5. — Relation with Radon transform.

In this section we make yet another digression to explain how the Pen­rose transform is related to other well-known transforms.

We recall first the classical Eadon transform which associates to a func­tion / on B3 its integral over affine planes. Thus to each / we obtain a trans­formed function <p defined on the space of planes n by:

 

 

The space of all planes in B3 is a 3-dimensional manifold which can be identified with P3(B) with a point removed. There is an inversion formula which defines / in terms of an integral of <p. The Eadon transform is closely related to the Fourier transform and is therefore useful for solving constant coefficient linear differential equations [20].

and the map 6 between them is given by <px. Thus if W= E, the fibre Ex — B°(PX, Wx) appears as the kernel of the homomorphism

An obvious generalization of the Eadon transform is to associate to / its integral over affine lines I in B3. Thus we define q> by

 

It is a function on the space of all lines in B3. This space is a subspace of the space of all lines in P3(B), namely the real Klein quadric (cf. Chap­ter III). This quadric has signature (3, 3) and contains two systems of real projective planes. Our subspace is obtained by omitting one plane of one system.

In (5.1) and (5.2) the measure on n or I, with respect to which the inte­gration is performed, is the usual Euclidean area or length respectively. Consequently both transforms are compatible with the group of Euclidean motions, this group acting naturally on the space of planes or lines. It is possible however to extend both transforms to the whole projective 3-space P3(P), compatibly with the action of #i(4, B), provided we replace functions by appropriate types of density. In general a fc-th root of a density will be called a /r1-density. Then in (5.1) 99, / must be taken as J and § densities respectively, while in (5.2) cp, f must be \ and I densities. More generally if we integrate over (p — l)-planes in        the relevant densities are

1 l(P + 0) and p/(p + q).

In this extended form (5.2) therefore transforms the ^-density / on P3{B) to the ^-density <p on the real Klein quadric QJ^B). Since dim = 4 > dim P3 it is unreasonable to expect a universal inversion formula, as for (5.1). Instead one expects 99 to satisfy some condition in order for it to be the transform of an /. In fact this condition is that <p be a solution of the con­formally invariant second order equation Aq> = 0, where A is the indefinite analogue of the conformal Laplacian [21], [45]. Note that Qt(B) is the con- formal compactification of the space Bl with the (2, 2) signature. Thus A is an «ultra-hyperbolic» operator, i.e. one which in flat space is given by

 

 

The group SL{4, B) acts on Qt(R) by conformal transformations and pre­serves the operator A.

In these transforms we have so far ignored regularity questions of /, <p. The precise correspondence holds either for Gm functions or for real analytic functions. Note that, unlike the positive definite case when A is elliptic, the equation Acp = 0 does not imply analyticity of cp.

If we deal with analytic functions then the transform (5.2) can be looked at in the complexification. Thus, if we extend / as a complex analytic func­tion to some neighbourhood of P3(B) in P3{C), then (5.2) becomes a Cauchy integral. This leads naturally to the sheaf cohomology occurring in the Pen­rose transform as we shall now explain.

To begin with let us recall that for the complex projective line Pt(C),

we have

 

 

In section 1 this isomorphism was given in terms of explicit representative cocycles, and this can be made more intrinsic by recalling that 0(— 2) ^ Q1 the sheaf of holomorphic differentials on I\(C) (note that the tangent bundle of Pi(C) is L 2). If co = f(z)dz is a holomorphic differential defined near the circle \z\ = 1, we can take <» = ct>0oo to define a 1-cocycle relative to the standard covering of I\(C) by the open sets U0, 1Jm given by \z\ < 1 + e, \z\ > 1 — e. The isomorphism (5.3) is then given by the contour integral

 

 

taken around \z\ = 1 (oriented in the conventional manner as the boundary of |z| <1).

We now return to the transform (5.2) in which /, as explained earlier, should be taken as a ^-density on P3(R). If we fix the orientation of P3(R) a density on P3(R) can be identified with an exterior differential 3-form and hence with a section of the real line-bundle L*. Thus / can be viewed as a section of L2 and complexified accordingly.

Next we fix a line l0 in P3(R) and consider all nearby lines I, parametrized by a small open set V cQ^R). The complexifications of the lines I fill out an open set U in P3(G) as one can verify. If we orient l0, and extend this orientation by continuity to all nearby I, we can define open subsets U0, U«, of V swept out by the (e-enlarged) upper and lower hemispheres of each Pi(G) in our family. If s is small enough i/„ n !/„ will be close to P3(R) and hence within the domain of definition of /. Hence taking /Ooo = / we get an element

 

 

By the Penrose transform, as explained in Section 2, (/) defines a ^-density <p, on the appropriate open set of Q4(C), satisfying A<p = 0. The value of <p at the point parametrizing a given Pi(C) is given by restricting (/) to this Pi(C) and using (5.3). Comparing (5.2) and (5.4) we see that they coincide, up to the factor 2ni. Thus the Penrose description of solutions of A<p = 0 via sheaf cohomology classes can be considered as the natural complex-analytic description of the integral transform (5.2).

Although the transform (5.2), taken globally on the whole of P3(R), is invertible (i.e. <p = 0 =>/ = 0), the same is not true locally. Thus if /, defined originally in U, extends holomorphically to U„ or Z7m, or more

linear Field equations

generally is the difference of two such extensions, then (/) = 0 and 99 = 0 in V. This clarifies the role of the sheaf cohomology group which precisely absorbs the local non-invertibility, in other words <p = 0 in V =>(/) = 0 in V.

It is perhaps worth emphasizing that the construction of the sheaf coho­mology class (/) from the f-density f depended on orienting the lines I, so that U0 and U^ could be unambiguously defined. Now for topological reasons we cannot coherently and continuously orient all real lines I in P3{B): the space Qi(R) is not simply-connected. Thus a global / does not define an ele­ment of the global cohomology group Jff1(P3(Cf), 0(—2)). This averts the contradiction that would otherwise arise from the vanishing of this coho­mology group and the global invertibility of (5.3).

with appropriate properties. Here dim W — fc, dim 'J = 2Tc + 2n, z = — {zlt ..., zt) and V has a non-degenerate skew-form. For each z^O the