Palatial twistor theory and the twistor googly problem

1. Introduction: basic twistor algebra

The central aim of twistor theory is to provide a distinctive formalism, specific to the description of basic physics. Space–time points are taken as secondary constructs in the twistor approach. An individual twistor can be regarded as representing the entire history of a free classical massless particle—usually with non-zero spin, of a definite helicity s. If s=0, such a history is a null geodesic—henceforth called a ray—which provides the most immediate picture of what is referred to as a null twistor. An individual space–time point is identified in terms of the family of rays through that point. In relativity theory, such a family has the structure of a conformal sphere, interpreted in twistor theory as a Riemann sphere, i.e. a complex projective line. Twistor theory gains much of its strength from complex geometry and analysis, bringing together many features of relativity and quantum mechanics.

So long as space–time curvature can be ignored, the basic twistor space is a complex four-dimensional vector space with pseudo-Hermitian scalar product of split signature (++−−). Geometrical notions are often best expressed in terms of the projective twistor space (a complex projective 3-space ). I use abstract indices, so a twistor Z (an element of ) can be written Z^α. Its complex conjugate (i.e. is a corresponding element of the dual twistor space , the scalar product between twistors Z and Y being written alternatively as

and the norm of Z is We say that a twistor is right-handed, left-handed or null according as ∥Z∥>0, ∥Z∥<0 or ∥Z∥=0, the respective portions of twistor space being , and , with projective versions , and .

Any projective null twistor Z (element of ) represents a ray (null geodesic) in Minkowski 4-space , where for the full correspondence we must include rays at infinity—generators of the ‘light cone at infinity’ in the compactified Minkowski space , a smooth compact conformal-Lorentzian 4-space, topologically S¹×S³, which can be identified as the space of lines through the origin in the zero-locus of a quadratic form of signature (++−−−−). The physical interpretation for a projective null twistor is thus the world-line of a classical (non-spinning) free massless particle, where idealized particles at infinity are also included. The orthogonality condition (or, equivalently, its complex conjugate between null twistors Y and Z has the direct space–time interpretation that the corresponding rays y and z in intersect (possibly at infinity). The twistor Zitself, up to the phase freedom Z↦e^iθZ (θ real)—so that a positive real-number measure of scaling is incorporated—provides a 4-momentum for the massless particle. We shall be seeing later that the phase also has a key geometrical role to play in the palatial twistor theory introduced in §5.

If the particle possesses a non-zerospin, which must be directed parallel or anti-parallel to its velocity—positive or negative helicitys, respectively—then its space–time trajectory is not precisely (relativistically invariantly) defined as a world-line, but can be specified in terms of its 4-momentum p_a and 6-angular momentum M^ab about some chosen space–time origin point O. These must be subject to

(curved or square brackets around indices, respectively, denoting symmetric or antisymmetric parts), where ε_abcd=ε_[abcd] is the Levi-Civita tensor fixed by its component value ε₀₁₂₃=1 in a right-handed orthonormal Minkowskian frame (with time-axis basis vector , so p₀ is the particle's energy, in units where the speed of light c=1). The quantity s is the helicity, positive for right-handed spin and negative for left-handed.

These quantities are explicitly represented algebraically in terms of a twistor Z (uniquely defined up to phase) and its complex conjugate dual twistor . To see this, we need the expression of a twistor Z in terms of its 2-spinor parts [1,2], where we write

The complex conjugate of Z is the dual twistor A general dual twistor W=(λ,μ) has the explicit index form W_α=(λ_A,μ^A′).

A massless particle's 4-momentum p_a and 6-angular momentum M^ab (in 2-spinor abstract-index notation [1]) can constructed from a twistor's spinor parts by

and all the above conditions are satisfied, provided that π_A′≠0. Conversely, the twistor Z (with π_A′≠0) is determined, uniquely up to a phase multiplier e^iθ, by p_a and M^ab, subject to these conditions. The helicity s finds the very simple (and fundamental) expression

Under a change of origin O↦q, the spinor parts of Z undergo

where q^a is the position vector . For a dual twistor W, with W_α=(λ_A,μ^A′), we correspondingly have This is consistent with the standard transformation of M^ab (and p_a) under origin change, where the position vector x^a of a space–time point x correspondingly undergoes This is consistent also with the incidence relationbetween a twistor Z=(ω,π) and a space–time point x, which is the condition for the complex projective line X (Riemann sphere) represented by x in , should pass through the point Z in . In matrix terms, this incidence relation is where (t,x,y,z) are standard Minkowski space–time coordinates (with c=1) for x. When Z is a null twistor, the real points x which are incident with Z are simply the points that constitute the ray z in (at infinity, in , if π=0). When Z is non-null, then there are no real points incident with Z. However, there will be complex points—points of the complexified space–time incident with Z (but if π=0, only at infinity, in ). This will have importance for us in §3.

2. Wave functions for massless particles

Up to this point, we have been concerned with classical theory. Quantum twistor theory involves commutation laws

and [2,3], where now the twistors are taken to be linear operators generating a non-commutative algebra (with an important role in §§5 and 6), acting on some appropriate space. We shall think of this space as a quantum ‘ket-space’ |…〉 of some kind [4], but it is best not to be specific about this, just now. We could alternatively think of our operators as dual twistors, subject to the commutation laws and which is the same thing as before, but with re-labelled as W_α.

These commutation laws are almost implied by the standard quantum commutators for 4-position and 4-momentum

but there appears to be an additional input related to the issue of helicity. By direct calculation, we may verify that the twistor commutation laws reproduce exactly the (more complicated-looking) commutation laws for p_a and M^ab that arise from their roles as translation and Lorentz-rotation generators of the Poincaré group. In this calculation, there is no factor-ordering ambiguity in the expressions for p_a and M^ab in terms of the spinor parts of Z^α and (owing to the symmetry brackets). Yet, the calculation for the helicity s (writing the operator as s) yields In accordance with standard quantum-mechanical procedures, in order to express wave function for massless particles in twistor terms, we need functions of Z^α that are ‘independent of ’. This means ‘annihilated by ’, i.e. holomorphic in Z^α (Cauchy–Riemann equations). Thus, a twistor wave function (in the Z-description) is holomorphic in Z and the operators representing Z^α and act Alternatively, we could be thinking of functions of that are ‘independent of Z^β’, i.e. anti-holomorphic in Z^α. Here, it would be better to re-name as W_α and consider functions holomorphic in W_α. Accordingly, in the dual twistor W-description, a wave function must be holomorphic in W and we have the operators representing and W_α, again satisfying the required commutation relations, but now with If we are asking that our wave function describe a (massless) particle of definite helicity, then we need to put it into an eigenstate of the helicity operators, which, by the above, is in the Z-description, and in the W-description. These are simply displaced Euler homogeneity operatorsso a helicity eigenstate, with eigenvalue s, in the Z-description requires a holomorphic twistor wave function f(Z) that is homogeneous of degree where I henceforth adopt . Then 2s is an integer (odd for a fermion and even for a boson). In the W-description, the dual twistor wave function is homogeneous of degree , where In ordinary space–time terms, the position-space wave function of a massless particle of helicity 2s [1] satisfies a field equation, expressible in the 2-spinor form for the integer 2s satisfying s<0, s=0, or s>0, respectively, where we have total symmetry for each of the |2s|-index quantities Some of these field equations are more familiar in tensor form. In the case |s|=1, we can write (in abstract indices, a pair of capital spinor indices, unprimed and primed, standing for a single tensor index) the antisymmetric quantities ε_AB and ε_A′B′ giving the symplectic structure of the two-dimensional spin spaces, related to the space–time metric by Then our field equations become Maxwell's free-field equations on F_ab (∇_[aF_bc]=0, ∇^aF_ab=0). When |s|=2 we write finding that K_abcd has the symmetries of a trace-free Riemann (or Weyl) tensor K_abcd=K_[cd][ab], K_[abc]d=0, K_abc^a=0, satisfying the vacuum Bianchi identities: ∇_[aK_bc]de=0. This gives us the weak-field (linearized) Einstein vacuum equations. Moreover, the case |s|=1/2 gives the Dirac–Weyl massless neutrino (or anti-neutrino) equation.

When |s|=1 or |s|=2 (or |s|=0) we can read our equations either classically or quantum mechanically. With quantum wave functions, we would normally demand a positive-frequency condition on the ψ and quantities; then the ψ-part would describe the left-handed (i.e. negative helicity) component of the particle's wave function, the -part describing the right-handed (positive-helicity) component. On the other hand, we can obtain the classical solutions of these field equations if, instead of demanding positive frequency, we demand that F_ab or K_abcd be real, i.e. that the ψ-part and -part be complex conjugates

The part of F_ab or of K_abcd involving ψ is called the anti-self-dual part and that involving the self-dual part.

What is the connection between the twistor wave function f(Z), or dual twistor wave function , with these space–time equations? In most direct terms, it is given (for constants k, k′) by contour integrals [2]:

and Here AB…E or A′B′…E′ are |2s| in number, and the 1-form δπ is The contour lies within the Riemann sphere, in , of twistors Z=(ω,π) satisfying the incidence relation ω=ix•π (which removes the ω-dependence, introducing x-dependence, and then the integration removes the π-dependence leaving us with just x-dependence). Satisfaction of the field equations is an immediate consequence of these expressions. The 2-form ‘d²π’=(1/2) dδπ=dπ_0′∧dπ_1′ is sometimes more appropriate to use, rather than δπ, the contour then being two-dimensional, lying in rather than . In the dual twistor description, there are corresponding expressions.

3. Twistor cohomology

For the above contour integral expressions to give non-zero answers, it is necessary that the holomorphic function f possess singularities in appropriate regions. The exact nature of the singularity and contour locations in relation to the domains of non-singularity of the resulting ψ and fields had initially been somewhat puzzling—as had the conformal/Poincaré non-invariance of such domains—until it later became clear [2,5] that twistor wave functions are really to be thought of as elements of sheaf cohomology. We shall need to understand just the basics of this. It is easiest in terms of a Čech description and this is the route that I follow here.

The issue is most clear-cut in the case of wave functions, and for these we require a condition of positive frequency. A convenient description of this is simply that our space–time functions (ψ or ) extend holomorphically into the forward-tube domain, which is the set of complex space–time points x+iy (points of ) for which the imaginary part y is timelike and past-pointing. In twistor terms, complex space–time points are represented by complex projective lines in ; those in by lines entirely within .

Accordingly, for a twistor wave function, we are interested, specifically, in holomorphic functions f whose ‘domain’ is in some sense the region . But what sense can this be? In order for the contour integration in the above expressions to work, we require that it have singularities in two disconnected closed subsets of which intersect all the complex lines—these being Riemann spheres—in . The function f is then holomorphic throughout the region that is complementary, within , to these two singular regions. Thus, any complex line R that lies in (R representing a complex point r in ) intersects , and it does so in an annular way, separating the two regions where f is singular. For any such line R in , the contour integral is then a loop within this annulus on the Riemann sphere, separating the two regions of singularity.

A more appropriate way to think about this geometry, however, is in terms of a covering of by two open sets and , the function f being taken holomorphic throughout their intersection, within which the contour is located:

If any function holomorphic throughout or else holomorphic throughout is added to f, this makes no difference to the result of the contour integral, each such additional contribution adding nothing to the answer, as the contour can be deformed away on one side or the other of the Riemann sphere. We can generalize this to coverings by many open sets , with functions f_ij (=−f_ji) defined on intersections , taken modulo functions h_k defined on entire sets in the manner of Čech (sheaf) cohomology, where refinements of such coverings and direct limits are taken when required. It is not necessary to go into the details of this here, as we shall find that 2-set coverings will be sufficient for our needs. The upshot is that twistor 1-particle wave functions should really be thought of, mathematically, as elements of holomorphic first (sheaf) cohomology on . For more details, see [2,5].

The matter that will concern us particularly, however, is not to do with detailed issues concerning this standard linear cohomology, but with the way in which such twistor cohmology ideas can sometimes extend to nonlinear generalizations that express basic physical interactions. Most specifically, I shall be concerned with the way that the linear massless fields for spin 2 (|2s|=4) can be generalized to nonlinear (vacuum) general relativity, and also to gauge-field interactions of electromagnetism and Yang–Mills theory (|2s|=2). The basic constructions, namely those of the nonlinear graviton and the Ward gauge-field construction, were found almost 40 years ago [6–8], but they dealt only with the left-handed helicity interactions (s=−2 and s=−1) in standard twistor conventions. I indicate both of these only very briefly in the next paragraph, but we shall come to the gravitational construction in more detail in §4. The googly problem has been the issue of extending such procedures to include, also, the right-handed helicity in a way that allows both to be taken into consideration simultaneously as part of the same procedure. (It may be noted that ‘googly’ is a term used in the game of cricket for a ball bowled with right-handed helicity using the apparent action that would normally give rise to left-handed helicity.) This issue will be addressed in §§5 and 6.

The procedures of Čech first cohomology, just referred to, involve taking a covering, by open sets, of the region of twistor space under consideration and regarding a twistor function (or family of functions) as something ‘passive’ that simply resides on the pairwise intersections of the open sets of the cover, the cohomology element provided by this family of functions having no active geometrical influence on the twistor space itself. However, under appropriate circumstances such functions can indeed be considered to be playing an active role, such as with the transition functions between ‘coordinate patches’ in the piecing together of a curved manifold, or by constructing a non-trivial bundle over a given manifold by gluing together pieces of trivial bundles. These two procedures are indeed what are involved in the nonlinear graviton and Ward constructions, respectively [6–8]. In the next section, I describe (in outline) how this works in the gravitational case.

4. Nonlinear graviton construction

The nonlinear graviton construction provides a twistor representation of a curved complex-Riemannian 4-space , where we find a very simple way of ensuring that is an Einstein manifold, i.e. R_ab=Λg_ab (for given cosmological constant Λ). This works only in the case where has anti-self-dual Weyl curvature but is otherwise completely general. In standard flat-space twistor theory (see §1), a complex space–time point r in is represented by a complex lineR (a Riemann sphere) in . A very similar situation holds for , applying to any small-enough open neighbourhood of a point , and we examine how an arbitrary point r of arises as a corresponding lineR (Riemann sphere in an appropriate topological family) in a certain complex 3-manifold , a deformed version of a tubular neighbourhood of a line P in .

A striking feature of this construction is that there is nolocal curvature information in the ‘curved’ twistor space (where is its projective version), despite the fact that there is actual local curvature in the space that is constructed from . This curvature arises entirely from global features of . Locally, would be identical, when Λ=0, to the twistor space of §1. When Λ≠0, it would be locally identical, instead, to the twistor space arising from de Sitter 4-space with (positive) cosmological constant Λ (or anti-de Sitter space if Λ<0). In order to clarify this distinction, I need first to be more explicit about the structure that actually acquires in order that the space–time metric can be encoded into 's structure, in these two cases.

Both in the case of Minkowski space and de Sitter space (and also in anti-de Sitter space) there are particular antisymmetric 2-valent twistors, fixing the metric structure of the space–time, which are referred to as infinity twistors [2]. These are I^αβ and I_αβ, taken to be both complex conjugates and duals of one another:

and where ε_αβρσ and ε^αβρσ are Levi-Civita twistors, fixed by their antisymmetry and ε₀₁₂₃=1=ε⁰¹²³ in standard twistor coordinates. For the infinity twistors, we have, in standard twistor descriptions For de Sitter space , the infinity twistors provide a complex symplectic structure defined by the 2-form: Also there is a symplectic potential1-form When Λ=0, this symplectic structure becomes degenerate, the matrices for I_αβ and I^αβ becoming singular. When Λ≠0, they are essentially inverses of one another: but annihilate each other if Λ=0.

To construct , we consider our above tubular neighbourhood , of a line P in . Let us regard the Riemann sphere P as made up of two slightly extended (open) hemispherical regions whose union is P and whose intersection is an annular region, as is appropriate for the contour integrals of §2. Each hemispherical region is to be locally thickened out into the ambient complex 3-space to open sets and , each of Euclidean topology , so that

the annular intersection having topology , a thickened up version of the overlap between the extended hemispheres, as for the contour integrals earlier. The idea is to glue to in a seamless way, so that the twistor-space complex-manifold structure matches on the overlap and so also does the structure—that I call the I-structure—defined by I^αβ and I_αβ, for a given value of Λ.

However, as things stand, this I-structure is not adequately defined, the infinity twistors being given in relation to the non-projective space , rather than . We need to phrase things in terms of ‘non-projective’ complex 4-spaces , where is some subregion of . Here, will be a -bundle over . (I am excluding the origin of in these considerations.) The -fibres of are integral curves of the Euler operator Υ. Thus, the ‘I-structure’ of really refers to , the Euler operator Υ being assumed given on for any in . We may note, also, that when Υ is applied to (i.e. contracted with) the 2-form , we get the 1-form , so that, in the presence of Υ, the I-structure of any such is simply equivalent to the complex symplectic structure (degenerate when Λ=0) restricted to .

We shall first see how to deform merely infinitesimally, where we start with the two regions and overlapping along to make , as in §3. We can then obtain a general finite small deformation, preserving the I-structure, by ‘exponentiating’ the procedure considered at the end of §2 for a linearized anti-self-dual gravitational field. There we took a ‘passive’ twistor function f(Z), homogeneous of degree +2, defined throughout the intersection region , where is glued to , but we are now to think of f(Z) in an active way, telling us how to ‘slide’ the patches and across each other, preserving the matching of their I-structures. Infinitesimally, we can achieve this by performing an infinitesimal shift along the zero homogeneity vector field:

We see, immediately, noting vanishing of the Lie derivative of I^ρσ with respect to this vector field (and its zero homogeneity, whereby Υ is also preserved) that this infinitesimal shift preserves the I-structure. When exponentiated, this provides us with a space which encodes a genuinely curved complex 4-space .

It turns out that any complex-Riemann coformally anti-self-dual complex 4-space solution of the Einstein (Λ-)vacuum equations can be locally described from such a deformed twistor space with a globally defined I-structure. How does this work? First, we need the ‘lines’ within a (obtained by such means). I cannot go into full details here, but theorems of Kodaira and others can be invoked to establish the fact that spaces , constructed in this way (and not deviating too far from the initial ) contain a complex 4-parameter family of ‘lines’ defined solely by the fact that they are imbedded Riemann spheres, with complex structure inherited from the ambient and belonging to the appropriate topological family, these lines being represented as the points of the space . An explicit geometrical procedure involving 's I-structure then naturally assigns a complex-Riemannian metric g to [6,8], where g turns out to be anti-self-dual and Einstein. (The complex-conformal anti-self-dual metric of is defined by 's complex-manifold structure, even without 's I-structure, simply from the fact that intersecting lines in correspond to null-separated points in [6,8].)

5. Palatial patching and twistor quantization

We have seen that not only can linearized vacuum general relativity be encompassed in a natural (albeit somewhat surprising) way by twistor theory, but so also can the nonlinear theory—albeit restricted, as yet, to the left-helicity case. This provides some encouragement for the twistor programme, as a means towards describing Nature. Yet, these positive features were profoundly offset by the apparent inability of the theory to resolve the so-called ‘googly problem’, whose resolution ought to allow both left- and right-helicities to be twistorially represented together, according to full nonlinear Einstein theory. It is therefore fortunate that a new outlook is to hand, offering genuine hope for a complete solution to this long-standing conundrum.

The key idea behind this altered outlook is to regard the underlying twistor structure as being modelled, in effect, not on the space but on the non-commutative (holomorphic) twistor quantum algebra referred to at the beginning of §2. For a (conformally) curved space–time, we would have a deformed such algebra that is, in an appropriate local sense, the same as , but whose global structure would encode the entire (conformal) geometry of a given curved space–time . The algebra itself is to be thought of as, in effect, the algebra of linear operators acting on (germs of?) holomorphic entities of some twistorial kind, constituting an imagined ‘ket-space’ |…〉 envisaged at the right of all these operators. An indication of this is given at the end of this section, but a full understanding of what is required has not yet become altogether clear. (The choice of ket-space is basically the notion, in standard quantum mechanics, of choosing a ‘complete set of commuting variables’. It is closely related to the issue of choosing a ‘polarization’ in geometric quantization [9].) In basic terms, is to be taken as the (Heisenberg) algebra generated by Z^α and ∂/∂Z^α, but where infinite series in these (non-commuting) operators would also need to be considered as belonging to . This raises issues of convergence and locality that need to be sorted out in due course, but for present purposes I shall ignore these more subtle issues and merely explain the general idea.

As with the left-handed construction outlined in §4, we attempt to build our entire structure from pairs of local pieces, analogous to and , each containing no local information specific to , whose intersection seamlessly continues the twistor structure from one to the other. Yet, the union encodes the (Lorentzian) conformal structure of a local region of . When this ‘twistor structure’ appropriately incorporates an I-structure, as in §4, then automatically satisfies the Einstein Λ-vacuum equations—completely generally! At least that is the idea.

We are now concerned with patching together two relevant ‘regions’ of the algebra, corresponding to two ‘pieces’ like and , rather than simply patching together two pieces of a manifold, as in §4. As yet, it is not altogether clear how this is to be interpreted, as we do not actually have a twistor space, like or , leading to or , above. Instead, we appear to have a form of non-commutative geometry [10] arising from some kind of deformation of the algebra .

Fortunately, for a real space–time , we can explicitly construct a global (real) 5-manifold analogous to the 5-space and, moreover, a non-projective 7-dimensional version analogous to the 7-space . The idea is to use to provide us with a key part of 's structure. The curved is the space whose points represent the individual null geodesics—or rays—in . It is best to assume that is globally hyperbolic, as this ensures that is a Hausdorff (5-real-dimensional) space [11]. If we include a (null-vector) momentump (index form p_a), pointing along each ray and parallel-propagated along it—to give what I call a scaled ray—then we get a 6-real-dimensional space , whose points represent these scaled rays. In fact, is naturally a realsymplectic 6-manifold, with symplectic potential 1-form Φ and closed symplectic 2-form Σ, given [2] by

(where, in coordinate notation, ‘dx^a’ stands for the coordinate 1-form basis, and in abstract indices dx^a is just a ‘Kronecker delta’ translating the abstract index on ‘p_a’ to conventional 1-form notation). When , so our symplectic 6-manifold is the canonically given , we can apply twistor notation, finding Equality of the two expressions for Φ follows from the nullity of Z^α, since .

At this point, the idea is to appeal to the procedures of geometric quantization [9], according to which a bundle connection is provided, the bundle being a circle bundle over the symplectic manifold (here ) under consideration, the curvature of this connection being the given 2-form Φ. In the present situation, we are fortunate in the fact that the appropriate circle bundle, namely , is already to hand, the circle being the phase freedom in π_A′, where the momentum null (co-)vector p_a, taken to scale the rays, is spinorially split:

The 2-spinor π_A′ is also taken parallel-propagated along each ray γ—providing a π-scaled ray Γ—the phase freedom applying to each ray as a whole. This defines our required bundle 7-space . In fact (up to a twofold sign ambiguity) the phase is geometrically realized as a null 2-plane (‘flag plane’ [1]) through the null-vector direction determined by p^a. In the flat-space case, this phase freedom is simply the Z↦e^iθZ noted in §1, our circle bundle over being .

The geometric (pre-)quantization connection [9] is defined by the 1-form , in the canonical flat case). The connection applies to helicity-charged fields on . This ‘helicity charge’ refers to a dependence on the phase e^iθ above, and would be determined by the eigenvalue of the homogeneity operator Υ. This connection is consistent with the flat-space replacements

and gives them geometrical meaning for operations within also for a curved .

This does not, however, extend unambiguously to curved analogues of and . The idea would be that, although locally possible, such extensions would be essentially non-unique, this freedom playing an important role for the palatial patching procedures. The key issue is the holomorphicity that the above replacements achieve, eliminating the anti-holomorphic quantity in favour of the holomorphic Z^α throughout. The intention is that by such means, the algebra would be maintained as entirely holomorphic, as required.

In the case of flat (or de Sitter) space–time, the above replacements are not only well defined but also fully consistent with the twistor formalism with regard to a notion of Hermitian inner product 〈g|f〉. This applies to twistor functions f(Z^α) and g(Z^α) (actually representatives of first cohomology), being bilinear in and f(Z^α), and is, in an appropriate sense, positive definite. Merely for notational reasons in what follows, it will be convenient to express this inner product in terms of a holomorphic bilinear scalar product {h|f} between functions and f(Z^α):

due to Andrew Hodges (2014, unpublished data; see also [3]), involving two eight-dimensional contour integrations lying in , the first having a seven-dimensional (S⁷) boundary in W•Z=0, the second being closed, where and are the natural complex 4-volume forms defined on (by ε^αβρσ) and on (by ε_αβρσ), respectively.

Writing ∂_α for ∂/∂Z^α, we find, from various integrations by parts, that

Accordingly, ∂/∂Z^α and −W_α have identical effects, when inserted between any ‘bra’ {…| and ‘ket’ |…}. This justifies the correspondence of §2, since in the Hermitian inner product, the variable W_α is playing the role of (though independent of Z^α for the purposes of the contour integration).

A similar calculation gives us the relation with (playing the role of , which at first sight appears to give the correspondence , with the wrong sign. However, this is a notational confusion, because if we think of the operator as being inserted between the {…| and the |…} in the expression , it would really acting to the left, as ; i.e. . From integration by parts, this is equivalent to inserted between the {…| and the |…}, acting towards the right.

This is an important point of the notation. In §2, we were thinking of all the quantities Z^α, ∂/∂Z^α, , , as acting on what follows to the right of them (ultimately on some unspecified ket-space |…〉), and the sign in the commutation relations in §2 is dependent on this. Here, we are taking of the ket-space as being represented by holomorphic first cohomology in the Z^α variables. Accordingly, we indeed get consistency also with the correspondence of §2. The bra space 〈…| would be represented by anti-holomorphic cohomology, in term of the conjugate variables, which would be consistent with the reverse operator ordering throughout.

6. Local twistors, palatial algebra and Einstein's equations

When it comes to the deformed algebra needed for a curved space–time, we have to allow that no specific ket-space |…〉 be singled out globally to represent the algebra, as in §5, but that we might have different such representations, each defined only ‘locally’, in some sense, but where the ‘Heisenberg algebra’ would be independent of the particular |…〉 representation, and isomorphic, locally, to that provided at the end of §5, namely . A central question arises here, concerning the appropriate meaning of ‘local’ in this context. Although these issues are not yet fully resolved, we do have the (Hausdorff) space defined globally, and we have a well-defined notion of local neighbourhoods within .

In fact, assigned to each single point of , i.e. to each ray γ in (and therefore also to each point Γ of , providing a π-scaling for γ), there is a canonically and conformally invariantly defined flat twistor space , which plays a role of a kind of ‘complex cotangent space’ to at Γ. This is obtained from the notions of a local twistor, and local twistor transport [2]. A local twistor is a quantity Z^α=(ω^A,π_A′), defined at a point q of the space–time , which transforms as

under a rescaling of 's metric, according to g_ab↦Ω²g_ab. Since a local twistor is a quantity that can be assigned separately to each point of , we need to be careful when relating this concept to the notion of twistor used earlier, in §§1–3, where the 2-spinor pair (ω^A,π_A′) referred globally to the (flat) space–time , assigning coordinates to the complex 4-space that is associated with . The notation in this section does correspond exactly to that of §§1–3, but only if we think of q as at the particular space–time origin point O.

Recall from §1 that, when the origin is displaced to a general point q (whose position vector from O is q^a), the twistor (ω^A, π_A′) defined with respect to O becomes (ω^A−iq^AA′π_A′, π_A′) when defined with respect to q. The local twistor perspective used here is that (ω^A, π_A′), defined at the point O, when carried to q by local twistor transport, becomes (ω^A−iq^AA′π_A′, π_A′), defined at q. When (or, indeed, when is any simply connected conformally flat space–time) the notion of local twistor transport is path-independent but this is not true in general.

The definition of local twistor transport along a smooth curve γ in with tangent vector t^a is

where Taking γ to be a ray—which is simply connected, with topology (by 's global hyperbolicity)—we use local twistor transport to propagate (ω, π) uniquely all along γ, thereby providing us with our canonical twistor space , assigned to γ. Correspondingly, we shall have spaces , and , just as in §1.

We ask what the relation between each and the global space of rays in might be. Within each , the ray γ, when π-scaled to Γ, can itself be unambiguously represented by (0,π_A′) all along γ, this being unchanged by local twistor transport along γ (since and π^A′π_A′=0). When is confomally flat (and simply connected), the integrability of local twistor transport allows us to achieve this globally for the whole of , where a π-scaled ray η in that meets γ in a point q would be represented at q by the local twistor (0,η^A′), in both and , where η^A′ provides the direction and π-scaling for η. In fact the spaces are all canonically isomorphic with each other, and (locally) with itself.

On the other hand, in the general case, the non-integrability of local twistor transport prevents such a global twistorial representation of in this way. Instead, the idea of palatial twistor theory is that the holomorphic twistor Heisenberg algebras associated with each , though not necessarily canonically identified with one another locally, would constitute a kind of -bundle over that is holomorphic in an appropriate sense. Such holomorphicity would not arise simply from the normal geometrical structures in , in the presence of conformal curvature, because the complex structure of is interpreted, in , in terms of the vanishing of shear in ray congruences [2], whereas the Weyl tensor provides the measure of change of shear along rays. Accordingly, when is conformally curved, we do not get a natural complex structure for (or, rather, natural CR-structure, like the one that inherits from ). In palatial twistor theory, we obtain a holomorphic structure by removing all places where complex conjugates arise in the normal geometrical procedures and translate them into holomorphic procedures in accordance with the twistor quantization rules of §2. Most particularly, in accordance with the above relations for local twistor transport, and the twistor quantization rules, we have (like the local twistor transport equations for , as the complex conjugate of (ω^A,π_A′))

Each of the twistor spaces will have its corresponding quantum algebra , but we have no reason to expect a canonical isomorphism between these algebras for different rays γ, when is conformally curved. However, I would expect that we can, for a topologically and holomorphically trivial open region of —that I refer to as a simple region—deform, continuously and holomorphically, the various so as to obtain a holomorphic ‘trivialization’ of the portion of lying above , thereby obtaining (by no means uniquely) a single algebra , continuously/holomorphically isomorphic to each for (and therefore isomorphic to ) appropriately consistently over the whole of . This notion of ‘consistency’ should demand that there be a consistent ‘ket-space’ |…〉 for the entire region . The point is that ‘locally’—in the sense of over any simple region of —we can obtain , isomorphic to , with a consistent ket-space, but this consistency would not, in general, be possible globally.

Each such ‘trivial’ is to be thought of in the spirit of a ‘coordinate patch’. Over the intersection of two simple regions and we require consistency of the algebras and in the sense of having a continuous/holomorphic deformation of one to the other, but we do not require a common ket-space to be present. Our global notion of the quantum twistor algebra for , and assigned therefore to the whole of , is obtained by taking a covering of by simple regions, with consistency over all (multiple non-trivial simple) intersections of the whole of of π-scaled rays in . Our requirement is that restricts to simple subsets , of in the way just described above. This algebra is, of course, to be taken in entirely abstract sense, the specific representations obtained from particular coverings having no special relevance to 's structure.

In order to identify the points of in terms of the algebra , we cannot do this by means of intersections of rays, because this condition, given in §1 as , involves a complex conjugation and does not directly survive within the local structure of . Instead, the points of have to arise by non-local considerations (as was the case with the nonlinear graviton construction of §4). Corresponding to any particular point r of there would be a locus R in representing r, namely the family of all rays through r, topologically S² ( acquiring a structure in ), the idea being that the consistency (i.e. trivialization, in the above strong sense of having a consistent ket-space) of the -bundle over R is what determines such an S² locus as representing a point of . For this to have a chance of working, as a sufficiently restrictive proposal for locating 's points in terms of such a twistorial construction, we must ensure that the bundle is, indeed, in an appropriate sense holomorphic, so that the rigidity of holomorphicity can be appealed to.

It does not seem unreasonable to me that , together with its conformal structure, would be obtainable in this kind of way. Although the nonlinear graviton construction, as outlined in §4, applies to complex space–times, so that the role of the space is not directly applicable, one can nevertheless see that in a complexified form the current construction reduces to it, in a degenerate case. The same would apply to the dual construction, where we must now employ a ket-space of dual twstor variables, this providing what has been referred to as a googly nonlinear graviton.

Yet, there is considerable mathematical speculation and conjecture in the descriptions above, aimed at the non-local coding of the structure of a general (presumably analytic) Lorentzian globally hyperbolic real 4-manifold . There are many mathematical issues that need to be sorted out, not the least being a need for some suitable generalization of the Kodaira theorem [6] that is central to the construction of §4. Moreover, none of this encodes the formulation of Einstein's equations.

It is perhaps remarkable, therefore, to find that Einstein's Λ-vacuum equations are themselves very simply encoded into this structure. For these equations provide precisely the necessary and sufficient condition that the local twistor spaces possess an I-structure (for given Λ), as defined in §4 (being constant under local twistor transport) and we now require that all the needed continuous/holomorphic deformations of the algebras preserve their nature as algebras on with this I-structure. If all these procedures (or something like them) indeed work as intended, with generalizations to the Yang–Mills equations and other aspects of physics, then there would appear to be significant openings for twistor theory in basic physics, not envisaged before.

Competing interests

I declare I have no competing interests.

Funding

I am grateful to the Leverhulme Foundation for supporting me with an Emeritus Fellowship during part of the time that this research was carried out.

Footnotes

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