f = 1 / ( ζ − ω ) , ζ = ω ] + πiRes [ f = 1 / ( ζ − ω ) , ζ = − ω ] = 0 . (B24)Meanwhile, for n ≥ I ± b = Z C ′ ζ n ζ − ω dζ = πiRes [ g = ζ n / ( ζ − ω ) , ζ = ω ] + πiRes [ g = ζ n / ( ζ − ω ) , ζ = − ω ] (B25)= πi
12 [1 + ( − n +1 ] ω n − , and I ± c = Z C ′ ζ − n ζ − ω dζ = 2 πiRes [ h = ζ − n / ( ζ − ω ) , ζ = 0] + πiRes [ h = ζ − n / ( ζ − ω ) , ζ = ω ] (B26)+ πiRes [ h = ζ − n / ( ζ − ω ) , ζ = − ω ]= − πi
12 [1 + ( − n +1 ] ω − ( n +1) . rkalian fields: ray transforms and mini-twistors I ± = 2 πiω ∞ X n =1
12 [1 + ( − n +1 ] h ω n − − ( − n ω − ( n +1) i J n ( u ± ) = ± πi ∞ X k =0 cos[(2 k + 1)( θ − φ )] J k +1 ( νr ) . (B27)The integral I ± reduces to I ± = − ( I ± a J ( u ± ) + ∞ X n =1 (cid:2) I ± b + ( − n I ± c (cid:3) J n ( u ± ) ) . (B28)Here I ± a = Z C ′ ζ ζ + ω ζ − ω dζ = 2 πiRes [ f = ζ − ( ζ + ω ) / ( ζ − ω ) , ζ = 0] + πiRes [ f = ζ − ( ζ + ω ) / ( ζ − ω ) , ζ = ω ]+ πiRes [ f = ζ − ( ζ + ω ) / ( ζ − ω ) , ζ = − ω ]= 0 , (B29)and since n ≥ I ± b = Z C ′ ζ n − ζ + ω ζ − ω dζ = + πiRes [ g = ζ n − ( ζ + ω ) / ( ζ − ω ) , ζ = ω ] + πiRes [ g = ζ n − ( ζ + ω ) / ( ζ − ω ) , ζ = − ω ]= πi [1 + ( − n ] ω n , (B30)and I ± c = Z C ′ ζ n +1 ζ + ω ζ − ω dζ = 2 πiRes [ h = ζ − ( n +1) ( ζ + ω ) / ( ζ − ω ) , ζ = 0] (B31)+ πiRes [ h = ζ − ( n +1) ( ζ + ω ) / ( ζ − ω ) , ζ = ω ]+ πiRes [ h = ζ − ( n +1) ( ζ + ω ) / ( ζ − ω ) , ζ = − ω ]= − πi [1 + ( − n ] ω − n . Thus I ± = − πi ∞ X n =1 [1 + ( − n ] (cid:2) ω n − ( − n ω − n (cid:3) J n ( u ± ) = 4 π ∞ X k =1 sin[2 k ( θ − φ )] J k ( νr ) . (B32)Then we obtain (75) using (B22, B27, B32) in (B18, B19).We also need sin( z sin α ) = 2 ∞ X k =0 sin[(2 k + 1) α ] J k +1 ( z ) , cos( z sin α ) = J ( z ) + 2 ∞ X k =1 cos(2 kα ) J k ( z ) , (B33)for the Divergent beam transform (76) of the Lundquist field. rkalian fields: ray transforms and mini-twistors
6. Gelfand-Goncharov’ s method
If we substitute equation (81) in (6) and interchange the order of integrations, we find F λ ( x ) = − π λν Z S θ D F λ ( θ , x ) I ( θ ) d Ω θ , (B34)where I ( θ ) = Z S κ θ · κ ) d Ω κ . (B35)If we decompose κ into components which are respectively parallel and orthogonal to θ : κ = κ k + κ ⊥ ( κ k = u θ , u = cos θ , and d Ω κ = sin θdθdφ = − dudφ ), then we find I ( θ ) = 2 πI, (B36)where I = Z u = − u du. (B37)The principal value of this integral can be easily calculated using a contour integral: I = −
2. This leads to (82).
APPENDIX C: MINI-TWISTORS1. Helmholtz equation: (mini-)twistor solution
We reproduce the following solution of Helmholtz equation from Ref. 46 which is partially based on Hitchin’ snotes.The solution of wave equation ✷ ψ = 0 , (C1) ✷ = ∂ /∂t − ∇ , ∇ = ∂ /∂x + ∂ /∂y + ∂ /∂z is given by ψ ( x, y, z, t ) = Z C g [ ω ( t + z ) + ( x + iy ) , ω ( x − iy ) + ( t − z ) , ω ] dω = Z C g ( p, q, ω ) dω, (C2)where g is holomorphic in each of its arguments: p = x + iy + ω ( t + z ), q = t − z + ω ( x − iy ) and ω . We can easilyverify (C1) using ∂/∂t = ω∂/∂p + ∂/∂q , ∂/∂x = ∂/∂p + ω∂/∂q , ∂/∂y = i ( ∂/∂p − ω∂/∂q ), ∂/∂z = ω∂/∂p − ∂/∂q .We can write the wave equation (C1) as ∂ ψ/∂u∂v − ∂ ψ/∂ζ∂ζ = 0 introducing new variables u = t − z , v = t + z , ζ = x + iy , ζ = x − iy and ψ ( x, y, z, t ) −→ ψ ( u, v, ζ, ζ ). This is immediately satisfied since: ∂/∂u = ∂/∂q , ∂/∂v = ω∂/∂p , ∂/∂ζ = ∂/∂p , ∂/∂ζ = ω∂/∂q . Then p = ζ + ωv , q = ωζ + u in (C2).For the Helmholtz equation, we suppose ∂ψ∂t = − ikψ. (C3)This leads to ( ∂/∂t ) g [ ω ( t + z ) + ( x + iy ) , ω ( x − iy ) + ( t − z ) , ω ] = − ikg using (C2), that is (cid:18) ω ∂∂p + ∂∂q (cid:19) g ( p, q, ω ) = − ikg. (C4) rkalian fields: ray transforms and mini-twistors
34A simple method for solving this equation is to use an integrating factor. For example we can use an integratingfactor that is a function of q so as to remove the right-hand side g ( p, q, ω ) = e − ikq h ( p, q, ω ) . (C5)Then (C4) becomes ω ∂h∂p + ∂h∂q = 0 . (C6)The solution of this equation is given by h ( p, q, ω ) = H ( η, ω ), where H is a holomorphic function of two arguments: η x ( ω ) = p − ωq = x + iy + 2 zω − ( x − iy ) ω and ω . Then g ( p, q, ω ) = e − ikq H ( η x ( ω ) , ω ) and (C2) yields ψ = e − ikt φ ( x, y, z ) , (C7)where φ ( x, y, z ) = Z C e − ik [ ω ( x − iy ) − z ] H ( η x ( ω ) , ω ) dω, (C8)satisfies the Helmholtz equation ∇ φ = − k φ . This reduces to solution of Laplace equation for k = 0.We could also use another integrating factor g ( p, q, ω ) = e − i k ( p/ω + q ) H ( p − ωq, ω ) . (C9)This would yield φ = Z C e − i k [ ω ( x − iy )+( x + iy ) /ω ] H ( η x ( ω ) , ω ) dω. (C10)Any solution of (C4) can be written in this form. Because introducing new variables: α = p − ωq , β = p + ωq ,the equation (C4) becomes ∂g∂β = − i k ω g, (C11)and the general solution of this equation is g ( α, β, ω ) = e − i k ω β h ( α, ω ), α = η ( ω ).
2. Contour integrals for Bessel functions
We need the following integrals J m ( νr ) cos mϕ = i m π Z πθ =0 e − iν ( x cos θ + y sin θ ) cos mθdθ, J m ( νr ) sin mϕ = i m π Z πθ =0 e − iν ( x cos θ + y sin θ ) sin mθdθ, (C12)where x = r cos ϕ , y = r sin ϕ . These lead to J m ( νr ) e imϕ = i m π Z πθ =0 e − iν ( x cos θ + y sin θ ) e imθ dθ. (C13)Hence rkalian fields: ray transforms and mini-twistors J ( νr ) = 12 π Z πθ =0 e − iν ( x cos θ + y sin θ ) dθ, J ( νr ) cos ϕ = i π Z πθ =0 e − iν ( x cos θ + y sin θ ) cos θdθ, (C14) J ( νr ) sin ϕ = i π Z πθ =0 e − iν ( x cos θ + y sin θ ) sin θdθ. See Ref. 36. One can prove these, for example following Ref. 5. We have: J ( − s ) = J ( s ), J ( − s ) = − J ( s ).Another expression is J ( β ) = 12 π Z π e iβ sin u du. (C15)
3. Solution with Laurent series
If we use f = (1 / ω ( x − iy ) + ( x + iy ) /ω ] with ω = iω ′ , ( k = ν ) and choose u = h ( ω ′ ) which has a Laurent series: h ( ω ′ ) = 1 /ω ′ ( n +1) , the equation (94) leads to F ( x ) = Z C h i (cid:16) ω ′ (cid:17) , − (cid:16) − ω ′ (cid:17) , − ω ′ i e − ν [( x + iy ) /ω ′ − ( x − iy ) ω ′ ] ω ′ ( n +1) dω ′ . (C16)The generating function e p ( t − /t ) = P ∞ m = −∞ t m J m ( p ) for the Bessel functions yields e − ν [( x + iy ) /ω ′ − ( x − iy ) ω ′ ] = ∞ X m = −∞ e − imϕ ω ′ m J m ( νr ) , (C17)with p = − νr , t = − ( x − iy ) ω ′ /r = − e − iϕ ω ′ , ( x + iy = re iϕ ) and J m ( − x ) = ( − m J m ( x ). The integrals in (C16) canbe evaluated using Z C ∞ X m = −∞ c m ω ′ m ω ′− ( n +1) dω ′ = 2 πic n , c m = e − imϕ J m ( νr ) , (C18)which is based on residues. We find F ( x ) = − πe − inϕ (cid:0)(cid:2) J n ( νr ) + e i ϕ J n − ( νr ) (cid:3) , i (cid:2) J n ( νr ) − e i ϕ J n − ( νr ) (cid:3) , ie iϕ J n − ( νr ) (cid:1) . (C19)This reduces to F ( x ) = 4 πie − imϕ (cid:20) im νr J m ( νr ) e r + J ′ m ( νr ) e ϕ − J m ( νr ) e z (cid:21) , m = n − e r = cos ϕ e x + sin ϕ e y , e ϕ = − sin ϕ e x + cos ϕ e y , using the identities: J n ( x ) − J n − ( x ) = − J ′ n − ( x ), x [ J n ( x ) + J n − ( x )] = 2( n − J n − ( x ), ( x = νr ). This is a circular cylindrical CK solution with no z dependence, upto conventions. rkalian fields: ray transforms and mini-twistors K. Saygili, J. Math. Phys. , 033513 (2010). V. Trkal, ˇCasopis pro pˇestov´an´ı matematiky a fysiky , 302 (1919); English translation by I. Gregora: Czechoslovak Jour.Phys. (2), 97 (1994). A. Lakhtakia, Czechoslovak Jour. Phys. (2), 89 (1994). M. A. MacLeod, J. Math. Phys. , 2951 (1995). M. A. MacLeod, J. Math. Phys. , 1642 (1998). K. Saygili, Int. J. Mod. Phys. A , 2015 (2008). S. Deser, R. Jackiw, S. Templeton, Phys. Rev. Lett. , 975 (1982). S. Deser, R. Jackiw, S. Templeton, Ann. Phys. , 372 (1982). J. F. Schonfeld, Nuc. Phys. B , 157 (1981). A. N. Aliev, Y. Nutku, K. Saygili, Class. Quant. Grav. , 4111 (2000). K. Saygili, e-print arXiv: hep-th/0610307. K. Saygili, Int. Jour. Mod. Phys. A , 2961 (2007). P. Baird, J. C. Wood,
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