## Abstract

A brief review is given of why twistor geometry has taken a central place in the theory of scattering amplitudes for fundamental particles. The emphasis is on the twistor diagram formalism as originally proposed by Penrose, the development of which has now led to the definition by Arkani-Hamed *et al.* of the ‘amplituhedron’.

## 1. Conformal symmetry

Twistor geometry was defined by Roger Penrose in the 1960s, with a first paper on ‘Twistor algebra’ in 1967 [1]. But until 2004, twistor geometry remained on the sidelines of quantum field theory. Only since 2008 have twistor methods emerged as yielding efficient new methods for calculating scattering amplitudes. This brief review will describe this recent sequence of developments, with an emphasis on the *twistor diagram* formalism. Twistor diagrams were conceived as roughly analogous to Feynman diagrams, as used since the 1940s for the calculation of scattering amplitudes. They have in fact led the way to a completely new picture of amplitudes. This review of the sequence of ideas is intended to be accessible to readers who are acquainted with the ideas of relativistic quantum field theory, but have not been closely involved in these new developments.

The central point is that twistor geometry makes manifest *conformal symmetry* within Minkowski space–time, and conformal symmetry has increasingly been recognized as a feature of fundamental physics. Conformal symmetry is a ‘hidden’ symmetry, not obvious in the usual space–time coordinates, and it has taken a long time to elucidate its significance. In what follows we confine ourselves to considering the physics of Special Relativity, i.e. flat space–time, with the usual coordinates. Then it can be briefly described as follows. The 1905 revolution can be summarized as saying that physical laws must be invariant under the transformation , where *L*^{a}_{b} is a Lorentz transformation and *c*^{a} defines a translation. This gives the 10-dimensional Poincaré group of transformations. There is a natural extension to an 11-dimensional group by including the scale transformation . Remarkably, there is a further transformation (an *inversion*) defined by
1.1where *t*^{a} is a unit timelike 4-vector, with the property that together with the Poincaré and scale transformations it generates the 15-dimensional conformal group. Details have been neglected here: the *L*^{a}_{b} should be restricted to the proper orthochronous component, and for proper definition the inversion map requires the compactification of Minkowski space with a null cone at infinity.

There is a very rich geometrical structure inherent in these ideas, expounded by Penrose & Rindler [2], but for brevity we can take advantage of the useful analogue with the simpler case of the Möbius transformations on the complex plane, to give a picture of how the conformal group behaves. The analogue of the Poincaré transformation is , with three real dimensions; scalings enlarge the group to , with four real dimensions, and now the inversion map , together with these, generates the six-real-dimensional Möbius group. This also requires compactification, but the simpler one of adding a point at infinity to give the Riemann sphere—or, equivalently, the projective space .

The concept of *angle* is invariant under this group; hence the concept of a Möbius transformation as a *conformal* (shape-preserving) mapping from the Riemann sphere onto itself. Likewise, there are conformal invariants in (compactified) Minkowski space. The simplest observation is that the concept of *null separation* is conformally invariant. A conformal transformation maps light cones into light cones. Intuitively, emphasizing conformal symmetry means that light cones (and so the causal structure of space–time) are regarded as primary, while metric and length scales are secondary.

Is the conformal group a hidden symmetry of some physical theory? As one might suspect, a geometrical transformation that preserves light cones is one that preserves the equations governing the propagation of light, namely Maxwell's equations without sources. This was known long ago, but this invariance acquires even more significance from a modern point of view, because Maxwell's equations are just one example (with spin 1) of a *zero-rest-mass* theory. Zero-rest-mass equations are conformally invariant.

This observation is even more important now that zero-mass fields have become central to the Standard Model of particles and forces. According to this picture, all the massive fields arise from the interaction of more primordial zero-mass fields with a Higgs field. It is also relevant that, since the 1950s, the use of coordinates based on light-cone structures has become a central idea in General Relativity. This is an example of the convergence of ideas which lies behind the recent developments. Another such convergence appears in the application of *2-spinors* as fundamental objects for the description of matter and forces.

## 2. Spinors

Standard textbooks on quantum field theory treat the massive Klein–Gordon equation, and the massive Dirac equation, as the fundamental cases. The Dirac γ-matrices, acting on four-component spinors, are an essential part of the formalism. This formalism is suitable for non-relativistic electrons, as in atomic theory. But if the Dirac formalism is used to describe zero-mass spin- fields, it is found that the four components naturally split into two parts, these two parts being equivalent to what is obtained by operating on the four-component spinor with (1±i*γ*_{5}). These two parts are helicity eigenstates, with helicity . For this reason, one finds that in standard texts, using the Dirac formalism, such (1±*iγ*_{5}) symbols pepper the description of weak interactions. But if zero-mass fields are regarded as fundamental, then the use of these awkward projection operators can be avoided. Rather than start with four-component spinors, which are then split into two parts, it is much simpler to work with two-component spinors from the start. This is done as follows.

Suppose the four-vector *p*_{a} is null, so that . Equivalently, the matrix
has zero determinant, so is of rank 1, so can be written as the direct product of a two-component row and a two-component column. These are the 2-spinors we need. We have , where the are connecting Pauli matrices, and the *A* and *A*' are indices ranging over {0,1}. We abbreviate this by letting the Pauli matrices be implicit, thus writing
2.1(Here we are following Penrose's notation as in [2] for consistency; the conventions used by physicists differ somewhat but generally they prefer the symbol *λ* and use Greek or small Roman indices for the spinors.)

Clearly, there is a freedom of choice when a null vector is thus factorized into spinors, by . Any function of the momentum *p*_{a} must be invariant under this rescaling. This is an important aspect of the spinor representation of amplitudes, and gives rise to a simple feature of spinorial ‘weight’. It is also important to know that, under a Lorentz transformation of the *p*_{a}, the *π*_{A′} transforms in the simplest possible way, by an SL2C transformation. (The transforms according to the conjugate representation.) This fact embodies the local isomorphism between the two six-real-dimensional groups.

If desired, the Dirac equation can readily be expressed in terms of 2-spinors, but it becomes a pair of coupled spinor equations, rather than a single equation. This might seem a loss of simplicity, but this separation of the electron into left- and right-handed parts is consistent with its modern interpretation in the standard model. It should also be noted that, when extending the 2-spinor calculus to general relativity (one of Roger Penrose's early achievements), there is no analogue of the 4-spinors. Thus for a number of reasons, 2-spinor calculus may be seen as the most fundamental to the modern description of space–time and matter.

Another beautiful feature of 2-spinor algebra is that the space of *projective* 2-spinors is simply , with the topology of a sphere. This gives a natural geometrical connection between SL2C (the Möbius group) and the Lorentz group.

In summary, the fundamental significance of zero-mass fields makes the expression of conformal invariance much more important, and left–right parity invariance much less important, than would have been considered natural in the past. This is the general background against which twistor geometry has come to prominence.

## 3. Twistors

Penrose's definition of twistors can be considered as doing for the conformal group what spinors do for the Lorentz group. The following gives a brief outline; detailed exposition is given by Penrose & Rindler [2] and Huggett & Tod [3]. Consider defined by (*Z*^{0},*Z*^{1},*Z*^{2},*Z*^{3})=(*ω*^{A},*π*_{A′}). Now consider the set of complexified Minkowski space points *x*^{a} such that the incidence relation *ω*^{A}=i*x*^{AA′}*π*_{A′} is satisfied. It is easily seen that SL4C acts on *Z*^{α} in such a way that this set transforms under the conformal group. To check this, note that the only non-trivial case is that given by the inversion (1.1) for which the element of SL4C is . The thus defined is *twistor space*, .

More precisely, there is a 4 : 1 local isomorphism between SL4C and the conformal group. We shall again neglect such details of the geometrical theory, and also leave it implicit that everything said here about finite points *x*^{a} will carry through correctly to the more complete theory involving the compactification of Minkowski space and the ‘points at infinity’.

Note that the set of *x*^{a} associated with a twistor is actually associated with the *projective* twistor. We can define projective twistor space , which as a can also be thought of as the Grassmannian Gr(4, 1). The *dual twistor* space is naturally defined in the normal sense of linear algebra, and its projective version is Gr(4, 3), the space of planes in . Minkowski space is regained as Gr(4, 2), the space of lines in . Intuitively, a space–time point becomes a secondary construct, defined by the space of twistors incident with it.

The complex conjugate structure is given by a map from twistor space to dual twistor space, such that . This defines a pseudo-norm on twistor space by . Twistors such that this pseudo-norm vanishes are called *null* twistors and have the property that they are incident with a real *x*^{a}, and indeed with a whole real null line. For this reason, twistor space can be considered as a complexification of the space of null rays. A remarkable feature of twistor space is that it divides so simply into two halves (of positive and negative pseudo-norm) with the null twistors as the common boundary. Note that to choose twistor space, as opposed to dual twistor space, is to make a strongly chiral choice.

One particular line in has special importance: it corresponds to the two-space spanned by the null twistors of form (*ω*^{A},0). Clearly, there are no finite points incident with any of these twistors. This is the line corresponding to the point in compactified Minkowski space which is the vertex of the null cone at infinity. The antisymmetric bi-twistor *I*^{αβ} describing this line is called the *infinity twistor.* The Poincaré transformations, as a subgroup of the conformal group, can be characterized as those which preserve *I*^{αβ}. And, very importantly, expressions which make no mention of *I*^{αβ} are manifestly invariant under conformal transformations.

On translating the standard relations of first quantization, *helicity* becomes an operator, which is none other than the Euler homogeneity operator. For a more precise statement, see Penrose [4], which also discusses a number of further topics in the foundations of twistor theory. This identification of helicity can be implemented concretely by means of Penrose's contour integral formula, which gives the zero-mass field of helicity *n* corresponding to a twistor function of homogeneity (−*n*−2)—or more properly, an element of a *first cohomology* group. The chirality of twistor space means that fields of opposite helicity are represented by quite different homogeneities, and there is no natural way of adding them, as one might wish to do to express a real Maxwell field. Another striking fact is that the property of being of positive (or negative) frequency can be expressed in terms of the geometry of the two halves of twistor space, without any reference to momenta. For such fields, there is an inner product structure, which can be realized in twistor space as a (many-dimensional) compact contour integral, without reference to space–time coordinates. As such, it is manifestly finite and conformally invariant. Moreover, in the particularly important case of spin 1, it is manifestly gauge invariant.

## 4. Twistor diagrams

This extraordinarily simple inner product structure can be considered as corresponding to *zeroth*-order interactions, i.e. no interaction at all. Penrose proposed [5] that an extension of such many-dimensional contour integration should supply a natural representation of the interactions of fundamental fields. This proposal was made within standard perturbation theory and was intended simply to re-express the standard predictions of Feynman rules. However, it would apply them to finite-normed in- and out-states, rather than to momentum eigenstates. Following this proposal, Penrose wrote down the first *twistor diagrams* for interactions of spin 0, spin , spin 1 massless fields at first order.

As a noteworthy example of this work, Penrose wrote down in [5] a twistor diagram for the first-order massless Compton scattering, with the form of a double box:

Here the black and white vertices represent, respectively, twistor (*Z*^{α}) and dual twistor variables (*W*_{α}) to be integrated over. The spin-1 fields *ϕ*_{1},*ϕ*_{3} are represented by functions of homogeneities (−4), and the spin- fields *ψ*_{2},*ψ*_{4} by functions of homogeneities (−3). An edge between a *Z*^{α} and a *W*_{α} vertex corresponds to a propagator which is the simplest possible object: a power of (*W*_{α}*Z*^{α}), consistent with the constraints of homogeneity. The convention used here, following Penrose's original notation, is that a label *n* signifies homogeneity (−*n*−1).

The diagram thus defines an integrand, and the integrand exhibits manifest conformal invariance. Provided genuine compact contours can be found, this should give a finite amplitude as a multi-linear functional of the external Fock space of states.

Penrose's scheme inspired many developments over the next 20 years and more, but various difficulties prevented any direct transcription of the Feynman diagram rules into a new formalism based on twistor diagrams, as might originally have been hoped. A first problem is that the fundamental example of the internal propagator lines arises when it has homogeneity 0, and this demands the definition of something like . This can be achieved (actually a contour with *boundary* on *W*_{α}*Z*^{α}=0 yields a solution) but, in general, there do not exist contours of this form for all channels, i.e. all choices of positive and negative frequencies of the external fields. This arises from a basic problem in the perturbative treatment of zero-mass-field interaction, namely that there is no sharp distinction between interaction at very low frequency and no interaction at all: this is the forward-direction problem. It corresponds to the existence of a *pole* in momentum space. When amplitudes are left as functions of momenta, this is normally thought of as harmless. But if the in- and out-states are to be genuine finite-normed states, one is faced with integrating this pole over phase space, which gives rise to an infrared divergence. Equivalently, the problem is that the Feynman rules are not in general sufficient to define a finite functional of finite-normed fields, so it is not clear what the twistor diagram is supposed to yield.

A further question lay in discerning what rule might be expected to generate the diagrams, analogous to the derivation of the Feynman rules from a Lagrangian. This was particularly problematic because the diagrams for an amplitude were certainly not uniquely defined. The Compton scattering amplitude addressed by Penrose could actually have been written less symmetrically, but more simply, as either of the two diagrams
which have only a *single* box structure. The details of the correspondence of these diagrams with Compton scattering are described in [6]. It was clear that twistor diagrams would only be specified up to some equivalence class, but little progress was made in understanding what the equivalence relation might be.

Despite these and other problems, the diagram formalism still suggested that objects of this nature, if only they could interpreted correctly, embodied the extraordinary possibility of using just the simplest possible ingredients in complex algebraic geometry, to define a conformally invariant scheme for fundamental scattering amplitudes. The main advance that was made was in the application to pure gauge-theoretic scattering, which by 1990 was recognized as the most fundamental of all. The four-field case was taken far enough to show new interesting features, with a connection to string theory noticed [7].

In retrospect, these investigations were over-influenced by the objective of finding explicit compact contours for all channels, defined for finite-normed external fields. It would have been more productive to concentrate on the formal momentum-space expressions, independent of channel, which had in fact been introduced in Penrose's original discussion of twistor diagrams [5]. These expressions embodied a vital feature—that of *manifest gauge invariance.* Penrose's original Compton scattering diagram showed strikingly that a single diagram could embody a gauge-invariant expression requiring the summation of two Feynman diagrams in the normal representation. Exploration of the four-field pure gauge-theory amplitudes had extended this observation. The extension to more than four fields was attempted and in fact the right expression written down for five gauge fields, but verification by full summation over Feynman diagrams seemed too daunting. Only a simplified version of this problem, double Compton scattering, was pursued [8]. Although, indeed, the manifest gauge invariance of these diagrams was notable, the very limited range of examples offered too slender a base on which to build a general picture. Unfortunately, these twistor-theoretic investigations were pursued in ignorance of the contemporary work by advanced field theorists on gauge field interactions. By using insights from super-string theory they had found important results which completely by-passed the summation of Feynman diagrams, and opened a much wider arena.

## 5. Gauge field theory

In the 1970s, when this investigation of interacting massless fields began, the natural interpretation was that of a relativistic limit of quantum electrodynamics. All that changed in the 1980s, and by the 1990s the theory of massless SU(3) gluons turned the detailed calculation of gauge-theoretic amplitudes, for large numbers of interacting fields, into a practical problem for particle physics. The 2-spinor helicity representation was being used by advanced field theorists, and a very useful notation introduced. Rather than write spinor contractions as (*π*_{1}^{A′}*π*_{2A′}) it is much simpler to write 〈12〉. In this notation, the amazing simplicity of the *Parke–Taylor amplitude* expression [9] is very striking:
5.1This is an expression for the tree-level ‘maximally helicity violating’ (MHV) amplitude involving *n* massless gluons, of momenta *p*_{1},*p*_{2}…*p*_{n}. More strictly, it is a *partial amplitude* or *colour-stripped component* of the full amplitude, a distinction that will be addressed later on. The unlovely term ‘MHV’ refers to the helicities of the gluons. An MHV process occurs when two gluons of negative helicity interact and the out-state consists of (*n*−2) gluons all of negative helicity. However, all the processes related by CPT symmetry are also MHV. So, by convention, the amplitude is expressed in a channel-independent way, as if all the momenta were ingoing. This means that the delta-function imposes the sum of momenta vanishing, and the helicities are such that just two are positive and all the others negative. In this expression, gluons *a* and *b* are the positive-helicity gluons. The reader is referred to the excellent account by Henriette Elvang & Yu-tin Huang [10] for detailed exposition of this and many of the topics touched upon in this review.

Expression (5.1) is enormously simpler than any Feynman diagram summation, which is difficult even for *n*=4 and quite impossible for general *n*. It is worth noting how the distinction between positive and negative helicity may be immediately read off from the *homogeneities* of the spinors: *a* and *b* are of degree +2, while all the rest are of degree −2. This is forced by the way that the spinors have the freedom of definition noted at (2.1) and is also closely related to the homogeneities of twistor functions. It immediately extends, via *N*=4 supersymmetry, to a more general formula applicable to super-fields of all helicities between −1 and +1. (The Parke–Taylor formula thus embraces all the massless field processes that had been studied in twistor diagram theory in the 1990s, which is why it is unfortunate that twistor theorists were unaware of it!)

The simplicity of this form naturally suggested that the calculation of non-MHV amplitudes could also be simplified. Although field theorists had developed many ways of improving upon the raw summation of Feynman diagrams, there was a particular impact on field theory at the end of 2003 arising from the discoveries of Edward Witten [11] on strings in twistor space. These brought together twistor geometry, string theory and advanced field theory. This work gave a twistor-space interpretation of the Parke–Taylor formula and proposed a powerful generalization applicable to more general amplitudes. In the actual calculation of amplitudes, it immediately led to the Cachazo–Svrcek–Witten formalism [12], which then stimulated an investigation leading eventually to the Britto–Cachazo–Feng–Witten recursive formula [13].

This ‘BCFW’ procedure gave an explicit algorithm for tree amplitudes involving any number of gluons, with any combination of helicities. Curiously, the proof of its validity rested on an essentially elementary idea in complex analysis, depending on the idea of partial fractions. It did not require twistors or string theory and could have been seen long before. However, a key idea lay in recognizing the fundamental importance of three-field amplitudes. Such amplitudes are meaningless for real fields with real null momenta, but make formal sense with complexified null momenta. This move into the *complex* was fundamental, and reflected the new influence of twistor space geometry which Witten had injected into the mainstream of thought.

## 6. Confluence of twistor theory and field theory

At the beginning of 2005, the BCFW formalism posed a puzzle. Its formulation seemed to have lost contact with the twistor-string structure that embodied conformal invariance. How could the conformal invariance of the BCFW procedure be made manifest? One answer was given in the observation [14] that the BCFW prescription is equivalent to a procedure in which two twistor diagrams are joined together by a *bridge* to make a larger diagram. As the diagrams are themselves defined in conformally invariant terms, and so too is the bridge, this gives a manifestly conformally invariant expression of the BCFW principle. This observation also meant that for the first time it was possible to write down twistor diagrams for any (tree-level) amplitude, going far beyond the special four- and five-field amplitudes hitherto considered. This was soon generalized to an *N*=4 supersymmetric version [15] (while super-BCFW was also achieved in parallel by other methods). Penrose's diagrams took on new life in this formulation, and old features acquired new interpretations. For instance, the vertices of twistor diagrams, which Penrose had written down in [5] as the generating elements, could now be identified as the three-field amplitudes which had been so vital in the BCFW theory.

The twistor diagram formalism by Hodges [14,15] was ill-defined: the diagrams were left as contour integrals that one could not actually perform. One could certainly say what they meant in terms of momentum space, by applying arguments going back to those of Penrose [5], and obtain some useful new results [16]. But even this procedure, though essentially discrete, finite and combinatorial in nature, was left as an *ad hoc* set of observations; there was no general formula or algorithm for the interpretation of a twistor diagram as an amplitude. This outstanding problem was identified and solved in 2008–9 by Nima Arkani-Hamed, Freddy Cachazo, and their collaborators. The first stage [17] was to study the twistor diagram formalism in the context not of Minkowski space, but of a four-dimensional space of (2,2) signature. The null structure of such a space is quite different and the twistors can be considered as *real*. The requisite integrals could then be stated precisely, without the difficulty of defining contours that pervades the Minkowski space theory. This argument validated the combinatorial structure of the twistor diagrams and in particular the claim that the BCFW procedure corresponds to joining diagrams with a bridge. (The disadvantage of (2,2) signature theory is that it lacks the physical content of scattering theory as a *process* running from past to future. Unfortunately, it seems unlikely that the real integrals of (2,2) theory can be ‘Wick-rotated’ into an integral formalism for genuine scattering in Minkowski space.)

The second stage was to introduce the concept of the twistor-space *Grassmannian* as the correct setting for the evaluation of amplitudes. This replaced the *ad hoc* rules for writing down the momentum-space interpretation of a diagram. Instead, an actual formula for this momentum-space expression could be given. (Note, however, that this new theory abandoned the goal of using finite-normed states in Minkowski space to obtain completely finite functionals; the external states are thought of as momentum states.) Furthermore, the formalism could be taken far beyond the diagrams generated by BCFW procedures. It naturally extended to a much more general picture which included all the leading singularities associated with loop diagrams [18]. Yet more rapid advances took the theory to the loop level itself, or, more precisely, to the *integrands* of loop integrals [19].

This development signalled a new place for twistor geometry in fundamental physical theory. Twistor variables became the primary objects of attention. Since 2008, Arkani-Hamed's group has adopted the view that the primacy of twistor coordinates is established and has taken the lead in expanding and modifying the concept of twistor space so as to encompass more and more powerful elements of physical theory.

## 7. Dual conformal symmetry

One very useful step in this direction came from the observation that another hidden symmetry—that of *dual* conformal symmetry—could naturally be expressed using twistor coordinates. The origin of this idea lies in the characteristic features of scattering amplitudes in SU(N) gauge field theory. Each of the interacting fields in such a theory has a colour, represented as a matrix *Λ*^{α}_{β}, where the indices live in the gauge group. The complete scattering amplitude, at any given order of perturbation theory, will involve these fields and their colours interacting in all possible ways. However, this amplitude can very usefully be broken down into a sum of simpler pieces, based on the topological properties of the Feynman graphs making up the complete amplitude. Some of these graphs will be *planar*, i.e. can be drawn without any edges crossing. All *tree-level* graphs (i.e. without any loops) will have this property. Such planar graphs have a well-defined ordering of the external fields, at least up to dihedral symmetry. We may call this a *ring-order*. It turns out that the dependence of the amplitude on the colours for such graphs is just given by the trace of the product of the *Λ* matrices, in the appropriate ring-order.

The non-planar graphs are just as physical as the planar graphs, and must be included to obtain a physical amplitude. But there are good reasons for concentrating on the planar graph sector as giving a *leading* contribution. It then becomes natural to concentrate on a particular ring-ordering of the external fields, and the *colour-stripped amplitude* associated with just that sub-sector of the total amplitude. (In fact, it is quite common for the colour dependence to be left implicit when discussing amplitudes, which leads to the danger of forgetting that a process with *n* coloured gauge fields actually requires a sum over (*n*−1)!/2 pieces, each corresponding to a different ring-ordering, even when only the planar graphs are counted. This consideration will be brought back to mind when we briefly consider gravitational scattering, below.)

In what follows we restrict our attention to a planar sector with a specified ring-ordering of the external fields. Without loss of generality, the ordering is (123…*n*), with momenta (*p*_{1},*p*_{2}…*p*_{n}). Now, it is easy to see from the nature of planar graphs that the momenta on the edges of any Feynman graph in this sector have a special property. They can involve the external momenta only in a particular manner: only through sums of *consecutive* momenta, like *p*_{2}+*p*_{3} or *p*_{4}+*p*_{5}+*p*_{6}, but not *p*_{3}+*p*_{6}.

These external momenta are constrained by conservation: . At this point, it is useful to recall an idea in elementary mechanics, the ‘triangle of forces’. If three vectors add to zero, then they can be represented on the page by a triangle, such that each edge represents one of the vectors. The same applies to *n* vectors, using a polygon of *n* edges; if the order of the vectors is specified then the polygon is also unique, up to the arbitrary choice of where it is drawn on the page. More formally, if we implement this idea for the momenta (*p*_{1},*p*_{2}…*p*_{n}), we are led to define a polygon in four-space, with vertices at *x*_{i}, where *p*_{i}=*x*_{i}−*x*_{i−1}. We may consider *x*_{n}=*x*_{0} as chosen arbitrarily, and then all the other *x*_{i} are determined. Functions of the *p*_{i} can be translated into functions of the *x*_{i}, and clearly must have the property of being invariant under , where *c* is any constant four-vector. If the functions of the *p*_{i} depend only on sums of consecutive *p*_{i}, we can say more than this. The corresponding function of the *x*_{i} can depend only on vectors of form *x*_{i}−*x*_{j}.

The property of dual conformal symmetry is that if a colour-stripped amplitude is written in this way, it is invariant not only under the translation , but also under any conformal transformation of the *x*-space. In 2008, this was shown to be valid at least up to one loop in *N*=4 planar gauge theory [20]. It is of course immediate from this fact that twistor variables can be used to render these functions *manifestly* dual-conformally invariant. In 2009, the concept of *momentum-twistor* variables was introduced [21], to perform this function, and they have since then been widely adopted for this purpose.

## 8. Polytopes and the amplituhedron

The introduction of momentum twistor variables immediately indicated a new and useful structure governing the tree-level amplitudes for gauge theory. At the simplest level, of MHV amplitudes, there is nothing new to say. The next simplest level is that of the NMHV amplitudes. A process in which *three* negative helicity gluons interact and produce (*n*−3) negative gluons is an NMHV process, along with all its crossing-related versions. For such amplitudes, momentum-twistor coordinates transformed the picture of the expressions that had arisen from use of the BCFW procedure. This procedure resulted in amplitudes being expressed as sums of terms, each containing *spurious poles*. These poles would all cancel in the total expression, but in a very non-obvious way. Moreover, there would also be many different possible versions of such sums, and verifying their equivalence was non-trivial. It may seem surprising that elementary algebra (nothing but identities in rational functions of many variables) should create any serious problem. The reason lies in the delta-function of momenta that is associated with an amplitude. While this is just linear in momentum *vectors*, it amounts to the imposition of four nonlinear identities in the *spinor* variables into which those vectors have been decomposed. For this reason, such identities had generally been checked by computer methods and were resistant to any intuitive interpretation. Momentum twistors revealed just such an intuitive interpretation.

The amplitudes can be regarded as *volumes of polytopes*. The different expressions for them arise through different ways of triangulating the polytopes into simplicial cells. Spurious poles arise from triangulations which introduce extra vertices. Using this insight, it became possible for the first time to see this highly non-trivial NMHV amplitude as a single geometric object. (However, it should be borne in mind that this polytope corresponds to one *colour-stripped component* of the amplitude. The momentum twistors encode momentum conservation very neatly, but the price paid for this elegance is that, in the total amplitude, each of the partial amplitudes requires different momentum-twistor coordinates for its expression.)

The polytopes associated with NMHV amplitudes are in fact four-dimensional, living in a . An extra dimension is added to twistor space to take care of the range of possible helicity allocations. This geometry was sketched by Hodges [21] and further developed by Mason & Skinner [22] and Arkani-Hamed *et al*. [23]. This work also showed that the new momentum-twistor picture is entirely consistent with the Grassmannian picture as developed in the original twistor space.

Naturally, the emergence of these concepts did not take place in isolation from other major developments. The influence of super-string theory, without which the application of twistor geometry to amplitudes would not have made progress, has already been noted. Another parallel development is the reformulation of twistor Lagrangians, or action principles. Penrose [5] had in fact derived the early twistor diagrams from a consideration of twistor Hamiltonians. The Witten twistor-string model stimulated a revival of this approach, notably by Lionel Mason and collaborators. This work has been able to draw on and extend the vast body of work in algebraic geometry built up between 1970 and 2000 to explore the twistor representation. It has also brought in another important idea from field theory, influenced by dualities in string theory, that of the *Wilson loop* [24]. This is important as Wilson loops are related to AdS/CFT formalism and so make contact with an actual *non-perturbative* scattering theory. These authors have shown momentum-twistor geometry to be particularly useful for Wilson loop evaluation and hence another way of generating amplitudes. Meanwhile the emergence of *Yangian* symmetry as the underlying structure encompassing both conformal invariance and dual conformal invariance underpinned all the developments after 2008. Yangian symmetry imposes very strong constraints and dominates the gauge-theoretic arena.

The four-polytopes, which can be identified as the *cyclic polytopes* known to geometers, have a remarkable combinatorial structure. They make the dihedral symmetry as manifest for the NMHV amplitudes, as the Parke–Taylor formula does for the MHV sector. In the simplest case of *six* interacting fields, for instance, the relevant polytope is as follows. In a with homogeneous coordinates *Z*^{a},*a*∈{0,1,2,3,4}, it is bounded by six hyperplanes *W*_{1a}*Z*^{a}=0,…*W*_{6a}*Z*^{a}=0. It has nine vertices, these being at the intersections of hyperplanes 1234, 2345, 3456, 4561, 5612, 1245, 2356, 3461. This polytope can be written as the sum of the three four-simplices defined by 12345, 12356, 13456 or alternatively as the sum of 23456, 12456, 12346. These sums correspond precisely to the three-term expressions given by BCFW for the six-field NMHV amplitude, and explain why they are equal: they are just different representations of the same polytope. The first sum introduces spurious poles associated with vertices 1235, 1345, 1356, which are not actually in the polytope; the second sum likewise has 2346, 1246, 2456. The polytope can of course be divided up in any way one likes, and in particular there is a six-term decomposition, specified in [25], which does not involve any spurious vertices at all, and thus shows how the formulae given in 1988 by Berends & Giele [26] are equivalent to the post-2004 BCFW-generated results.

It proved to be much more difficult than originally expected to continue this line of thought into the description of more general amplitudes. At this point, a completely fresh and radical direction was taken by the group led by Arkani-Hamed. Making contact with leading geometers, a much bolder question was asked and answered. What underlying principle could explain the cyclic polytopes, and what would such a principle suggest for amplitudes in general? The answer, at first sight surprising for a theory based on complex analytic structure, is *positivity.*

At the end of 2012, Arkani-Hamed *et al.* [27] announced the theory of the *positive Grassmannian*. Although in 2009 it was only an isolated observation that certain polytope volumes gave the amplitudes, this new theory offered an explanation of the polytopes from an underlying principle, which then extends to all levels. This positivity principle is so fundamental that *locality* and *unitarity* become derived properties. It should be noted that ‘positivity’ refers to positive conditions on (generalized) twistor coordinates and has nothing to do with reality or positive quantities in Minkowski space–time. It is a concept which is based on the simple ordering of real numbers, on the one-dimensional level, but exploits the fact that this has remarkable higher dimensional analogues.

Notably, this paper refers to *on-shell diagrams* as the primary objects. They are essentially the same as Penrose's original twistor diagrams, though now considered more abstractly and endowed with a well-defined graph-theoretic structure. In particular, the question of the correct equivalence classes of twistor diagrams is completely solved. It will be recalled that this problem was inherent in the diagram formalism from the outset, exemplified by the different possible diagrams for Compton scattering. This example now becomes an illustration in miniature of the structure generated by the graph theory. The equivalence of the two single-box diagrams is in fact a case of the ‘square identity’ which defines the equivalence classes. The validity of Penrose's original double-box structure, however, is an aspect of the capacity of the full diagram theory to represent loop-level amplitudes and the reduction of these to their leading singularities. On top of this, the graph theory exhibits an amazing one-to-one relationship between graphs and the permutation group. The classification problems involved with this question turned out to be intimately related to advanced topics in the pure geometry of the Grassmannian. This analysis also involves an identification of the diagrams as giving a complete analysis of Yangian invariants.

In subsequent papers, Arkani-Hamed & Trnka [28,29] have given the name of the *amplituhedron to the idea of a space encompassing all the amplitudes, for **N*=4 gauge theory in the first instance, but not restricted to that arena. The enlargement from to , needed for the NMHV polytopes, is a foretaste of a much larger extension to complex spaces of arbitrary dimension. The on-shell diagrams are superseded by a more abstract approach, in which the last vestiges of the Feynman diagram calculus disappear altogether. The most striking prediction of the new theory lies in yielding new expressions for loop amplitude integrands which have nothing to do with the joining up of trees into loops. There is much about this theory that is still exploratory. For instance, it is not clear yet whether the concept of ‘volume’, which was so useful for starting off the polytope picture, extends in general. But it can confidently be anticipated that further discoveries will soon transform the picture. As an example, a very recent development describes momentum-twistor diagrams which introduce a new dual diagram structure [30].

## 9. Gravity

Experimental questions have driven the efficient calculation of gluon amplitudes, and there is no such motivation for studying amplitudes in quantum gravity. But theorists have still devoted attention to the problem, with a sense that, despite its apparent difficulty and complexity, there must be radical simplifications and extraordinary properties yet to be unearthed, which would in turn transform other fundamental problems.

Penrose's application of twistor methods to gravity began in the same very early period that saw the emergence of twistor diagrams. It might be remarked that it requires considerable confidence in twistor geometry to believe that it must be relevant to expressing the world of Einstein's gravity. We live in a galaxy with a large, real, phenomenally nonlinear black hole at its centre; we experience the milder but still real and macroscopic effects of curved space–time in our earthly life. But twistor geometry is essentially complex rather than real. Twistor representation makes positive and negative helicity fields behave like quite different entities. This is already puzzling in electromagnetism, where there is no natural twistor mechanism for describing a real field, as the sum of complex-conjugate helicity fields. It is even more puzzling to see how the macroscopic curvature of real space–time can arise within twistor geometry.

Penrose's *nonlinear graviton* construction of 1976 [31] showed that twistor geometry could readily incorporate non-perturbative curvature. But it was restricted to just one helicity. The ‘googly’ problem of incorporating both helicities together in a single geometric structure has resisted attack over the decades. As a general point, one might say that, if twistor geometry is fundamental, it is hard to see why anything is real, rather then complex, and why anything in physics should have even approximate parity symmetry at all.

Another important feature of gravity is that it breaks conformal invariance. The coupling constant (Newton's constant) has a length dimension. At first sight, this does not favour twistor representation. However, this is too faint-hearted an attitude. Twistor variables are good not only for conformally invariant structures, but also for making manifest the *breaking* of conformal invariance. The breaking of conformal invariance in linearized gravity is then made manifest through the appearance of the *infinity twistor*. This breaking is of a very mild kind: only through *numerator factors*. Although they make the formulae a little more complicated, these numerators do not interfere with the essential ideas developed for gauge-theoretic amplitudes.

As in gauge theory, three-field amplitudes can be defined, and then BCFW recursion can be used to build up any required tree amplitude. This was done at an early stage by Cachazo & Svrcek [32]. It is straightforward to define a supersymmetric extension, and, as in gauge theory, corresponding twistor diagrams can be written down. The difficulty lies in making any real sense of the resulting sums of terms. The puzzle is particularly acute with the *N*=8 supersymmetric extension, for which an amplitude involving *n* super-fields is completely symmetric. Yet, even for *n*=4, it is hard to see any symmetry in the expressions obtained. In the case of MHV processes, a number of expressions were known [33,34,35], but without any manifest symmetry. This situation changed in 2012 with the observation of a new formula for gravitational MHV amplitudes [36], which made their total symmetry manifest. This arose out of an investigation using *N*=7 supersymmetric twistor diagrams, exploiting the simplicity of the numerator factors expressing the breaking of conformal invariance.

The key feature of this new formula is that the symmetry arises from the elementary properties of a *determinant*. The formula was soon developed into a much more powerful formalism for all tree amplitudes [37], and since then the determinantal structures have been greatly generalized. This has prompted investigations relating gravitational to gauge-theoretic amplitudes in new ways, and there are indications that twistor geometry will emerge as the underlying principle behind these new structures.

It cannot yet be said that twistor geometry has been established as the definitive foundation for gravitational scattering. As already noted, it should be appreciated there are considerable difficulties to be faced up to. Besides the breaking of conformal invariance, linearized gravity also differs from gauge theory in that it has no analogue of the colour-ordering that makes the definition of momentum twistors so easy and useful. For gravity, it is the symmetric group, not the dihedral group, that dominates. There is scope for considering functions of momentum twistors which are order-independent [36], but this has not yet yielded significant progress. It is also striking that other leading approaches to gravitational scattering have no immediately obvious contact with twistor geometry. As an early consequence of string theory, the Kawai–Lewellen–Tye (KLT) relation [38] showed how to obtain gravitational amplitudes by a ‘squaring’ of gauge-theoretic amplitudes. But this ‘squaring’ is expressed in standard momentum-space terms. So is a more recent development, based on the Bern–Carrasco–Johansson (BCJ) relations and colour-kinematic duality [39].

However, it is also notable that the development of the determinantal formula has tied in closely with insights from the twistor-string perspective. Mason & Skinner's [34] derivation of the gravitational MHV amplitude came from a twistor Lagrangian approach, and this perspective has been maintained in the more recent work following up the determinantal formula. The work of David Skinner [40] goes back to the twistor-string picture and in particular brings in the remarkable fact, which so far has not been much exploited, that twistor geometry can represent de Sitter or anti-de Sitter space just as well as Minkowski space. The only modification necessary is that the ‘infinity twistor’ becomes non-degenerate. This non-degenerate *I*_{αβ} fits elegantly into a determinantal structure. In this approach, gravity is naturally formulated within this more general setting, and flat-space properties extracted by taking a (non-trivial) limit. Since then, further work on *ambitwistor strings* by Geyer *et al*. [41] has forged a strong connection between twistor geometry and the work of Cachazo *et al.* [42] which has given the most powerful extension of the determinantal structure, including contact with both KLT and BCJ relations.

The questions raised by this investigation are not in fact restricted to the gravitational setting. Even in gauge theory we are faced by the problem of how to escape the restriction to ring-ordered partial amplitudes, so as to express the relationships between components from different ring-order sectors, and to express the non-planar components of amplitudes. The situation seems to call for some striking new idea that will connect these various questions and approaches.

## 10. From the 1970s to the 2010s

Roger Penrose's original motivation for twistor theory was that it would give a radically new framework for the description of physics. Although twistor-geometric descriptions of known physical phenomena must necessarily be consistent with standard space–time theory, the twistorial reformulation would suggest new ideas which would not or could not have been formulated in space–time. The concept of the positive Grassmannian, as developed by Nima Arkani-Hamed and his collaborators, has already given an example of such a new idea.

Contact with experiment has been transformed since the 1970s, by virtue of the importance of quantum chromodynamics, established as the theory of the strong nuclear force. At tree level, where QCD is indistinguishable from *N*=4 super-symmetric gauge theory, one can say that twistor methods are already practical and very efficient, for reasons explained in detail in [43]. At loop level, the super-symmetric gauge theory can only be regarded as a first-step towards QCD. The remarks made in [27] suggest there are good reasons why the positive Grassmannian approach to loop amplitudes is not actually restricted in application to the super-symmetric theory, and that there will be ways of systematizing the ultraviolet divergences of QCD within this structure. It must be said, however, that so far it is the loop *integrands* that have been completely transformed by the new developments. Actually performing the (regularized) loop integrals to extract amplitudes is another matter. However, several different approaches using twistor variables are being undertaken, for example by Lipstein & Mason [44].

There is a general question about how the breaking of conformal symmetry enters into fundamental physics, which has been a feature of the twistor philosophy from the outset. In standard space–time physics, both this symmetry and the breaking of it are obscured. Once it comes into clear focus with twistor coordinates, one may hope to distinguish and inter-relate the different aspects of conformal symmetry breaking. There is the breaking of symmetry due to the fact that amplitudes are measured ‘at infinity’. Length scales associated with infrared and ultraviolet regularization also require breaking conformal symmetry. Gravity breaks conformal symmetry in a different way. The Higgs field, in the Standard Model, is the source of mass, which also breaks the symmetry. We now also need to take seriously the enormous length scale given by the cosmological constant. All these topics are on the agenda.

One of the approaches to regulating divergences involves studying a ‘deformation’ of amplitudes which retains conformal (and indeed Yangian) symmetry. The authors of a recent paper [45] remark that ‘We feel very encouraged by the interesting work of Penrose and Hodges, who considered the very same deformations already since the early days of the twistor approach’. This is over-generous, since in the early days there was no systematic way of thinking about loop-level amplitudes at all, whereas now the Grassmannian provides such a framework. Nevertheless, there may still be material from the early period of twistor diagram investigation, especially its use of many-dimensional compact contour integration, that is relevant to current questions. One idea from the early days is that regularization might make use of the *scale* dimension of twistor space [46]. So far, the theory of scattering amplitudes has greatly developed the linear algebra and combinatorics of *projective* twistor geometry. The paper by Roger Penrose [4] brings out the importance of structures in *non*-projective twistor space, building upon the 1970s definitions of deformed twistor spaces.

There also remains the original idea of ensuring that the external states are described entirely within a twistor-geometric setting, rather than borrowing the space–time concept of momentum eigenstate. A 1970s idea of ‘elemental’ states, which are formally parametrized by twistors rather than by momenta, has had useful application in recent work [24] and is important in making sense of crossing symmetry in a twistor-geometric setting. The physical interpretation of this idea deserves more attention, and it might be helpful also in generalizing from Minkowski to de Sitter geometry.

More generally, it would be consistent with the radicalism of Penrose's original programme, and with the revolution introduced by the amplituhedron, if attention were paid not only to computing the interactions of quantum fields, but also to probing the very foundations of quantum field theory. The paper by Roger Penrose [4] indicates a possible direction.

This review is intended to illustrate the importance of persistence in following bold new ideas, but also of openness to other approaches and the willingness to synthesize. Given the rate of progress since 1967, even more impressive advances may be expected by 2061.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Footnotes

One contribution of 13 to a theme issue ‘New geometric concepts in the foundations of physics’.

- Accepted April 27, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.