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1.3.8 A 11011' on the wo\'e equation 1.3.9 SpiJlon anr! comJllc)\' Sl'ometr), in ltigher dimen~ iol1l
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in\'ari{1nt Ollcratoh 1.3.13 Ou ~ l j('ha"l ~ l u rrIlY '. twistor correspuudenc,' 1.3.1 '1 On the tOI)()logy of familicl> of tIIallifold~
and 1 h4! pul1b,u:k mN'halli~1J1 1.3. 1:1 The '~('nroSe: Ltllnsfof m for ('oml,lcx
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Ih ecaH'nfS(,(1 .:2 )
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Cba p~cr 4: Twistor throry and elementary particle physic.. ..
1.4 .0 '1 he twlsl or Jlan icle l,r0lo;r'UlIlI)l' . s\lpCrsYl11l1H'U)·. Sl. riug t}lcor,)' antl conformf'lilidil LhC'Ories
1.<1.1 a I l l~w 10 ,row (or ,n :Cihl ~ ;rllw) lIew I wi$1 or. from oltl
l. 'l. lb Some relllark.!; 00 the tWQ twi510r dr5('fiption or (·It-~t rOlllagnetir alld Iill("ari~ .-d
gravitational pOlt'n\ i;,l ~
1.4.1.: T it.: twisLor di.'~c rij)li()n of lu\\ I~ins haryon ,,111lei 1.'1.2 Dyolllll lcally brokell ~~·ntmrlr~ I'Inll global
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IA.G TI", spin .5'lll iln~ 11 opc::rntor$ autl Su{.t) ('uimlr operator,; for a/l 'l. poi ul sy~tl!JIl
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It. T.Shull
:l:!6 :,l :1O
~Iaulf'~$ fields (including ~ l l'Ix\I,,.1I ~tf'orr and linear gravity) are CDnfcrlulI.l1y invariant.
In the s tl\lIdnr.1 1110.1,.1. llu': b" ~k fi('!d~ ('xcerH th,' lI iggs ba5on ) ;UI! all llli\.!lsle$,s in t he
lim it. tha.l int{~raction l are , witched ofr. and pick up their ma,., "in their interaction with
Llac lI iggs fidd. ludeetl, tilt: fillile 11l~M:" they do IlllI' !! a.re \'er~' ,.mall compared IU lin:
P!and mu",. -rhe mechan"m b) .... hich conformal s)'mmeur is broken is. surdy. a
fumjamelltal part. of plly.iCl lOud lUI undehll\lIding o r this Ilv"chal1ilim w/)uld gil"(" iusight
illto the nature of relit mass. iu rdation (0 g rl'lViL), Rnd-wc mny Ilopc-il.$ :.pec:trllm un,ler
quantization . TWistOr theory prol'idcs a conform ally invariant formalism, so lhal any
conformal symmetry brel\king is mAde cxplicit-reduction to Poincare invariaru:c Oil
twist or ~pace mllst hf' "fff'ctf'd by 1Mroducing thl' 'infinity t\\'i~tQr',
S,JCClnt <.:o ll .",dcr(lIiOIlS: ~ I :.n~' of Ihe m;lth~malical mUlhod$ or twi5tor t hcory ran !I<'
extended to other llignalur..:s and dlIllC I\~iolll-illll~d in many difftrcill \\'a~· s . l"';e\"l~ rtf1e ·
t(:ss, c(:rtain f ... atures o( twistor theory arc 'p!.'cial 10 Lorculzial1 ,signature ill four
dimClujolls. and onc IIlhY rcgnrd tll('~e (e{lllIteS ,U Burfidclltly 'e5Scnlial' Ij.S to Ilrol"idf! an
expbnation vi tl1(' dimension and lig l1ll.1 urc of $1)IICC· t iull', f irs t. the nOIl·linenr
CVllstruct iulii do nOL appear to have fUIH.logucl appropriate for pll)'sics in ~pllce.time
dimension greater t hun four. Seconrl, the Lotent-z. signn t ur,. of 'pace· time arises from dll'
rea l codimension oue hypcrsu rface PN In twls tor spnce. llypcrsur facE'$ of tl1is type ellierge
as II,.. nalllr't! o.."unullri l '~ uf of the domain oi dcfmhioll of tile coh(llllolotp' dl\$«'~ ... tld non­
linear C"OIl$trut:tiolla used to dc"ribe ph)'sical fiellis Oll I .... istor space.
Fixing twistor r pat;e ill primar)! . rather thall dual t ..... istor rpace. I1Itrooucl:$ lcft.rij!;h\
llS)'mmctry. Thllrdore th .. rc IS the I)oulbllh )' or pf(!\' ic.lh,s sOllle explallMion o( parit~
\'iolMing phYSics. Till. hO\\'c\'cr turns {Jilt 1I0t 10 be a bonu. for twistor lhf"Ory, but in"'"ud
olle vf it. most uuyidlling problem I . It I,allo rar been Jirficuh to undcu'tand sa.tisfaclorily
till" widcs.-It.le left-right sYlll nu:try prc:I<!llt in llh,vli<:s (Iuch ill ill gra\' jUl. t ioll lind
c1ectronIOllgnetism) in the <.:onlt'X I or t ..... i:uor theor.\" .
Originally, idelUl (ro111 ,pin uct .... ·orks wcr'> 1nflucn t ;'t! III the de\'I~lopll1en t of twistor
tllf'ory. l1 (1WC\' C"r. their rclatiollshif) to twi.tor lI,eury 11l't~ ~ I ncl.' u..:t:omc leu cleilr. 1'he$Cl"
ideas arc of more than mere historical intercit and tlu~ reader iii referred 10 Pel1ro~
(197"'). (19 il h). (10 i:l ) 8tltJ (1079 ) (or detailed Jilcu$lliou5 or spin UCl\\'orks and their
possible fC"lalion ship 10 l\\'i ~ lor lhoory.
The prr!!H~ ll t $Ialtu ()/ h uu{IJ, ' ,/' eu 1'Y f~ fi ll 0T1IJ1 'f}(leJi /0 TI/' y!>ics, Tllese aSviraLicmli are ~til1
n Ions way from being rea liud. and in the tIlO~1 parI. the twiSlor formulation i~ equivitlent
lO the standard space'Lime formulat ion. T wo rlot.abl ... e>:ct!plionl arc \he ~\\'iUu r part icle
8 I 1. An overview of twistor theory
ds' = ~.b dx·d3;' =( dx')' -( dx1)' - (dx')' -( dxs), .
Here x2.=(xO,;,X2,X3) are the standard coordinates on R4 and the Einstein summation
convention is used, as it will be throughout these volumes. The vector space ~ is the
tangent space to M. Indices wiU be raised and lowered using 71d and its inverse 71 Gb where
J1a.b'7be=6 ~ (6~ is the Kronecker delta).
An abstract index A,B or a on some qua.ntity ~A or va is regarded as a label signifying
t he vector space SA or r, to which it belongs. Only when we take a basis of the
appropriate vector space, say E A ~ =(e A 0 ,e A 1) for the 2-dimensional vector space SAl can
we translate the abstract indices into concrete indices which take on numerical values:
e~l=(eo,eJ)=(eAtAO,eACAl)=~AEAcd. Tensor algebra is performed in p recisely t he same
way as if t he indices were concrete. Unless otherwise stated , indices will be taken to be
abstract. Notationally, concrete indices, which do take on numerical values, are
distinguished by underlining. Square brackets around a clump of indices denotes skew
symmetrization and round brackets denotes symmetrization.
The Lorentz group 0(1 ,3) is the group of group of linear transformations of the
tangent space --r to M which preserve the inner product va vl''1ab ' If we code V- as the
2x2 matrix:
(1)
. .AA' we have 2det( v~- )== Va. 0!, Prcmuitipiying yodd' by unit dete rmin ant complex 2x2
matrices, and postmultiplying by the complex conjugate matrices induces all the Lorentz
group. This yields the isomorphism of the complexified Lorentz group
SO( 4 ,C)" SL(2,C) x SL(2,C)'
or, alternatively, the 2-1 covering of the real identity connected component of the Lorentz
group 50,(1,3) by SL(2,C).
The spi n spaces, SA and SA" are the two dimensional complex vector spaces on which
SL(2,C) and its complex conjugate act, We can write:
§Ll. 2 Mathematical background I 9
-here & AB=t:[ABJ and III the basis above E01 =1 (sim ilarly fOfE A' B')' E AB and E AB are
"be Levi-Civita spinors. Spinor indices can be raised and lowered using £ AB and its inverse
:.A B which satisfy E:BA,BC=£AC=CA C . Since cAB is skew, the ordering of indices is
Important and we have the following conventions:
klgether with their primed cou nterparts. If ~A and f]A are proportional,
{A 1]B E A8=~A'7A = OJ conversely {A7JA =0 implies that {A and '7A are proportional.
Equat ion (1) defines the isomorphism between TtJ. and the tensor product of the 2
complex dimensional unprimed spin space SA on which the group SL(2,C) acts and its
complex conjugate SA" primed Spin space: Ta=TAA,=SA®SA' . Complex conjugation
defines a map from SA to S A,i
{A-(",
Some useful spi noe t rans lations are:
where t"abcd = E[abcd) is the Levi Civita symbol for r-, for which in a right-handed time­
oriented orthonorm al fram e t"Ol23 =l. If Fab 1s antisymmetric, then
where the complex conjugate relation, A
I ' F' If d aI 'If d I ' f 1 ""d 'F comp ex two- Iorm Clb IS se - u or antl-se - ua 1 it Clocdr =1 ab or -iFab
respectively. In spinor terms th is becomes Fab=¢ A' B,e AB for FClb self-dual , and
Fd =¢ASt"A'B' for F anti-self-dual. The spinor translation of the source free Maxwell
equations, V[aFocJ= O=VCl Fab on Fab=F,ab) is:
where V AA' is the space-t ime derivative V Cl=8/ 8xCl ,
A vector kCl is nu ll and future-pointing if a nd only if it can be written as the product of
a spinor and its complex conjugate ka="A;cA', The future pointing null vector kCl
determines KA up to phase, "A -+ eil "A, The phase of a spinor up to sign may be realized
10 11. An overview of twistor theory
geometrically by consideration of the two·form (flag plane):
F a.b = I\. A ,.. BE A'B'+"A,R Bit AB I
which satisfies
These conditions imply that Fab is the skew product of two covectors, ka=KAK-A , and Ta ,
F'ab=k[QTbl' where k 2=O and rOka=O. The flag plane 'encodes ' the phase of ",A up to sign;
under I\:A _e" ",A, the flag plane fotates by 20 .
2. Conformal geometry. The Poincare group is the group of dirreomorphisms of M
preserving the Lorentz metric. It is the semi-direct product of t.he Lorentz group 0(1,3)
with the group of translations of R4. It is useful to consider the Poincare group as a
subgroup of the conformal group C(l,3), the group of motions of Minkowski space
preserving the metric up to scale. The conformal group does not act on 1\.1 1 as such. One
must enlarge !WI to M by inclusion of the light-cone at infinty 3, since the inversion
interchanges 3 with the light-cone of the origin.
Compactified Minkowski space M = M 1U3 is the projective light-cone of a. metric of
signature (2,4) Oil Re. In the language of projective geometry, M is a quadric hypersurface
in Rps. Consider coordina.tes (u ,v,xa) on Ri5 with the metric:
ds'=dudv+dx. dx· (2).
The quadric is the set of points in Rp5 for which uv+xaxo,=O. Compactified l\·tinkowski
space M, is th us {(u,v,XU ) such that uv+xoxO'=O}/{(u,v,XO) ..... (>.u,>.v,..\xa) with ..\;#0, >'ER}.
The quadric M, in Rp5 has a natural conformal structure: the points x and yare null
connected if x lies in the plane in Rp5 that is tangent to M at y, and vice-versa. The
identity connected component of the conformal group Co(1,3), arises from the action of
SO(2,4) on the projective light-cone M, in R6. This is a 2-1 covering: SO(2,4)-Co(1,3) .
We can represent M I as the subset of M coordinatised by xo. obtained by setting u= 1
and u=-xo.xo.. The metric on H6 reduces to the Minkowski metric on MI. The conformal
factor , is specified by the hyperplane (with null normal vector) u=1. The intersection of
u=O wit h the projective light-cone is 3.
§1.1.2 Mathematical background I 11
The spin group of 50(2,4) is 5U(2,2) . Thus 5U(2,2) is a 4-1 cover of Co(1 ,3). The
basic representat ion s pace of SU(2,2), C4. together with a pseudo·Hermitean inne r product
of signature (++-- ), is twistor space la, the fundamental inner product being given by
!:(Z,Z)=ZO Zo o
3. Zero rest mass particles and twistors. A twist~r contains the information of the linear
and angular momentum of a zero rest mass particle. The linear and angular momentum of
a massless particle is represented by a future·pointing null vecto r pD., the linear
momentum, and a skew tensor field MOb(:t)=MabJ(x), the angular momentum of the
system about the space·time point xD., The position dependence of Mob(x) is:
The intrinsic spin is measured by the Pauli-Lubanski spin vector:
which is position independent. For massless particles occurring ill physics Sa=sPa, where 5
is t he helicity of the particle and lsI is the spin .
Since po is null and future-pointing , we have po='Ir A,1f A for some constant spinor ;r A'
determined up to phase. The condition So=sPa yields:
M ilO . ( A _ B ) A' B' ._(A' B' ) AB
=IW 'If f: -lW 11' t
where wA (x)=w A(O)_i.0 A ' iT A" The space of such spinors (w A (O).7r A') a.re coordinates on
four dimensional complex (non-projective) vector space r) of twistors ZOo 'vVe write, using
abstract indices , ZO = (wA I 'if A')' za ETo. Note that the mass less particle determines ZO
only up to a freedom of a phase, ZO -- eil za. The massless particle determines also the
complex conjugate pair of spinors : - - A' -
(if' AI W )=Z"ET " I
a dual or (complex conjugate) twistor . "Ve have
05 (8- + -8')-- a= W if'B 'irS'W 7r A'ir A,
so that the helicity is given by:
( A- - A') -25= W'irA+'irA,W =ZOZ".
12 11. An overview of twistor theory
This defines the fundamental pseudo- Hermitean inner product of signature (2,2) between a.
twistor and its complex conjugate. The group of conformal motions of M are reali zed on
TO by complex linear transformations preserving E. SU(2,2) .
4. The twistor equation. It was noted above that the spinar wA has position dependence
associated to that of M ao(x). We have:
This is the general solution of the conform ally invariant twislor equation:
(A B ) V A'w =0.
This is also the (one index) 'Killing spinarl equation and the vVess-Zumino equation-in
supersymmetry th is is the condition that w A defines a global s upersymmetry (see chapter
1.4). The vector field kO = wAw A' is a null conformal killing vector, and the general
conformal killing vector can be obtained as the sum of such vector fields. The relationship
between solutions of the twistor equation, symmetries and conserved quantit ies is explored
in more detail in chapter 11.1 in connect ion with t he quasi-local mass construction.
5. Geome'trical interpre tations of twistors in M, null geodesics and Robinson congruences.
When ZO'ZQ=O, t he massless part icle has a well defined world-line where Mab , or
equ ivalently wA, vanishes. This is a n ull geodesic aligned along 11' A,1f A th rough the points
"AA' such that ",A(x)=O.
When t he particle has non-zero spin (ZO'ZQ,cO) the particle no longer has a well defined
world-line. It can nevertheless be represented by the congruence of nu ll geodesics
determined by the vector field wAlJA' =1/". This is a Robinson congruence. A picture of
this congruence at a given it.lstant can be obtained by projecting the null vector kIJ. so that
it is tangent to a flat space-like constant time hypersurface in M. When project.ed into the
bypersurface, kIJ. is tangent to a foliation by circles which form the Hopf fibrat ion of S3
over 52. As time evolves, the picture mOVeS along the axis at the speed of light.
6. Geometrical interpretation of lwislor's in CM and the Klein COrTcsponde nce. The
geometrical description o f a twistor simplifies in complexified Minkowski space eM . .lust
as M was defined as a quadric in Rp5, eM is a complex (non-degenerate) quadric in CpSj
eM I is then C4 , the compiexificat ion of M I~R4. A twistor ZO with spinor field wA then
\1.1.2 Mathematical background 113
res ponds to the complex two-dimensional plane on which w A vanishes. The plane
sjsts of those (complex) values of x AA ' foc which:
AA' "these values of x ,the twistor ZO is incident with x IJ • When 'ITA' vanishes, the two-
ne lies on !l.
Tangent vectors to this 2.--plane are all of the form 7f A,>'A for various "A " Such planes
,an totally null in that all tangent vectors are null and mutually orthogonal. The two-form
at is orthogonal to the 2-plane 'IT A'lI' B'~ AS is self-dual. Totally null self-dual 2-planes are
otaIled a-planes. Dual twistors a re similarly represented by tot ally null 2-planes that are
ti-self-dual. These arc ,B-planes.
Since ZO and >.zcr (>.ec*) determine the same Q-p!ane, the space of a-planes IS
dect ive twistor space, PT =To /{ZO ...... tZO}. Similarly the space of p -planes is projective
~ual twistor space, PT*. PT is thus CpJ, and PT* is its dual , a.lso CpJ. A point of nOIl ­
projective twistor space is represen ted by an O'-plane together with a covariantly constant
-pin or ;r A' aligned along it as above.
A point :tEeM is represented in projective twistor space by the Riemann sp here or cpt of
projective twistors that are incident with x. This is a two plane through the origin in TO.
Choose a.n origin in eMI; then a point in eM I can be represented by its position vector,
rtA'. The twistors incident with this point are ZO=(w(O)A,1I'" A,)=(ixAA'1!' A" 1t' A') as 11'" A'
vanes. NOll-projectively this is a complex 2-plane. Projectively, 11'" A' are homogeneous
coordinates on Cpl. Note that the space of non-projective twistors through the point xAA '
is t he primed spin spa.ce at xAA'.
The picture just described is tbe Klein correspondence in which projective lines in CpJ
=PT) are represented by points on the quadric CM in CP5. The correspondence can be
coordinatized as follows. Lines in PT can be represented by skew two-index twistors ~11
satisfying the simplicity condition >"AofJ X-r6]=O. This implies that X.~" is the skew product
of two twistors: ~"=A[a BP] for some AO,BaeTo. T he line is then the set of [ZOI such
that ZO ='xAo+IlBo for some 'x,Il, or alternat ively such that. z.0 x"-r] =0.
The space of skew two index twistors ~" is CG • Such Jt>" are simple if and only if
Q(X,X)=~"X-r6EQ"-r6=0! where E a .h6=E[ap,.6]' Q defines a quadratic form on C6 . The
twistors ~11 and ,X~fJ. 'xEC*, determine the same line in PT and conversely a line in PT
determines ;tr11 up to scale. Thus the space of lines in PT is the quadric Q=O in CP5,
14 I 1. An overview of twistor theory
which is eM. (For further discussion of the I<lein representation a.nd its applications to
twister theory see Penrose & Rindler 1986, pp. 307-332, and Hughston & Hurd 1983.)
7. Breaking conformal invariance; the i.nfinity twistor. Conforma l in variance can be
broken by introducing a skew simple two index twister 10 /3' The twister 1013 determines
t he line in PT corresponding to t he vertex of the null cone at infinity, j , in eM. In the coordinates introduced above o n R6 we have ~fJ 108=11., and the metric can be
retrieved as follows. Fix the scale on ~.8by setting ~.8 10,8=2; then
defines the metric on the region ~{J 1 0
,8#0, which is M I in eM.
The coord inates on TO, (wA(O), iI" A') ' were obtained relative to a choice of o rigin and
infinity in M. With these coordinates, the origin is represented by the matrix OQfJ and
infinity by lo fJ as follows:
~ ] where Oop=~COfJ"6 0"6 is the duaJ of OcrP and fcr {J is the dual of rP. T he o nl y nontr ivial
AIBI AB 1 components of ~oP,.' are ~ AB~ and ~ AI Hie . A point in CM with position vecto l'
0 A' is given by the matrix ~fJ
J x •• =[ <AB , A'
~ IX8
The Minkowski interval between two points , XU and yU is given by the expression
The point x is real ,r>P X- , X"
A - o(J='ieQP,.' .
if and only i f, under t he conjuga.tion ZO--+Zo we have
This implies tha.t the tine {ZO such that ZO Xop=O} lies entirely
in the real hypersurface ZOZo= O.
8. The future tube and positive frequency massless fields. The open region in CM+ in CM
whose points have posit ion vectors with past-pointing time- like imaginary parts is called
§I.1.2 Mathematical background 115
forward or future tube. Similarly CM- is the region In eM whose points have position
lors with future.pointing time-like imaginary parts.
The Fourier transform of a massless field is supported on the light-cone p2=O in
A massless field is said to be of positive frequency when its Fourier
;za.nsform is supported on the future light-cone in momentum space, and negative
quency when its Fourier transform is supported on the past light-cone. It is clear from
Fourier representation that a (normalizable) massless field has a unique splitting into a
it ive and a. negative frequency part .
A field defined on M is of positive f7-equenclJ if and only if it has an analytic extension
r CM+. This can be seen from the fact that the e-ip':Z: kernel in the Fourier transform
exponentially small for p future-pointing null, and the imaginary part of x past-pointing,
,.I!. for xECM+.
Points of CM+ can be characterized by the property that the corresponding line in
projective twistor space lies entirely in PT+ . This follows from the fact that Z · Z restricted
a line L:z: is given by Z.Z=i(.:0 A '_x AA ')lI' A';;' A' This is positive for all 11' A' if and only
the imaginary part of x is past-pointing time-like.
We will see that posit ive frequency massless fields are in 1-1 correspondence with
'COhomology classes on PT+ .
. Twistor quantization and the zero rest mass field formulae. Twistor space restricted to
Z·Z =s modl,do the phase Z-.e i9 Z, TO!z.z=,/{Z ...... e i9 Z}, is the phase space of zero rest
mass part icles with helicity s. The standard quantum mechanical commutation relations
i>r ( Pa.,Ma.b)' i.e . those of ftx the Poincare Lie a lgebra, is implied by the following:
This suggests that one should treat the coordinates za as position coordinates, and the ZO
as momentum coordinates1 and consider wave functions depending holomorphica1ly on 2f'.
On such wave functions the operator za is represented by multiplicat ion and .20 is
represented by -fl.lJ~ ' We represent the helicity operator symmetrically so that :
T hus a state function f(Z) is in an eigenstate o f helicity if fis an eigenrunction or the Euler
"homogeneity' operator T=ZOlJ/8ZO. Eigenfunctions of r with e igenva.lue n are
16 I 1. An overview of twistor theory
homogeneous function s of homogeneity degree n. For in teger values of n they can be
thought of as sections of the line bundle O(n) on projective twistor space PT. The line
bundle O(n) is the n1h power of the hyperplane section bundle on CP3:=PT, or
alternatively the _nth power or the line bundle T -+PT and has first Chern class n.
From here on we shall put h:=l. For helicity s>O, wave functions should be defined as
solutions of (Y+2+2s)f-0 on TOlz.Z=2.s>0/{Z ...... eilZ} and therefore as sections of
O( -2-2s) on PT+:={[ZIEPT. Z·Z>O}. However, the re are no global holomorphic
sections of O( -2-25) on PT+. Instead the wave functions are cohomology classes on PT+i
they are elements of the sheaf cohomoLogy group H1(PT+,O( -2-25) .
Elements of Hl(U,C1(n», with U some open set in PT, can be represented 111 a variety
of ways. The most popular methods are Cech, where the cohomology classes are
represented by equiva1ence classes of collections of holomorphic functions on Stein subsets
in V, and DoLbeault where the classes are represented by a-dosed (0,1) forms on U modulo
a-exact forms.
We shall use Cech representatives here (see Woodhouse 1985 for a discussion of twistor
methods using Dolbeault cohomology) . Consider the open neighbourhood U in PT that is
swept out by the lines corresponding to points of an open ball V' in eMI. The set V can
be covered by two Stein open sets, Vo and Vl . Typically Vo and Vl can be chosen to be
the complements of 11'0,=0, and 11'1,:=0 in V respectively for some choice of primed spin
fra.me. A Cech cohomology class (j]EHl( V,O(n» is an equivalence class of hotomorphic
sect ions /01 of O(n) defined on von VI modulo the addition of holomorphic sections go and
91 defined on Uo and Vt respectively. For the purposes of the integral formulae below we
shall take J to be some representative of a fixed class.
It is possible to make contact with the standard quantization of massless particles by
means of an integral formula. Recall first that on space·time, wave functions for massless
particles are represented by means of solutions of the conformally invariant zero rest mass
field equations:
A' V A tp A' 8' ... £' := 0
where the splllor field tp A' 8' ... L':= I.fJ(A' 8' ... L') has 2s indices and conformal weight -1.
Heiicity zero wave functions are represented by solutions of D ip:= 0 (again ip has conformal
weight -I), and negative helicity wave functions are solutions of the complex conjugate
wave equation with -2s unprimed indices.
For lJ1EH1(U,O(-2-2s») with s~O, we restrict fto the line Lr in U corresponding to
h 0 U'b 0 0 j{A )1 j{0AA' ) t e pomt xE y wntmg w ,'1r A' Lz = lX 11' A,,1r A' . The integral formula is as
§I.1.2 Mathematical background I 17
ows ( Penrose 1968, 1969i Penrose and MacCallum 1972):
The integrand has homogeneity degree zero. That '.pA ' .. . solves the massless fi eld
equa tions can be demonstrated by differentation under the integral sign. The contour of
tegrat ion links the intersection of the line Lx with von 0 1 which is an annulus. If one
chooses a different Cech representative, 1+90 say, t hen the difference is the integral of 90'
This difference is zero since the contou r of integration can be contracted to a point within
r.on L:!: (although not within von UI) '
For s<O and UlEHI( U,Oe -2-25» we have the following formula ( Hughston 1973 •
• nrose 1975b) :
here n = -25, and "p has n indices. The remarks made above for the s~O formula apply
10 t his formula a lso. The replacement of the operation of multiplication by 7r A' by the
derivative operator 8/8wA is an example of twistor quantization-the complex conjugate
of iT A' is 1i' A which acts on twistor functions by tJ / tJw A via the Z III _ -8/ tJZO prescription.
Furtlaer reading. For elementary twistor theory, chapter 6 of Penrose & RindJer (1986) is
perhaps t he most comprehensive and up to date account. This also contains an elementary
a.ccount of Cech cohomology. See t he general art icles by Penrose (1968, 1975a, 1977 &
1983). and Penrose & MacCallum (1972) and the in troductions by Hughston (1979).
Huggett & Tod (1985) and Ward & Wells (1990). (A brief resume of basic twistor theo ry
is also presented by Hughston & Ward in SO.I.)
References for chap ter 1
Hughstoll, L.P . (1973) Birkbeck College seminar, London , January.
Hughston, L.P. ( 1979) Twistors and particles, Springer LNP, 97.
Hughston, L.P. and Hurd , T .R. (1983) A Cpli calcul us for space-time fields, Physics
Reports, 100 no . 5, 273-326.
Huggett, S.A . & Tad, K.P. (1985) An introduction to twiS'tor theory, L.M.S. student texts
4
~Iason, L.J. & Sparling, G.A.J. (1989) Nonli near Schrodinger and Korteweg de Vries are
reductions of self-du al Yang-M ills, Plays. Lett. A, 137, no. 1,2,29.
Penrose , R. (1968) Twistor qua.ntization and curved space-time,
18 I l. An overview of twist~r theory
Int. J. Theor. Phys. 1, 61-99
Penrose, R. (1971a) Applications of negative dimensional tensors, pp. 221·244 in
Combinatorial mathematics and its applications, ed. D.J.A.Welsh (Aca.demic press).
Penrose, R. (1971b) Angula.l· momentum: an approach to combinatorial space·timc, pp.
151·180 in Quantum theory and beyond, ed. T .Bastin (C .U.P.).
Penrose, R. (1972) On the nature of quantum geometry, in Magic without magic: J.A.
Wheele,- fests chrift, ed . J.R.Klauder (Freeman San Francisco)
Penrose, R. & MacCallum, M. (1972) Twistor theory: an approach to the quantization of
fields and space·time, Phys. Repts . 6C, 241-315.
Penrose, R. (1975a) Twistor theory, its aims and achievements, in Quantum gravity, an
Oxford symposlUm, eds. C.J.lsham, a .Penrose and D.W. Sciama, O.U .P .
Penrose, R. (1975 h) Twistors and particles, in Quantum theory and the structure of time
and space, eds. L.Castell, M.Drieschner and C.F. von Weiszacker (Carl Hanser
Verlag).
Penrose, R. (1977) The twistor programme, Repts. math . phys. 12,65.
Penrose, R. (1979) Combinatorial quantum theory and quantized directions, §0.5.14
Penrose, R. (1983) Physical space-time and non realizable CR. structures, Proe. Symp. Pure
Math. 39, 401-22.
Penrose, R. (1986) On the origins of twistor theory, in Gravitation and Geometry (l.
Robinson festschrift volume) cds. W.Rindlcr & A.Trautman (Bibliopolis , Naples).
Penrose, R. & Rindler, W. (1984, 1986) Spinors and space-time, volumes I and II, C.U .P.
Ward, 11..5. (1985) Phil. Trans. Roy. Soc. A 315 451.
Ward, 11..5. & Wells, 11..0. (1990) Twistor geometry and fields, C .U.P (to be published).
Woodhouse, N.M.J. (1985) Real methods in twister theory, Class. & Quant. Grav. 2, 257-
91.
19
'God forbid thlJt truth should be confined to mathematical demondration!'
- William Blake
1I.2.1 Contour integral formulae old and new by L.J.Mason and L.P.Hughston
This chapter is concerned with concrete formulations of the Penrose transform and its
eneralizations. The next chapter is concerned with the development of the associated
~stract mathematical machinery. In th is context the Penrose transform is t he one to one
correspondence between solutions of the massless field equations on regions in eM and
cohomology classes, represented either by Dolbeault forms or Cech cocycJes on twist~r
space (see §O.2.2, a nd Eastwood, Penrose & Wells 1981). This map is realized by t he
integral formu lae
( l - I f J{ . AA' l d M' ffJ x A'B' ... L'-2rri il"A,1I'"B' "' 'irL' IX iTAI,1r'A' 7r/ll' 7r
and
1 f an! I ' x - - 'It d'lt A "'( lAB ... L-2;ri 0 A. B 0 L A"
uw UW " ·vw Lr
Many art icles develop integral formulae that yield the solution of analogues of the
massless field equations (for example the wave equat ion) in higher dimensions in terms of
free fun ctions. T his is the aim of §I.2,3, 7, la, 12 and 13,
The first systematic investigation of the Penrose transform in higher dimensions
appears in Hughston (1979a) , In that case the group SU(n,n) generalized the SU(2,2) of
~linkowski space twistor theory, and analogues of the zero-rest-mass equations were solved
by use of appropriate new integral formulae, An SU(n,n) analogue of the l<er r theorem
was also established (d. Hughston 1979b, §10,6). These results were ve ry suggestive, and
were followed, later, by a good deal of interesting work in higher dimensional
rep resentation theory (cf. chapter 3) . A 'problem' with th is development was that the·
analogue of t he 'hclicity 0' case for SU(n,n) was not t he wave equation, but rather a set 0
20 I 2. Concrete approaches to the Penrose transform
higher dimensional second order equations. So a new investigation was launched in mid
1979 with the aim specifically of solving the wave equation in higher dimensions- the
result is described here in §I.2.4 , a formula that may (in modified form) have been
appreciated much earlier by W'hittaker and Bateman. A deeper investigation over the
following two years concluded that the space of 'pure spinors' for the complex orthogonal
group 0(m+2) was the ' correct' twistor space for analysis of the wave equation (and other
linear systems) in dimension 711. See article §1.2 .8. Since the pure spin spaces are
homogeneous spaces (cf. Pctrack §T.3.9) th is paved the way back again to representation
theory (see chapter 3).
The articles §1.2.2, 3, 15 and 16 are concerned with extending the transform to include
different field equations on CM (see also §I.4.5,7). The articles §1.2.7 and 13 are concerned
with reducing the Penrose transform to lower dimension. dimension 3, using ' mini­
twistors'.
PA = {([Z"J. [W. j)EPTxI'T*. Z" W. = O}
are used, rather than twistor space. The description is less economical , but solutions of tbe
massive Dirac equation and the non-linear ~4 equa.tion are characterized on PA. In §I.2 .15
and §I.2.16, tensor fields rather than homogeneous functions are used and shown to give
rise to generalizations of the massless field equations which take values in the local twistor
bundle on space-time. This is a concrete version of the Penrose tra,nsform for
homogeneous bundles §II.3.11.
Articles §I.2.5, 6, 9, 11 , 14, 15 are concerned with investigating the Penrose transform
for massless fields in further detail. In §5 and 6 the geometry of t.he Penrose transform is
studied in the context of the embedding of CM into CP5 as a quadric hypersurface. In §9
the correspondence is shown to be purely geometrical, in that massless fields are shown to
be in correspondence with afline line bundles on twistor space. In §11 the problem of
describing positive frequency Cech cohomology classes relative to a two se t cover is
discussed. In §14 the d'Adhemar integral formula is shown to extend to anti-self-dual
space-times and in §15 the d 'Adhemar integral formula is shown to be a special case of the
Dolbeault version of the massless field integral formula in which the Dolbeault
represent.ative is constructed out. of characteristic data for the field.
References
Eastwood, M.G., Penrose, R. and Wells. R.O. Jr (1981) Cohomology and massless fields ,
§1.2.1 Contour integral formul ae old and newl 21
Comm. Mat h. Phys. 78305·351-
~ston , L.P. (1979a) Some new contour integral formu lae, Pi>. 115-125 in Complex
manifold techniques in theore tical physics, cds. D.Lerner & P .D.Sommers, Pitman.
~ston, L.P. (1979b)Twistors and particles. Springer L.N.P., vol. 97.
rose, R. (1969) Solutions of the zero rest mass equations, J . Math. Phys.l0, 38·9 .
. 2 Ambi-twistors by M.G.Eastwood (TN 9, November 1979)
• e usual twistor correspondence may be regarded as follows.
Grassman ian) space of 2-planes in C" (through the origin)j PT is Fit t.he space of lines in
: an d t hen the correspondence is pictured by means of the following diagram:
F 1,2
F, = CM
-here Fl 2 is the (fl ag) of lines in planes in C4 and the maps 0' and P are the appropriate ,
iorgetful' maps. A point :t in eM gives rise to a line L;J:=O(p- l(X» in PT. A poin t Z in
PT gives rise to an a-plane ZE/J(a- 1(Z) in eM.
This set- up is good for studying spinol" fields on portions of eM with just primed
dic.es. This , of course. spr ings from the fact that F 12 is the projective pr imed spin
und le. A more exot ic reason for this is that
/3.0(m) = OA' B", ·D' (m indices)
/J~O( -m-2) = (} A' B' ... D' (m indices)
"'here OA' ... (0 A"") denotes the sheaf of germs of hoiomorphic spinor fields, symmetric in
' he upstai rs (downstairs) iud ices and f3,. is di rect image. T his is why twist.or integral
fo rmulae describe right. handed massless fields directly , whereas left- handed fields are best
described via potentials.
22 I 2. Concrete approaches to the Penrose transform
By use of obvious notation, the dual twist~r correspondence is given by
F'3 ,
j \ PT*=Fs F,=CM
Of course, there are many useful spinor fields involving both types of indices. For example
the Dirac equations of a. spin ~ massive field. In o rder to incorporate such fields it seems
natural to combine these two pictures and introduce F 1,3 = PA, the space of projective
'amhi·twistors" and a correspondence given by the following diagram
F I,2 ,3
\b F,=CM
Here a point r in eM gives rise to a quadric Q:Ea(b-1(x» in PA, and a point p in PA gives
rise to a null geodesic b(a-1(p» in eM. PA is nothing ncw in twist~r theory. 1t may be regarded as the space {Z· W= O} inside
PTxPT*. From this point of view the null geodesic obtained by b above is simply the
intersection of an a-plane Z and a ,a-plane W wbere they are guaranteed to intersect
precisely by the condition z· W= O. It now makes sense to talk about homogeneous
holomorphic functions on PA (or F) ,2,3) that have homogeneity (m,n). and we denote the
sheaf of germs of such by O(m,n). The direct images (under b., b~ and b!) of t.hese
sheaves give spinor fields with mixed indices (and either up or down). Por example
b.O(l,l) = OAA', the tangent bundle
b;O( -3,-3) = 0 AA" the cotangent bundle
etc.
We should expect by analogy with the usual Penrose transform that if Y is a suitable
region in eM and V is the corresponding region in PA (Le. V= a( b- 1 ( }-~ ») then the
§1.2.2 Ambi-twistors I 23
rohomology groups FI( V,C1(m,n» for various ],m,n may be interpreted as solutions of
differential equations involving spinar fields on Y. This is indeed the case. The results
ay be derived from the following short exact sequence of sheaves on F 1,2 ,3:
'KA'nAV 1 " AA' 0-.- O(»I,n)_O(»I,n) , 0(m+1,n+1)-0 (0)
here 0- 10 denotes the sheaf of germs of holomorphic functions that are constant up the
:..b res of a and (l1Al{A') are coordinates on PT*. Really 1r A
'1] A V AA' is differentiation up
e fibres of a and may be invariantly regarded as such, The conditions on Y that enable
• to be employed concern the topology of the intersection of null geodesics with Y.
ertainly if Y is convex then * may be applied without trouble. In the usual Penrose
a nsforlll only no's (-twistor equations) and HilS (-massless field equations) are of
terest. . F'or ambi-twistors H'!'s are interesting too. In particular,
B'( V,O( -p-2,- q-2)) - {<PA' A' A' B B holomorphic on Y and symmet.ric in 1 2"" P 1'" q
the primed and unprimed indices respectively such that
BIA; } V .pAl A2 ... A pB J •• • S q =0
This result gives a cohomoiogieai interpretation of the Penrose integral formula in
Quantum Gravity. As observed there, this may be regarded as part of the Dirac equations
i>r a spin (p+q)/2 field. More surprising is that the full Dirac equations may be
corporated merely by using a subsheaf of O( -p-2,-q-2). The result is that (if Y
eonvex, for example):
{Dirac fields on Yofspin (p+q)/2 and mass m}"H'(V,'lI(-p-2,-q-2))
, 8' here ~(m,n)=ker~D+m ):O(m,n)- O(m,n) where 0 = T A,l'1A8{ lJ I the twistor wave
A' WA operator. (Note that introducing 0 involves the infinity twistor and so we are breaking
~nformaJ invariance).
This may be proved by regarding the Dirac equations as applying b! to
O(-p-2,-q-2)
Ell
0(-p-1,-q-3)
and taking the kernel to obtain
8\1+1 V, Ap+l
the Dirac equations for a. spin p~q field .
The usual massless fields may also be represented on PA by lilts. For example
H'( V,O( -p-2,O))",{Soins of V AA' ¢ A' ... Ap = 0 on Y}
It is now natural to ask whether it. is possible to couple field s into an electromagnetic
background as is the case (or the standard Penrose transform. There, this is accomplished
merely by tensoring the 'flat' sheaves O(k) with Ward's ' twisted photon', In that case the
construction necessarily only gives coupling with left-handed fields . Since PA is noL biased
with respect to handedness we should expect that this restriction should disappear. \VhaL
we need is an 'ambidextrous photon' on V. For this we should calculate HI( V,O) . From.­
we obtain AA'
i.e. gauge freedom '
potentiaJs
Unfortunately a general potential q;AA' only gives rise to a 2·form that is closed , and not
necessarily co.closed, so MaxweUts equations are not all automatici but, in any case, we
have a n inclusion
{Electromagnetic Fields on Y} --. HI( V,O).
As usual exp2;ri ,. o 10 identifies HI( V,O) with the topologically trivial holomorphic line
bundles on V. Such a line bundle is an ambidextrous photon and tensoring this into the
above description gives rise to general electromagnetically coupled fields. Note that an
ambidextrous photon is left-handed if and only if it pushes down t o PT.
Finally we should ask what happens if we try to deform V as we would in the non­
linear graviton construction. Penrose has observed that this is possible and gives rise
(locally) to a general complex 4-manifold with conformal structure (see LeBrun'S articles in
chpater 11.5 to see why the quadrics remain under deformation).
The interpretation of H},s of a deformed twis tor space tells us what the definition of
massless fields on a curved but half-nat background should be. It is reasonable to expect
that a similar analysis of the analytic cohomology of a deformed V should say what
§I.2.3 The ¢' equation I 25
sasslcss (or even massive) fields on a general curved background should be. Alternatively,
nrose has suggested that it may be possible (in line with general twistor philosophy) to
ard a. deformed \I as more fundament",.! than the space-time it represents and to take
cohomology as providing a definition of a field whether or not it is possible to link it
-lh equations on space-time (thus side-stepping any Buchdahl conditions).
Additional notes: The ambidextrous photon on V is a special case of the Isenberg-Yasskin­
::ireen- \Vitten twistor description of general (not necessarily self-dual) Yang-~lills fi elds
Re, for example, Isenberg, J. & Yasskin , P. in Complex manifold techniques in theoretical
- "sics, eds. D.Lerner and P .D.Sommers, Pitman Mathematics Notes 32) .
JI.2.3 Note on the </>' equation by N.P.Buchdah/ (TN 11 , February 1981)
In a recent paper entitled 'Twistor description of classical Yang-Mills fields' Henkin &
llanin give a reformulation of the classical coupled Yang-Mills-Dirac equations in terms of
obs tructions to extensions of bundles and cohomology classes on regions of ambi-twistor
$pace PA. The purpose of this note is to demonstrate how the same trick can be applied to
the ¢J4 equation . (For relevant information concerning PA, see Eastwood §I.2.2.)
Let U be an open subset of compactified complexified Minkowsk i space M. By the ¢4
equation on U I mean the coupled system
l a) J1 = ).¢l/J
(b) 04> = I'¢
(e) O.p = 1''''
Let VCPA be the open set associated with U in t.he usual way . If U is such that. its
intersection with every nu ll geodesic is connected and has Hl(C) = 0 = fP(C), t hen the
fo llowing isomor phisms hold :
26 I 2. Concrete approaches to the Penrose transform
HI( V,O( -l ,- l),d( U,O[ -1)[ -1]')" H'( V,O( -2,-2))
H'( V,O( -1 ,-3))" r( U,O[ -2)[ -1]'), H'( V,O( -3,-1))" r ( U,O[ -1)[ - 2] ' )
If q"t/J correspond to ¢E HI ( V,O(O ,- 2», ;Pe el ( V,oe -2,0» respectively , and J.I corresponds
to jJE Hi (V,O( - 1,-1)) and p'E H2( V,O( -2,-2» under these isomorphi sms , then equation
(a) above becomes
where u is the cup product, whilst equation (b) (resp. equation (c» is in terpreted on Vas
the statement 'The obst ruction to t he extension of ~ (resp. tp) to the first infinitesimal
neighbourhood of P/\ in Pxp· is eq ual to jiU¢ (resp . p.U4».'
§I.2.4 The wave equation in even dimensions by L.P. Hughston (TN9 , November 1979)
J would like to record here a new contour integral formula that generates solutions of the
wave equation in even-di mensional com plex fiat space. The formula generalises Penrose's
well-known twistor contour integral formula for t he wave equation in Minkowski space­
t ime. When interpreted cohomologica1ly t he new formula gives rise locally to all analytic
solutions of the wave equation , modulo a few technical it ies. The technicalities are of the
same nature t hat a ri se in the Minkowski case, a nd t hus a re not very serious. T his
investigation arose to some extent as a. consequence of a seminar given in May 1979 by
Eastwood, who was in vestigating the Dirac equation from a twistoria l point of view (see
§1.2.2) . His work suggested looking at the spin operator for a. system dependent upon a
twistor pair (ZO', Wo). T his operator (cf. Penrose 1975) is:
where
j= - zo O}- W.8~. -4.
cd and I noticed subseq uently that it sufficed to consider functions satisfy ing
and that a ll relevant eigenfunctions of S2 could be built up from such
ctians in a strrughtforwa rd way. It turns out that t his equation can be solved very
ply by means of a contour integral formula, wllich can be described as [oJlaws. Let
J.. Uo,x"I1) be a ho lomorphic function depending on A. n ° and ;rr", with )('11 skew.
J(ZO,W.)= f F(W.n·,na,ZlanPI),u/ ,
--.b GJfl=ealh6 f1 °dnP/l.dn 7,.,d f1 6. It is not difficult to see that j{Z,W) does indeed
' sfy t he wave equation {}2// 8~8 W Q = O. Furt.bermore, with an appropriate adjustment
'he definition of GJU it is ev ident. that the range a of is irrelevant, a nd that the form ula
us solves the wave equation in any even dimension. It is not hard to check that F
ods upon t he right number of variables needed in order to describe a solution of the
\·e equation, namely 2n-1, where n is the range (n2::2) of cr.
tf {>.,llo ,~,8 } is thought of as an (n2 +n)/2-d imensional complex projective space, then
subvariety defined by X{afJO-rl=O has dimension 2n-1, and this is the space upon
P is act ua ll y defined. Let V denote this variety, a nd let V denote a subset of V
cpt o ut by ll-plane as za and WCl' are varied locally. Then the relevant cohomology
up is If"-'( V ,O( -n)).
In the case of eight dimensions (i .e. the twistor case) there is a curious link with E.
artan's principle of triality that appears to be of considerable s ignificance and wi ll be
cribed in detail elsewhere. (Cf. E. Cartan, pp. 116-122.)
fe rences
.artan, E. (1981) The theory of Spinors, Dover.
Pen rose, R. (1975) Twistor theory, its aims and achievements, in Quantum g1·avity, an
Oxford symposium, eds. C.J.lsharn, R.Penrose and D.'W.Sciama, D.U .P.
28 I 2. Concrete approaches to the Penrose transform
S1.2.5 Extensions of massless fields into CP' by L.P.Hughston & T.R.Hurd (TN 12, July
1981)
The wave equation O¢(x) = 0 in flat space-time is manifestly invariant under the action of
the lO-para meter Poincare group, but is in fact invariant under t he larger 15-parameter
conformal group. The conform al invariance of the wave equation can be made directly
evident by regarding 0 as the rest rict ion of a su itable manifestly S U(2.2) invarian t
differential operator on Cps.
1. Notation. For homogeneous coordinates on Cp5 we writ e )(,P, with L~fJ = -x:'P,
Space-time is t he quadric X'J::= 0, where X 2 : = ~tJ X(r fJ = ~£tlJ' "(~"~P X"y ~ . This quadric will
be called n. The sheaf O(n) on Cps is defined in the usual way. Its restriction On(n) to
{} is t he sheaf of germs of holomorphic spa.ce-time fields with total conformal weight t1j for
convenience here we ignore the distinction between 'primed ' and 'u n primed ' conformal
weights (cf., however, our remarks in §1.2.6) . In what follows V will be a region in a t and
U will be a region in Cps such that una = v. By Pn we denote t he rest riction map to fl .
' Infinity' in n is de fi ned as usual to be the cone 3 = nnt' , where .t : = {lo fJ X'1J = O} is t he
tangent hyperplane of the 'absolute' point f' fj En. The wave operator 0 is defin ed to be
\l().\la on Q-j , and to be identically zero on j. Let ¢(~# ) defin ed on U be holomorphic.
and homogeneous of degree n, representing an element o r r( U.O(n)). " te shall use the
. X-, 8/8''''P d , . , X-, oPX-, notation Q(J = A I an .a. = A = cr {3 '
Proposition 1: If n = -1, then PnU tP = 0 if and only if OPn¢ = O.
Proof The generators or infinitesimal U(2 ,2) t ransformations on Cps are ~ = _2 .. '(l fJ .. y fJ"(l
satisfying the commutation relations [Ep ,~] = 6~ ~-6; Eg_ The total mass o perator for
the Poincare group acting on ClPs is accordingly given by i.f".!. = IcrJjf" EP £'!. The mass
operator on n is indu ced naturally by restriction-i .e. if on(.;t" tJ) represents a n element of
r( V,On(n») and if ¢(~p ) is any extension of <PnPC'} {j ) into Cps su ch that Pno = 6n• then
the space-time mass operator AliJ( = - 0) is defined by t he action Mhc?n = Pni1'J.6 . It is
Dot difficult to convince oneself on physical gro unds that. Mh is in dependent of t he choice
of extension of ¢n into CPl> . Using the e lementary identity
and its diffe ren tial counterpart , one can establish that the Cps mass operator can be
§I.2.5 Extensions of massless fields into cps I 29
expressed in the form -2 1:2 . - 11 - 2
Ai = -;(l' X) o+(N+J)]·X]·X -IX (l.X) ,
here N is the Euler operator ~yoftX"'ft. For n= -1 it therefore follows that
Consequent ly if a field rp with n= -1 satisfies PnO¢ = 0, one has D¢n = O. Conversely, if
:Jon = 0 t.hen Oq, must vanish everywhere on V, evidently, except possibly on vn!l;
.owever , if 04> vanishes on V-( !l n V) thcn it must vanish on Vn!l as well , by continuity. 0
2. Solutions to the wave equation. Proposition 1 is very useful in t he analysis of certain
d asses of solutions of the wave equation. We list a few examples below:
J) elementary state" ¢(X·')=(KP)-' with P· P=O .
2) 'higher order' elementary states (see Eastwood & Hughston 1979) :
¢(x"')=(XQ.)(X.P)-' with Q.Q= Q.p= p· P=O.
3) Ijr potential fields (and certain generalizations thereof):
In each of t.hese cases one can verify very directly, using Proposition 1, that the wave
equation is satisfied. Note that in cases (1) and (2) t)~ = 0, whereas in (3) only
PnO~=O.
3. A contour integral formula. Solutions to 0; = 0 can be gcnerated using a two twistor
technique. Lct f(ZO, Wo ) be homogeneous of degrees (-1, -3). Then the field
where ~ \lV= ,r»"-rA WodWp .... dW,. .... dW" is homogeneous of degree -1 and satisfies O¢ = O.
To prod uce e lementary states one puts f(Z, W) = {(A .Z)( B· W)( C· 11')( D· IV)) -', where A,
B, C, & D are constant twistors .
O ne can also generate solutions o f alP = 0 directly from a 'twistor function ' j{Z)j see
Hurd (1981) for detai ls.
30 I 2. Concrete approaches to the Penrose transform
4. Fields of arbitrary helicitv. Solutions of the zero rest mass equations with positive
helicity correspond to symmetric fie lds q,0,,,p(X) on Cps homogeneous of degree -I, .' o.···fJ 0 d Q· · ·fJ satisfYing PnO¢ = an PnX.,oq, = O. When these conditions are imposed the
In-part' of Pnt/J°"'fJ is a symmetric spinar field ,pA' .. oB' satisfying VAA'q)A1 ''' 8,=Oj the
remaining Cparts' of Pnt/J°"'{J are uniquely determined in terms of ,pA' ... B" Negative
helicity fields are generated analogously by dual symmet.ric fields <Po ... p(X) sat. isfying
Pn-UtPo ... p =O and PnX1a¢o"' P = O. In the case of Maxwell fields there is a close
connection between our work and some results of Dirac (1936) . [Gratitude is expressed to
Tsou S.T. for bringing Dirac's paper to our attention.]
5. New solutions of the ¢4 equation. Proposition 1 can be applied also in t he analysis of
certain non-linear equations. In particular if 2oq'l = >.t/>3 +t/J with Pnw = 0, then tPn satisfles
04> =).4>'. An example of such a field is generated by putting <i>=(A.X)-1 with A non­
simple, i.e. A·A#O. A 15~parameter class of solutions is generated by putting
where OCl P,.6 is any 0{6,C) pseudo-orthogonal matrix, sat isfying OCIP -,60-,6 pa = 6~Cl6~1.
References
Dirac, P.A .M. (1936) Ann. of Math, . 37, No.2, 429-442
Eastwood, M.G. & Hughston, L.P. (1979) §O.2 .15.
Hurd, T.R. (1981) Conformal geometry and massive fields, unpublished essay,
Mathematical Institute, Oxford.
§1.2.6 Conformal weight and spin bundles I 31
iL2.6 Conformal weight and spin bundles by L.P.Hughston & T.R.Hurd (TN 12, July
981)
Conformal weights. An important simplification that arises from t he Cps approach to
!:pace-time is the natural description of the various conform ally invariant space-time
bu nd les. Recall that ;(l" = _x"Ct (cx,P = 0,1 ,2,3) are homogenous coordinates on Cps and
"bat Minkowski space-time may he regarded as the 4-quadric defined by the relation
.. ,.f.a/l X'Y6] = O. The conformal group is the subgroup of the projective group that preserves
this quadric.
We shall call the Cps sheaf CJ(n): = CJ(n,O) the sheaf of germs of holornorphic sections
.. -ith primed conformal weight n . Consider now the trivial twist~r bundle 0 00 , all its
powers. and all tensor products with O(n) (e.g. O{Q.Bl(l): = 0° I\0" ®0(1»). Because of the
dimension of twistor space (4), the bundles 0 and O[~.t"..s l are isomorphic; but not
nat urally so; a choice ofa global section of 0[0.81'6 ] must be made. Let O[oP'Y 6J:=O(_1/,1)
be called the bundle with ' primed weight' - 1 and ' un primed weight' 1.
Spinor' sheaves on CP5. One can proceed to define what we shall call spinor sheaves.
Consider the fo llowing infin ite CP5 sheaf sequence (which is exact only when restricted to
il) :
( J)
and in particular examine the images of these maps ".j' p( -1), 1°(0). j'[~ ,hJ(l). ". which
are defined by exact sequences:
O~(-l) ~1°(O)~O (2)
These image sheaves (not locally free), tensored with powers of 0(1' ,0) and 0(-1',1), will
be called spinor sheaves. There is some redundancy in their defi nition because of the many
canonical isomorphisms, such as
32 [ 2. Concrete approaches to the Penrose transform
1[oP'[(1 )®O( -1 '.- 1 )0:1 p( -I).
Thus, a primed SpinOl" field, for example, lS represented on Cp5 as a. twislor lJolu ed
fun ction ¢O(X) that is of the form ~~~fI(X) for some function ~ fJ ' Out does this really
make sense? To see that it does, restrict attention to the quadric n.
Primed and unprimed conJormal weight. The space·t.ime conform ally weighted bundles are
given by 0n[n]/[m]: = PnO(/l',m), where
D(n'.m): = [O(l'.O) ®n+m] ®[D(_l'.l)®m].
When restricted to 0 , the sequence (1) becomes exact (since, on il , q,Q = ~f1 ~ f1
~<p[Q X"..,J = 0, etc.). allowing images to be identified with kernels. In other words,
sequence (1) spl its int,Q a series of short exact sequences on [}:
(3a)
(3b)
Note too that the sequence (3a) tcnsored with PnO(l) is the natural dual of (3b).
Infinity. The usual picture of spin bundles requires a breaking of conformal invar iance by
the choice of infinity twistor
and the restriction of attention to affine Minkowski space. ('We do not fix ~# since tbis
would give us a sect ion of O{a,8l"~J).
A sect ion of Pnj'O(O) is a twist 0 twist~r valued fie ld ~O( .. .\) such that dJfa XlJ l"l = O. Let
Then , since t/J [o xP..,] = 0 , the map is invertible on 11-.9:
11.2.6 Conformal weight and spin bundles I 33
" e can therefore make the identification Pn1°(O) :;;::: SA'[O]'[O).
The dual of Pn:t°(O) is Pn1o(1), a section of which may now be thought of as an
~uivalence class of twist 0 dual twistor valued fields [~o(.X)] = ~Q(A') mod
X·" II'hJ(X)}. On fl-l let:
wbich is independent of the choice for [{a] . Moreover, we can recover [eo] from €A' since:
The inner product between ,p°and [eo1 is:
T he relation between the primed bundle and its dual is as follows:
A sim ilar (but slightly trickier) exercise leads to the identifications:
and
The canon ical isomorphism SA'{OJ'[O];;;;SA,[-l)'[O] may alternatively be regarded as a
consequence of the existence of a canonical isomorphism between S[A' 8'1[0]'[0] and
0$1[-1]'[01. This isomorphism is executed by multiplic.ation by ~yo /3 (the canonical section
of S[A' 8'][1]'[0]) a nd is a direct rein terpretation of index raising using e. A'81 .
The choice of 'epsilon'. Care has been exercised througho ut th is note to avoid the
introduction of £0/3-,6' this being a. choice of global section of 0[0/31'6]' 'rhe lack of a n £
requ ires the distinction between primed and unprimed conforma l weights : a choice of E
34 I 2. Concrete approaches to the Penrose transform
affords their identi fi cation.
The distinction between 'primed' and 'unprimed' conformal weights is a subt.le one, and
may form suggest ively t he basis - we are tempted to speculate- for all identification
some of the quantum numbers of e lementary particle phys ics.
§I.2.7 Mini-twistors by P.E.Jones (TN 14, July 1982)
Mini-twistor space MT is a two dimensional complex manifold t hat arose in connection
with solving the Bogomoln'yi equations on R3 , Hitchin (1982). This note will explain its
relationship to the standard twister picture and also how stationary zero rest mass fields
can be described in mini-twister terms.
Consider a unit time-like vector field gt on eM I , Its integral curves correspond to
quadrics in PT I each having an a-generator (the line' J') removed. 3, induces a vector
field..., on each such quadr ic. T he integral curves of "I are the p-generators of the quadric.
In homogeneous coordinates . AA' a
1= It 11" A'-----:4 Ow
where tAA ' = diag (1,1). MT is obtained from PT r On factoring out by the ,a-generators,
resulting in a rank 1 holomorphic fibration p:PT I-+MT.
Explicit ly
Thus (=~, Zl = C'Z', so MT is the 0(2) line bundle on PI'
As far as stationary systems on eM I are concerned, o ne can conside r them as ex isting
on C 3 ? wh ich is equipped with coordinates
xA' B' = (-(X+iY) . z ) z X-ty
'We have the fo llowing correspondence diagram:
( A'B' ~l') x , , ~ o
§I.2,7 Mini-twistors I 35
.here:l is the projective spin bundle on C 3 , The image of a space point under 11~-1 is a
ine (a. holomorphic sect ion of 0(2). The image of a mini·twistor (<,ZO) under vjJ-l is a
u11 plane in C 3 .
0'\' 1'0' Q'o' """"jTi"i If one restricts to real space points ($ :::::z: ,x = -x ) then one obtains real
tines in MT in t he sense of t he reali ty structu re introduced in [1], and mini-twisto rs
correspond to oriented geodesics in R3 .
Stationary zero rest mass field on eM f correspond to solutions of the zero rest mass
equations
on C', Define the sheaf0 JlofT ( - n-2) on MT by 0MT(-n-2)=n*O(- n-2) where n is
the projection MT -Cpl , Call an open set UCC3 regular if its intersection with any null
plane N is (i) connected and (li) satisfies HI< UnN, C) = O.
Applying the techniques of [21 to the above double fibrat ion, one obtains the ro llowing
result:
Theorem: If UCC 3 is reg ular, then
whe re the spinor field tP 8' . . L' has n indices .O
~!ow let VCRs be a geodesically convex open set, and tP 8' ... L' a real analytic solution of
t he zero rest mass equations on V. Since these equations are ellipt. ic on t.hree space, there
must be some neighbourhood of V, N( V,{II B' ... L,)CC3, such that (II 8' ... L' extends to a
solution on N( V,IP 8' ... L')' Without loss of generatily one can take N( V,I/> B' ... L') to be
regular. Applying the preceding theorem, one has
H1(pv -1 N(V'¢B'",L')' 0MT( -n-2»)=<{real analytic spin g ZRM fi elds on V which
extend to N( V,¢ B'" ,L,J} ,
36 I 2. Concrete approaches to the Penrose transform
Choosing progressively smaller neighbourhoods of Vand taking direct limits one gets
following :
Theorem: If VCIR:3 is geodesically convex and open, then
H1(JiIl-1 V,OMT( - n-2)~{real analytic solu.tions to VA' B' q, B' ... L' == 0 on V}
where the spinor fie ld q, B' ... L' has n indices.O
The relationship between min i·twistor cohomology and cohomology on PTf can be deduced
from the following short. exact sequence of sheaves on PT f
O-OsC -n-2)i..OC -n-2)2.0( -n-2)-O
where 0sC -n-2) is defined to be the kernel of the twistor time·translation operator 'Y
acting on O( -n-2) and i is inclusion. The corresponding long exact sequence is in part
-n<'(PT',O( -n-2))-H'(PT ',OsC -';-2))-H'(PT ',O( -n-2))2.H'(PT ',O( -n-2))-.
Since the first term vanishes in the cases of interest (n2:0), we have
H'(PT ',0 s( -n-2))",keq:H'(PT ',O( -n-2))-H'(PT' ,O( -n-2)).
Now 0s( -n-2) = P- 10MT(-n-2) where p:PT1 -MT is projection and p-l means
topological inverse image. Using [3] we have Hl(PT1 ,Os(-n-2)~H!(MT,C1MT(-n-2».
Thus, the relevan t mini·twistor cohomology corresponds to twistor cohomology annihilated
by the twistor time·trans lation operator i.
Thanks to M.G .Eastwood for helpful suggestions .
References
[1] Hitchin, N.J. (1982) 'Monopoles and geodesics, Commun. Math. Phys. 83579
[2] Eastwood, M.G., Penrose, R. & Wells, R.O. , Jr , (1981 ) Cohomology and massless
fields, Gommun. Math. Phys. 78,305·51.
13J Buchdahl, N.?, The inverse twistor function revisited , §I.3.5.
§1.2.8 A connection between the wave equation and pure spinors I 37
11.2.8 A remarkable connection between the wave equation and pure spinors in
!Ugher dimensions by L.P.Hughston (TN 14, July 1982)
§I.2.3 I showed bow the (complex) wave eq uation in even dimen sions could be solved by
of a contour integral formula. The prescription is as follows: Let 0' = 1·· on, and let
~ .1r0,...r"p) be a holomorphic function of t he Hn2+n+2) variab les {>. ,1r'\.,yo" ) , where
~a = _xP° . Let GJll denote the natural holomorphic (n-1) form of t wist n on the
JROjective a·space Cpll-I, Then
(1 )
sat is fies the wave equation 82¢/DZOaWo ;;;; O. We require F to be homogeneous of degree
- n collectively in the triple {>'.'lf0,~p}. In the case n=2 one can verify that. ( 1) reduces
the standard twistor contour integral formula for scalar ze ro rest mass fields on space­
-i.me.
In dimension six the wave equation can be solved in a related but disti nct manner that
l5 of considerable interest. Let a = 1· · ·4 , and let F(zn I Wo) be a function of the pair of
~~;istors {ZOO, Wo- }, homogeneous of degree -4 in the pair , but not (necessarily)
~mogeneous in the twistors individually . We assume F is defin ed on a domain away from
{\"o ::::: O. Let ~fJ = _xP o- be coordinates for C6 , and put :
(2)
c here "] W= €o-~..,6 W(>dW~dW..,dW6' One can verify t hat a24>/8 ,1YO-~aXo-" = O. which is the
6-dimensional complex wave equat ion. The contour in (2) is, for each val ue of y:o- fJ ,
Sl XS1XSI, and it is straightforward to work out numerous explicit examples.
Note that in formula (2) t he domain on which F needs to be speci fied is not, strictly , a
;·dimensional region in the CP7 {ZO, Wo }, but rather a reg io n in the 6-dimensional quadric
ZO Wo = O. But th is quadric in CP7 can be regarded as the 'pure' projective sp in-space (in
-he sense of Cartan 1937) for the complex orthogonal group O(8,C).
A pattern begins to emerge: The invariance group of t he wave equat ion o n (affine)
space-time C 4 is O(4,C) . T o solve it we cO llsider holomorphic funct ions on regions of a
pure spin-space (which of the two spin-spaces is taken does not matter) for the group
0(6,e) . Similarly, the invariance group of t he wave eq uat io n on C IS is O(6.C), and to solve
it we consider holomorphic function s on regions of a pure spin-space for the group O(8,C).
Rema,rkably, the pattern continues. The wave equat ion on C8 , for example , is solved
38 I 2. Concrete approaches to the Penrose transform
by the use of holomorphic function defined on regions of a. pure spin-space ' for t.he group
O( lO,C) .
To see this explicitly let us go back a step and examine t he s pin-spaces for O(S,C) in
greater detail. For coordinates on C8 we write {)..~ .tJ 'Il} with )...a fJ skew. The
fundamental quadratic form preserved by 0(8,C) is <P = ~.tJ Xo ,6 -4Ajl. The pure Spin­
spaces in this case consist of the two distinct families of isotropic 4-planes through the
origin in C8 • ' Isotropic' means 'lying in the cone ~ = 0'. The ' ll'-planes ' are parametrized
by pairs of twistors {Z\ WQ } satisfying ZC' WQ=O. For fixed {ZO,Wo } the equations for
the corresponding a . plane are:
(3)
Note that these relations imply ifJ = O. Similarly ,p-planes ' ate parametri zed by pairs of
twistors {PO ,Qo} subject to po Qo = 0, for which the corresponding equations are:
( 4)
which likewise imply f/J = O. An Q-plane aJways intersects a p- plane in C8 . If they arc in
general position the intersection is 'weak" i.e. in a line. If the a-plane is { Z. W} and the fJ­
plane is {P,Q} . the intersection consists of points in C8 proportional to the tri ple:
(5)
which gives a ray in the cone f/J = O. If the intersection is ' st rong' then expression (5)
vanishes , and the intersection consists not merely of a ray , bu t rather of an ent ire 3-plane
through the origin , lying in f/J. Moreover for an Q-plane { Z. W} and a .a-plane {P,Q}
strong intersection is a necessary and sufficient condition for there to exist points in C8
(not necessarily in (/j = 0) such that
(6)
Thus, for a fixed .a-plane {P,Q} , the space of a-planes th at strongly intersect it ca n be
parametrized by points in C8 ,
A pure spinor for the group O(lO,C) can be regarded as a kind of ' hyper-twistor ',
composed of a pair of spinors for the group O(8,C), each subject to lhe condition of purity,
and furthermore subject mutually to the condition of strong '"t~N~d.on. The 'semi-
§1.2.8 A connection between the wave equation and pure spinors I 39
spinors' for G(ID,C), which form a linear space, can be represented by quadruples {Z. Wi
P,Q}, and among these, the ' pure ' ones are those which satisfy Z· W=O , p·Q=O, and the
vanishing of (5).
Suppose FeZ, Wj P,Q) is hoiomorphic, and homogeneous of degree -6 in {Z, W;P,Q}.
\Ve shall construct from it, by restriction to the space of pure OCID,C) spinors. a solution
of t he wave equation in eight dimensions. We parametrise F as in (6) above, and write
where ~7(p, Q) is the natural 7-form, homogeneous of degree 8, on the projective {P,Q}
space CP7, The contour is taken to surround the pole P' Q=O, so that after the resulting
Cauchy-type integration F is restricted down to the pure spin·space. It is straightforward
to verify t.hat. rP satisfies the wave equation:
8' ) 0>:8P ~=O. (8)
thus the ' purity ' conditions 011 spinors III ten dimensions are intimately related to
properties of the wave equation in eight dimensions. These ideas generalize to higher
dimensions .
Gratitude is expressed to T.R.Hurd, R.Penrose & S.B.Petrack for useful ideas and
suggestions, some of which date back to the summer of 1979 .
References:
Hodge, W .V.D. & Pedoe, D. (1952) Methods of algebraic geometry Chap . XIII , C.U.P.
Cartan, E. (1981) The theory of spinors, Dover.
Hughston, L.P. (1979) in Complex manifold techniques in. theoretical physics (cds.
D.Lerner & P.D.Sommers) pp. 115·125. Pitman.
40 \ 2. Concrete approaches to the Penrose transform
§I.2.9 The Penrose transform without cohomology by M.G. Eastwood (TN 14, July 1982)
If U is an open subset of compiexiiied cornpactified Minkowski space and U" is the
corresponding open subset of projective twistor space then t he Penrose transform is the
homomorphism:
GJ : HI( U" ,O( - n-2»- {holomorphic solutions of the massless field equations of heJicity n/2}.
Under simple topological conditions on U. GJ is an isomo rphism (with a potent ial modulo
gauge interpre tation if 11..::;-1). In case n= -2 (left-hand ed or an t i-self-dual Maxwell ) the
transform can be seen much more geomet.rically as t.he twisted photon construction - a
special case of the Ward correspondence. The purpose of t his note is to give a similar
geometric interpretation for n~O.
In general if V is a holomorphic vector· bundle (over any complex manifold ) then an
element wE Ht(O( V) may be regarded alternatively as either
(i) an extension 0_ V~ E~l-O of the trivial bundle by ~'. or
(ii) an affine bundle A with underlying translat ion bu ndle \ '.
The links between these three concepts are straightforward. For example (i) - (ii) is given
by setting A = 'If-1(1 ) the fibre·wise action of V being given by
o
Given A, w is the obstruction to finding a global section. T.h~ ~.ions ( i) & ( ii)
have appeared already in the context of twistor theory (e.g . 1 ~ ~ 12 and chapter
I1.4, (ii) Penrose §O.3.7 & Hughston §O.3.8, especially (li ..
photon. From now on I shall consider only the case n~ O.
_ for a googly
To produce a. field from either an extenion of 0 by O( - "1 - _ 01 froa .... affine-bundle
with underlying translation bundle O(-n-2) is straightforwani: tk of __ field at x
11.2.9 The Penrose transform without cohomology I 41
. obtained by restricting the extension or affine bundle to L%. To describe the inverse AA'
procedure suppose 'At" .. An,Ercu,crAl""An,r-lJ ') satisfies 'il ItPAt""An'=O and that
ZE U" , Pick any :cEZ, the corresponding O'.p lane in U and cons ider the equations
For any fixed x these a.re just algebraic equations for the unknowns f &. g. In case (i) we
kave homogeneous system of equat ions with a two-dimensional linear solution space. Case
ti) is an in homogeneous system so its solutions form an affine space under the action of
BOlulions of the homogeneous equation 0 = 9A'."D,7rD'. In these equations f is a complex
.um ber and 9 is a symmetric s pinor homogeneous of degree - l in If. Solut.ions of (i) gi\'e
aD extension of 0 by 0(-n-2): h-!f=O,9A' .. _D,=h7f A,···'lf D, ], (j,gA1 ... d J- f. Solutions
of (ii) give an affine space over O( - n-2) because gA' . " 0' = h'lf A I· " 1(' 0' is the general
solut ion or the homogeneous equation. To obtain objects 011 U" it is necessary to see that
solut ions of ( i) and (ii) are in some sense independent of x. To this end it suffices to
assume t hat Z is connected and simply connected and to exhibit flat connections to effect
parallel transport of solutions. In case (i) we may take
~1 any thanks to P.E.Jones for useful discussion.
42 I 2. Concrete approaches to the Penrose transform
11.2.10 Integral formulae for D~ = 0 by R.S. Ward (TN IS, January 1983)
I want to make some remarks about various integra] solution formulae for t he wave
equation in n-dimensional flat space, and about whether these integrals generalize to non­
linear equations.
The first formula is as follows. Let:r:tl denote the coordinates of complexified flat rr
space en (so the index a runs from 1 to n). Let va = 110 ( 1 •.... (n-2) parametrize the null
directions; the ('s a.re complex variables and are allowed to take on the value 00. Then
(W)
solves the wave equation. The integral is (n- 2)-dimensional.
Whittaker [Math. Ann. 57 (1903) 333-335] in effect gave this formula for n=3 and 4.
He said nothing about higher n, but Bateman [Proe. Lond. Math. Soc. (2) 1 (1904) 451-
458] clearly knew that it worked for all n (although he did not write it down).
Example: dimension three. For n = 3 , (W ) can be rewritten as
contour 51 (W3).
Here 11'" is a 2-spinor a.nd ,r-4 B = xBA a.re the coordinates on C3 . This is exactly t he formula
tl1&t one gets from mini-twistor theory (see e.g. Jones in §1.2.7).
Example: dimension four. Here the formula looks like
contour S' (W4).
This is, I think , the Kirchoff integral that gives ¢ in terms of its DuD-da.tum on~. rt is
what one gets from a. 'twistor theory' in which the 'twistor space-' is the set of all null
hyperplanes in C4 •
Bateman (see reference cited a.bove) reali~ed tha.t in even dimensions there is an
a.lternative formula. (He only dealt with dimensions four and six explicitly). Again we are
in dimension n, but now n is restricted to be even. A null cone is 'made up' out of totally
null ~n-planes. We have to choose a (~n-l)-parameter family of t.hese !n-planes that fill
up the null cone of the origin . Thus. choose v~«(), ... ,v~/2(O such lbat
(i) ( denotes a (!n-l)-tuple of complex parametersj
11.2.10 Integral formulae for O¢ = 0 I 43
(i i) for each (, ti!..()xQ = 0 for j= l ,,,,, ~n defines a totally null ;n.planej
( iii) as <" varies, t hese sweep o ut the nu ll cone of the origin in en.
¢(X) = f!(.i.«)X'.()d( (B)
. en gives the solution of DIP = 0 in en, One way of implementing this const.ruction was
ven ted by Hughston (see §1.2.4). In his notation, (Zt, W o ), Q = 1,2 •.... ~n. are the
II)Qrdina.te5 on en, the metr ic is dZOdWo 1 and t.he formu la reads
Exa mple: dimension 4. Here the fo rmula reduces to the standard twisto r formula
f . AA' I q,(x) = J{tX 1r A"1!' B,)d'ir , contour S . (84)
Remarks and questions
fj) The 'twistor space' on which the integrand f of (8) is defmed , is a space whose points
correspond to some (but not , for n = 6, all) of the totally null ~n-planes in en, There are
also formulae which use all the cr-planes. as was pointed out by 8ughston in §1.2.8. For
example, in dimension six, this formula takes a function of 6 variables and does a 3-
dimensional integral , whereas 86 takes a function of 5 variables and does a 2-dimensional
integral. Presuma.bly one can get from the first formula to the second by doing one
dimension 's worth of integration.
( ll) The main interest for me of all this is the question of whether a.ny of these formu lae
generalize to non-l inear equations. We know that W3 and 84 do: 'W3 generalizes to solve
the Bogomolny equations, and B4 to solve the self-dual Yang-Mills and self-dual Einstein
equations. In each case, an 'ordinary' HI generalizes to a 'non-linear' lIl. Now in
di mension 2:5, it seems to me that a solution formula for a hyperbolic equation could never
be of 8 1 type, because there are too many null directions. The formula might be of HP
ty pe for p2:2 (can the formulae Wand B above be interpreted cohomologically?). But I
don't know of any Inon-l inear' H P fo r p2:2. So I wou ld conject.ure t.hat in dimension ~5
there are no non-linear hyperbolic equations which are ' nice ' (in the sense that the self­
dual ity equations in four dimensions are nice; for example, one ca.n write down large classes
of solutions of them) . [Exercise fo r the reader: define ' nice" and then prove or disprove
t his conjecture.] Perhaps t.h is is another reason why space-t ime is four­
dimensional - equations in higher dimensions are too hard to solve!
44 I 2. Concrete approaches to the Penrose transform
Note added, 14 April 1989: Murray has produced a comprehensive modern treatment of
the formula (W) (Math. Ann. 272 (1985) 99·1I5J
§I.2.11 On closed-set coverings for PT+ - correction by R. Penrose (TN 16. August
1983)
In a note in Advances in Twistor Theory, §O.2.2 pp. 35-36, I made the suggestion that the
(;,
would suffice for all positive-frequency analytic massless wave fun ctions (defined on CM+),
claiming that an explicit inverse twistor function construction would show this. However
this is actually not so, it being necessary that the field be defined in a somewhat larger
region, and the claim , as stated, is false.
Any twistor function defined in a neighbourhood or e1nC!2 would yield a field at. any
point of eM corresponding to a line in PT which meets ~I nC!2 in an SI. It is nol hard to
see that, for example, the line joining (1+),,0,1->',0) to (O.l+p.O.l-p) has this property
provided that 1>.21+11'21:52, whereas it extends into PT- whenever 1>'1>1 or Ip l>O,
If I
• '. 0 :--.-'. :.' I •••. ::..:..~
1::=\ .: 00 \
• ,0 0
.' 0 • . "\ o .... ___ ....:.' ... :.; • ..; ....... _~~~
\1.2.12 Applications of the geometry of 50(8) spinors to Laplace's equation I 45
the resulting field extends to a fixed region outside CM+. But it is clear that not all
assless wave functions defined on CM+ can extend to slich a region (e.g . the posit.ive
aeq uency part of a nowhere analytic field on the real compactifjed space M does not
extend outside CM+, and by an arbitrarily small conformal transformtion. CM+ can be
o:-tansformed to CM+ together with an arbitrarily narrow open 'collar' extending into
C\tt-). No such simple 2-set cover will work generally (and to have a reasonable chance
igh t ha.ve to be based on sucb as the 'snail contour' of TN I, p.22. B.D .Bramson &
R..Pe nrose- unfortunately not reprinted in Advances in Twistor Theory). Thanks to M.G.
Eastwood for his perceptive scepticism a.nd to A.Pilato for reminding me of this problem .
1l.2.12 Applications of the geometry of SO(8) splnors to Laplace's equation In SIX
dimensions by L.P.Hughston (TN 17, January 1984)
In a series of articles I have shown that in even dimensions the conform ally invariant
laplace's equation
can be solved by consideration of the appropriate 'pure spin-space' for the group
O(2v+2). Here the case of six dimensions wiJI be discussed in more detail. J treat the
matter elsewhere at greater length [6}.
1. Lap/ace's equation in six dimensions. A conformally invariant formulation will be used
here , analogous to the treatment for four dimensions described in (2,3,4.5]. Let Xi
U= 1···8) be coordinates (or C8 (or R8 ). and homogeneous coordinates for the associated
p1. Let [}jj).t1 Xi =O be a non-singular quadric in this p1. Then, if¢(x') is homogeneous
of degree -2 in >r, the condition
(1)
where Pn denotes restriction to il, depends only upon the