Twistor Theory began as a subject in the late 1960's with the appearance of Penrose's two papers (1967, 1968a). A more definitive statement of its aims and accomplishments was ‘Twistor Theory: An Approach to the Quantisation of Fields and Space-time’ (Penrose and MacCallum), which appeared in 1973.

What is apparent from the start is the breadth of application which Roger Penrose saw for it. In the decades since then the subject has grown in different direction to the extent that different traditions have emerged.

There is a purely mathematical strain which refers to the Penrose Transform and is interested in its geometrical and complex analytic features, frequently in a positive definitve setting.

There is a quantum field theoretic strain concerned with elementary particles and their interactions in Minkowski space.

There is a modest point of view which simply holds that the theory is useful for solving some non-linear equations and the object is to discover which ones, and there is a full-blooded strain which holds that the repeated occurence and usefulness of complex numbers and complex analyticity tells one something fundamental about the physical world.

The full-blooded strain of twistor philosophy may be seen in for example (Penrose 1975). One place where the diversity of twistor theory is manifest is the Twistor Newsletter, an informal publication produced by the Oxford group about twice a year. The content of the first ten Newsletters was published as ‘Advances in Twistor Theory’ (Hughston and Ward 1979) and later articles were collected in ‘Further Advances in Twistor Theory’ (Mason and Hughston 1990).

In this book, our aim has been to provide a brief introduction to twistor theory, sketching the mathematical background and indicating where it makes contact with the physics of space-time. It is possible to carry on from here in directions which we may characterise as inner or outer, and mathematical or physical.

By inner we mean developments which make rigorous and treat more thoroughly what has been treated informally here. For spinor theory the main reference is the two volume Spinors and space-time of Penrose and Rindler (1984, 1986). This is a mighty compendium of results for spinors in flat and curved space-time, and has more detail, in volume 2, on the quasi-local mass construction described in chapter 13. For more on sheaf cohomology and the twistor theory of massless fields and the active, curved-twistor-space constructions of chapter 11 see Twistor Geometry and Field Theory (Ward and Wells 1990).

By outer we mean developments that go beyond what has been covered here. Other parts of the theory of z.r.m. fields have been given a twistor description, and the formalism of chapters 8 to 10 has been much extended. Hyperfunction z.r.m. fields are discussed in Bailey et al. (1982) and fields with sources, which can be described by relative cohomology, by Bailey (1985). This material is reviewed in the article of Bailey and Singer in Bailey and Baston (1990).

In the eight years since the first edition of this book was published the literature of twistor theory has increased very substantially. There have been several books published whose subject matter overlaps to a greater or lesser extent with ours: Ward and Wells (1990) contains a mathematically more thorough and rigorous account of sheaf cohomology and the machinery from complex analysis necessary for twistor theory than we have found space for; Baston and Eastwood (1989) is a mathematical account of the ‘Penrose transform’ and its generalisation as part of representation theory; Bailey and Baston (1990) is a collection of commissioned reviews which go on in many directions from topics which we touch on more briefly; finally, Mason and Hughston (1990) follows on from Hughston and Ward (1979) by being a collection of articles which had previously appeared in Twistor Newsletter.

None of these precisely duplicates our book and so we have prepared a new edition taking account of some of the developments of the past eight years. The principal changes for the second edition are as follows:

1. - chapter 9 has been rewritten with a slightly different emphasis. Our intention is to provide a much clearer and more detailed motivation for the notion of a sheaf, with readers new to the subject in mind. Then (as in the first edition) we go on to discuss the elements of íech cohomology.

2. - chapter 13 has been extended to describe the present status of the original quasi-local mass construction. There has been a lot of work in this area, and also a proliferation of other definitions of quasi-local mass. We describe one of them, that of Dougan and Mason, in detail.

3. […]

In general relativity there is a long-standing problem of defining momentum and angular momentum in a general curved space-time. In this chapter, we describe an application of twistor theory which aims to provide a solution to this problem. As we shall see, this aim is not entirely achieved and the programme has both successes and failures.

We recall that in general relativity, all the local matter content is described by the stress–energy tensor Tab. Gravitational energy, whether in gravitational waves or in the form of gravitational potential energy, is notoriously non-local and one cannot expect to characterise it by a local density. Instead, Penrose (1982) has suggested that one should seek a non-local invariant associated to any two-surface S and representing the total momentum–angular momentum flux through that surface. This non-local invariant will be constructed by twistorial techniques tailored to give the right answer for linearised general relativity where there is a clear right answer.

We begin by reviewing the definition of momentum and angular momentum in special relativity and in linearised general relativity.

A material system in special relativity is defined by its stress–energy tensor Tab which we may suppose for definiteness to have support within a world tube W in Minkowski space, M. (The case of say electromagnetic fields spread throughout M is a simple generalisation.)

Given the powerful constructions of chapters 8–10, which show how the Penrose transform describes zero rest mass fields and neatly geometrises the concept of positive frequency, it is natural to seek a twistor description of interactions. One might expect such a description to arise out of a twistor dynamical principle, replacing the usual space-time Lagrangian, and indeed the early work (Penrose and MacCallum 1972) proceeded in just this way, the resulting twistor integrals being referred to as twistor diagrams. Twistor diagrams share with Feynmann diagrams a simple combinatorial structure, but are not supposed to share their divergences! The analogy between twistor diagrams and Feynmann diagrams has been extensively explored, for which see Hodges (1990) and references therein. In this work twistor diagrams are regarded as determining integrals of holomorphic functions over compact contours, and it is only fairly recently that the diagrams have been reinterpreted as functionals on the cohomology classes representing the fields. This latter question, which is reviewed in Huggett (1990), is our subject here, and in order to discuss it adequately we will need to draw on some more cohomology theory. We start, however, by making a relatively simple observation on the geometry of products of twistor spaces.

In the product space {(Zα, Wα) ∈ PT × PT* : ZαWα ≠ 0} the following holomorphic differential forms arise quite naturally. Let DZ = ∈αβδγZαdZβdZδdZγ, where ∈αβδγ is as in (7.7), define DW similarly and let DZW be their wedge product.

We saw in the last chapter that in order to make the relationship between twistor functions and zero rest mass fields precise we seem to need to study certain collections of these (holomorphic) twistor functions. These collections are actually well-known to the mathematician: they arise naturally in the context of sheaf cohomology.

We start by reviewing the essential ideas behind analytic continuation of complex functions. This will quickly lead us to discuss Riemann surfaces and sheaves.

Consider an analytic function f : D → C, where D ⊂ C is a domain (that is, an open connected set). Under what circumstances can we extend f so that it is defined (and still analytic) on some larger set than D? Well, as usual, the analyticity allows us to push out the boundaries of D until the ‘natural boundary’ of f (a singularity of some sort) is reached. It frequently happens that in doing this the function f gets continued back to a point z ∈ D, but takes a different value there from its original one. The archetypical example, of course, is the logarithm. We therefore seem to be obliged to study multivalued functions, unless we can somehow absorb this behaviour into the space on which f is defined.

More precisely, we define a function element to be a pair (f, D) where f is analytic on the domain D.