Octahedron of Complex Null Rays and Conformal Symmetry Breaking

PHYSICAL REVIEW D 99, 104064 (2019)

Octahedron of complex null rays and conformal symmetry breaking

Maciej Dunajski,1 Miklos Långvik,2 and Simone Speziale3 1DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2Department of Physics, University of Helsinki, P.O. Box 64, FIN-00014 Helsinki, Finland, and Ashöjdens grundskola, Sturegatan 6, 00510 Helsingfors, Finland 3Centre de Physique Théorique, Aix Marseille Université, Université de Toulon, CNRS, 13288 Marseille, France

(Received 7 February 2019; published 28 May 2019)

We show how the manifold TSUð2; 2Þ arises as a symplectic reduction from eight copies of the twistor space. Some of the constraints in the twistor space correspond to an octahedral configuration of 12 complex light rays in the Minkowski space. We discuss a mechanism to break the conformal symmetry down to the twistorial parametrization of TSLð2; CÞ used in loop quantum gravity.

DOI: 10.1103/PhysRevD.99.104064

I. INTRODUCTION All these constraints are conformally invariant when M A twistor space T ¼ C4 is a complex-four-dimensional expressed in the Minkowski space . The incidence vector space equipped with a pseudo-Hermitian inner constraints have a natural geometric interpretation in terms product Σ of signature (2,2) and the associated natural of four twistors in a single twistor space: they describe a T symplectic structure [1,2].In[3] Tod has shown that the tetrahedron in , whose vertices correspond to twistors and symplectic form induced on a five-dimensional real surface faces to dual twistors. This configuration is self-dual in a of projective twistors which are isotropic with respect to Σ sense to be made precise in Sec. II. In the Minkowski space coincides with the symplectic form on the space of null this tetrahedron corresponds to an octahedral configuration geodesics in the 3 þ 1-dimensional Minkowski space M.In of 12 complex null rays. the same paper it was demonstrated that the Souriau Our main motivation to perform this analysis is to further symplectic form [4] on the space of massive particles in explore the mathematical relations between twistor theory – M with spin arises as a symplectic reduction from T × T.In and loop quantum gravity (LQG) [6 12]. The building ’ SU 2 a different context it was shown in [5–8] that a cotangent blocks of LQG are Penrose s ð Þ spin networks, with an bundle TG to a Lie group G arises from T if G ¼ SUð2Þ important conceptual difference. Penrose regarded the and T × T if G ¼ SLð2; CÞ. In these references the Darboux quantum labels on these networks to describe only the coordinates were constructed from spinors and twistors, conformal structure of spacetime, specifically angles [13]. respectively. To introduce a notion of scale, he envisaged extending the SU 2; 2 The aim of this paper is to extend these constructions to theory to the Poincar´e group, or better to ð Þ that is the case when G ¼ SUð2; 2Þ, the covering group of the semisimple. The associated conformal spin networks and their geometric interpretation have never been used in conformal group SOð4; 2Þ=Z2 of M. The starting point for our construction will be the 64-dimensional real vector quantum gravity models, but these ideas then flew into the SU 2; 2 space consisting of two copies of T 4 ≡ T × T × T × T. The construction of twistors, which are ð Þ spinors. In symplectic reduction from T 4 × T 4 to the 30-dimensional LQG, on the other hand, the use of Ashtekar-Barbero TSU 2; 2 variables underpinning the theory allows one to interpret manifold ð Þ will be realized by imposing a set of SU 2 constraints: the second-class incidence constraints stating the ð Þ Casimir directly in terms of areas, thus intro- 4 ducing scales. It is nonetheless still an open and intriguing that the four twistors in each copy of T are nonisotropic question to develop Penrose’s original program and show if and pairwise orthogonal with respect to Σ and the first-class and how a notion of scale relevant for quantum gravity can helicity and phase constraints (see Sec. III C for details). be introduced via the translation and dilation generators of SUð2; 2Þ and how it can be compared with the one used in LQG through some mechanism for conformal symmetry Published by the American Physical Society under the terms of breaking. To that end, one needs to establish a precise the Creative Commons Attribution 4.0 International license. relation between SU 2; 2 spin networks and the SU 2 Further distribution of this work must maintain attribution to ð Þ ð Þ the author(s) and the published article’s title, journal citation, ones used in LQG. As a first step in this direction, we and DOI. Funded by SCOAP3. consider the classical counterpart to this question. Recall in

2470-0010=2019=99(10)=104064(8) 104064-1 Published by the American Physical Society DUNAJSKI, LÅNGVIK, and SPEZIALE PHYS. REV. D 99, 104064 (2019)

L2 G;dμ PT* PT fact that the spin network Hilbert space ½ Haar with its holonomy-flux algebra is, for any Lie group G, the quantization of the canonical Poisson algebras of the W cotangent bundle TG. We can thus ask how the classical W Z phase space TSUð2Þ used in LQG can be embedded in Z TSUð2; 2Þ. Our work answers this question. We provide a uniform parametrization of TSUð2Þ, TSLð2; CÞ and TSUð2; 2Þ in terms of twistors. The embedding is identified by a hypersurface where the dilatation generators M match. This matching breaks conformal symmetry in a way C that, unlike in standard twistor theory, does not require introducing the infinity twistor. The paper is organized as follows. In the next section we shall introduce the twistor space T of the Minkowski space, Z W and to prepare the ground for the constraint analysis we shall construct an octahedral configuration of complex rays in MC out of four non-null incident twistors. In Sec. III we FIG. 1. Twistor incidence and null rays. shall consider a set of constraints on a product T 4 × T 4 of eight twistor spaces and implement a symplectic reduction is a dual twistor corresponding to a two-plane CP2 in the to the canonical symplectic form on TSUð2; 2Þ. Finally in projective twistor space PT. The plane Z¯ intersects the Sec. IV we shall comment on the conformal symmetry hypersurface PN in a real three-dimensional surface—the breaking of our construction down to TSLð2; CÞ and Robinson congruence in the Minkowski space. The point Z TSUð2Þ and on the physical applications of our results. lies on the plane Z¯ iff Z ∈ PN. Then the complex α plane Z ¯ meets the complex β plane Z in a real null geodesics in MC. II. TWELVE COMPLEX NULL RAYS FROM Assume that this does not happen. A TWISTOR TETRAHEDRON Let Z1 and Z2 be two non-null twistors. They are ¯ incident if Z1 belongs to the plane Z2 in PT and Z2 The twistor program of Roger Penrose [1] is a geometric ¯ ¯ ¯ belongs to the plane Z1. The two planes Z1 and Z2 intersect framework for physics that aims to unify general relativity ¯ and quantum mechanics with space-time events being in a holomorphic line X12 in PT. Now let us add a non-null derived objects that correspond to compact holomorphic twistor Z3. It will be incident with Z1 and Z2 only if it lies ¯ curves in a complex manifold known as the projective on the holomorphic line X12 above. Thus, given an incident twistor space PT. There are now many applications of non-null pair Z1, Z2, there exists a one-parameter family of twistors in pure mathematics and theoretical physics (see Z3 ∈ PT such that Z1, Z2, and Z3 are mutually incident. ¯ ¯ [14] for a recent review). Our presentation below focuses The plane Z3 intersects the line X12 in a unique point Z4 on the simplest case of twistor space corresponding to the and the four twistors Zi, i ¼ 1; …; 4, satisfy flat Minkowski space. 4 Σ Z ;Z 0;i≠ j: A twistor space T ¼ C is a complex four-dimensional ð i jÞ¼ ð2Þ vector space equipped with a pseudo-Hermitian inner It is not possible to construct a set fZig of more than four product Σ of signature (2, 2) twistors such that (2) holds: the four twistors correspond to four vertices of a tetrahedron in PT, see Fig. 2. The dual ΣðZ; ZÞ¼Z1Z¯ 3 þ Z2Z¯ 4 þ Z3Z¯ 1 þ Z4Z¯ 2; ð1Þ twistors are the faces of this tetrahedron. A fifth twistor Z5 1 2 3 4 cannot be added in a way that makes all sets of three points where ðZ ;Z ;Z ;Z Þ are coordinates in T. ¯ collinear (or such that the plane Z5 intersects all faces of the Let T be the dual vector space, and let PT ¼ CP3 be a T PN Z ∈ PT; Σ Z; Z 0 tetrahedron). projectivization of . Let ¼f ð Þ¼ g be X ≅ CP1 PT PT PN Let ij be a holomorphic line in joining two a real five-dimensional surface in . The points in Z Z X¯ ≅ CP1 are referred to as null twistors and correspond to real twistors i and j, and let ij be a holomorphic line PT Z¯ Z¯ null light rays in Minkowski space [2,15]. A dual twistor in arising as the intersection of the planes i and j. Then W ∈ PT W≡ Z∈PT; corresponds to a projective plane f ¯ ¯ WðZÞ¼0g⊂PT. We say that Z and W are incident if Z lies X12 ¼ X34;X13 ¼ X24; etc:; on the plane given by W. In this case the α plane Z and the β X plane W in the complexified Minkowski space MC intersect which resembles the self-duality condition. The line ij in a null geodesic. See Fig. 1. corresponds to a unique point of intersection of two α planes ¯ Let Z ∈ PT be a non-null twistor. We can use Σ to Zi and Zj in MC. Similarly, the line Xij corresponds to a point ¯ ¯ ¯ ¯ identify the conjugation Z with an element of PT . Thus Z of intersection of two β planes Zi and Zj in MC.Ifi ≠ j, then

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Z 1 III. SYMPLECTIC REDUCTION TO TSUð2;2Þ Our strategy will be to pick two linearly independent sets X13 of four twistors and construct an element G of GLð4; CÞ X mapping one set to the other. We shall then impose a set of 12 Z Z 4 2 Z3 constraints on both sets which will guarantee that G is X Z 14 unitary and has a unit determinant. Some of these constraints 3 will be first class and some second class with respect to the X23 twistor symplectic structure, and we will show (by explicit X Z Z1 34 computation of Poisson brackets) how the symplectic 2 structure on TSUð2; 2Þ arises as a symplectic reduction X24 from the symplectic structure on eight copies of T. Z 4 A. Notation FIG. 2. A tetrahedron in PT. Vertices are the incident twistors, In what follows we shall denote components of a twistor α faces are the dual twistors and the edges are lines corresponding Z ∈ T by Z , α ¼ 1; …; 4, and components of the corre- _ to points of intersections of α planes in MC. ¯ ¯ β sponding dual twistor by Zα ≡ Σαβ_ Z , where ¯ 0 1 the α plane Zi intersects the β plane Zj in a complex null 2 Σαβ_ ¼ ð3Þ geodesics (a light ray) Rij¯, as shown in Fig. 3. 12 0 This leads to the octahedral configuration of 12 complex null rays arising as intersections of incident α and β planes, is a matrix of the (2, 2) inner product Σ from Sec. II. The Σ Fig. 4. The six vertices of the resulting octahedron O in MC imaginary part of gives the twistor space a Poisson correspond to the six lines Xij in PT. The 12 edges of O are structure: complex null rays. α ¯ α fZ ; Zβg¼iδβ; ð4Þ

R14 which is invariant under SUð2; 2Þ transformations of T. Z These are generated via a Hamiltonian action: 4 R24 Mab;Zα Γabα Zβ; Mab ≔ Z¯ Γabα Zβ X12 f g¼ β where α β Z1 a; b 0; …; 5: Z2 and ¼ ð5Þ The matrices Γab ≡ ð1=2Þ½Γa; Γb are constructed out R of the six generators Γa of the Clifford algebra in (4 þ 2) Z3 23 R13 dimensions, and they form a representation of spinð4; 2Þ. They also form 15 out of the 16 generators of uð2; 2Þ. The ¯ last one is the trivial identity element (normalized by 1=2) FIG. 3. An intersection of an α plane Zi with a β plane Zj is a and corresponds to the helicity null ray Rij¯. 1 U ≔ Z¯ δα Zβ s: 2 α β ¼ ð6Þ X12

23 B. Unitary transformations 14 24 4 X Let ðZ1;Z2;Z3;Z4Þ ∈ T ≡ T × T × T × T be four X 21 24 34 32 13 twistors such that the holomorphic volume 31 43 1 X13 Z ≔ ϵijklϵ ZαZβZγ Zδ ≠ 0: 12 X14 αβγδ i j k l ð7Þ 42 4! 41 α Here, for each fixed i ¼ 1; …; 4 the symbol Zi denotes the 32 four components of Zi with respect to the standard basis of T. We shall use a summation convention with the Latin X34 indices α; β; …, and our formulas will be SUð2; 2Þ invariant FIG. 4. The octahedral configuration of 12 complex null in these indices. The Greek indices i; j; k; … are reminis- rays in MC. cent of the internal twistor indices in the twistor particle

104064-3 DUNAJSKI, LÅNGVIK, and SPEZIALE PHYS. REV. D 99, 104064 (2019) program [16,17]. Parts of our construction will break the C. Symplectic structure on TSUð2;2Þ internal symmetry, in which case we will write explicitly Before presenting our main result, let us fix some the sums over internal indices. This makes the resulting notations and provide explicit expressions for the sym- formulas somewhat ugly. We set plectic manifold TSUð2; 2Þ ≃ SUð2; 2Þ × suð2; 2Þ. Let Mab ¼ −Mba form a basis of the Lie algebra suð4; 2Þ: i 1 ijkl β γ δ Zα ≔ ϵ ϵαβγδZj ZkZl ð8Þ 6Z ab cd ac bd ad bc bd ac bc ad P ½M ;M ¼η M − η M þ η M − η M α i α i α i and verify that i Zi Zβ ¼ δβ, ZαZj ¼ δj and also abcd ef ≕ − f efM ; 13 ¯ iα αβ ¯ i ð Þ Z ≔ Σ Zβ. The condition (7) guarantees that the twistors 4 ab Zi form a basis of C . We require it to be orthogonal with with a ¼ 0; …; 5, and η ¼ diagð− þþþþ−Þ. We para- respect to the inner product Σ, i.e., metrize the base manifold with a Σ-unitary unimodular α 4 × 4 matrix G β and the algebra with the generators in the I∶ ρ ≡ Σ Z ;Z 0 ∀ i ≠ j; ij ð i jÞ¼ ð9Þ fundamental irrep, which can be writtenP as a traceless 4 × 4 α ab abα matrix themselves using M β ¼ a

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D. Symplectic structure on T 8 and reduction to and a further reduction by Φ brings it down to TSUð2;2Þ 30 ¼ dimðTSUð2; 2ÞÞ. For the symplectic reduction to Let us consider T 8 and split the eight twistors into two work, however, we have to remove some regions of the α α initial phase space. First, our construction of the group sets Z and Z˜ , i ¼ 1; …4, with Poisson brackets i i element requires non-null twistors and linearly indepen- α ¯ α ˜ α ˜¯ α dent in each sector. Any parallel pair will imply the fZk; Zjβg¼iδkjδβ; fZk; Zjβg¼−iδkjδβ: ð18Þ vanishing of either Z or Z˜ and thus det G ¼ 0. Under these brackets, the scalar products in each set form a Furthermore, the counting above shows that we want closed glð4; CÞ algebra,1 the incidence conditions to be second class; therefore, we must exclude within each sector twistors with the fρmj; ρklg¼−iδmlρkj þ iδjkρml; same helicity. 8 8 Let T ⋆ be the subspace of T satisfying the following fρ˜mj; ρ˜klg¼iδmlρ˜kj − iδjkρ˜ml; ð19aÞ P P anholonomic restrictions: ˜ whose centers are U ¼ i si and U ¼ i s˜i. The other (i) The twistors within each group of four are linearly conformal invariant quantities, the holomorphic volumes Z independent and non-null. and Z˜ , commute with the off-diagonal scalar products, (ii) The twistors within each group of four have different s ≠ s s˜ ≠ s˜ i ≠ j whereas any helicity shifts the phase: helicities, i j and i j for . 8 Proposition.—The symplectic reduction of T ⋆ by the fρmj; Zg¼−iZδmj; f2sm; Zg¼−iZ ∀ m; helicity matching and incidence constraints and similarly for the tilded set, but with opposite signs. We now look for constraints capable of reducing this hi ¼ si − s˜i ¼ 0 ðfour real; first classÞ; ð21aÞ 64-dimensional symplectic manifold to TSUð2; 2Þ. The unitarity discussion earlier has already identified a ρij ¼ 0 ¼ ρ˜ij ∀ i ≠ j ð12 complex; second classÞ; ð21bÞ candidate set of constraints: the incidence conditions I, the helicity matching conditions h, and the unimodular describes a symplectic space of 32 real dimensions iso- condition Φ. The constraint algebra is given by (19a) morphic to TUð2; 2Þ, parametrized by above together with X4 Z˜ αZ¯ X4 α i iβ ab ¯ abα β i i G β ≔ pffiffiffiffiffiffipffiffiffiffiffiffi ;M¼ ZiαΓ βZ ; h ;h 0; h ; ρ − δ ρ δ ρ ; 2s 2s˜ i f i jg¼ f m jkg¼ 2 mk jm þ 2 mj mk i¼1 i i i¼1 i i X4 h ; ρ˜ − δ ρ˜ δ ρ˜ ; M˜ ab − Z˜¯ Γabα Z˜ β; f m jkg¼ 2 mk jm þ 2 mj mk ð19bÞ ¼ iα β i i¼1 h ; Φ 0; Φ; ρ 0 ∀ m ≠ j; X4 X4 f m g¼ f mjg¼ ˜ U ¼ si; U ¼ − s˜i; ð22Þ Φ; ρ˜ 0 ∀ m ≠ j: f mjg¼ ð19cÞ i¼1 i¼1

α ab ˜ ab ˜ These brackets are all zero on the I surface, except for with G β ∈ Uð2; 2Þ, and ðM ;UÞ and ðM ; UÞ, respectively, left-invariant and right-invariant vector fields iso- fρmj; ρjmg¼2iðsm − sjÞ; fρ˜mj; ρ˜jmg¼−2iðs˜m − s˜jÞ: morphic to the uð2; 2Þ algebra. A further reduction by the additional constraint ð20Þ Φ ¼ arg Z − arg Z˜ ¼ 0 ð23Þ Therefore, hi and Φ are always first class. The inci- dences are generically second class; some or all become imposes det G ¼ 1, removes U from the phase space, and first class on measure-zero subsets of the phase space describes a symplectic space of 30 real dimensions iso- where two or more helicities match. In the generic case, morphic to TSUð2; 2Þ, parametrized by (22) above. symplectic reduction by h and I gives a space of 8 Therefore, T ⋆== C ≃ T SUð2; 2Þ with C ¼ h ∪ I ∪ Φ. dimensions Proof.—To prove the symplectic reduction we need to show that the reduced variables commute with all the T 8 − 4 2 − 12 − 12 32 TU 2; 2 ; dimð Þ × ¼ ¼ dimð ð ÞÞ constraints, are all independent, and generate the Poisson algebra of TSUð2; 2Þ. Because of the presence of second- 1For the reader familiar with spin foam models, we point out class constraints, which we have not explicitly solved, the that ρij are used to construct the holomorphic simplicity con- reduced algebra is defined a priori through the Dirac straints introduced in [19]. bracket

104064-5 DUNAJSKI, LÅNGVIK, and SPEZIALE PHYS. REV. D 99, 104064 (2019) X −1 G M fF; Gg ≔ fF; Gg − fF; ρijgfρij; ρ¯ijg fρ¯ij;Gg shows also that and are gauge invariant with respect D 8 i≠j to all first-class constraints in T ⋆. We are left to check that −1 they satisfy the right algebra, namely (16). þfF; ρ¯ijgfρ¯ij; ρijg fρij;Gg α α This means that ðG β;M βÞ span the 32-dimensional ¯ −1 ¯ þfF; ρ˜ijgfρ˜ij; ρ˜ijg fρ˜ij;Gg reduced phase space. We have also already proved that G is ¯ ¯ −1 unitary, and we now show that on shell of the constraints it þfF; ρ˜ijgfρ˜ij; ρ˜ijg fρ˜ij;Gg: relates M and M˜ via the adjoint action, since The only nonvanishing entries of the Dirac matrix are X ˜ abI;h ¯ −1 ab α β −1 α M ¼ − ZiαðG Γ GÞ βZi ¼ −ðGMG Þ β: ¯ fρmj;ρ¯mjg¼2iðsm − sjÞ; fρ˜mj;ρ˜mjg¼−2iðs˜m − s˜jÞ; i thus, the Dirac matrix has zeros everywhere except on 2 × 2 It remains to show that they satisfy the right brackets. blocks along the diagonal. The inverse is then easy to To that end, we compute compute, being given by a matrix with the same structure X ˜ α ˜ γ ¯ ¯ Z Zj Zkβ Zjδ and elements given by minus the inverse of the original Gα ;Gγ ffiffiffiffikpffiffiffiffi ffiffiffiffi ; ffiffiffiffi f β δg¼ ps˜ s˜ ps ps entries. kj k j k j For the algebra generators, we have Z¯ Z¯ Z˜ α Z˜ γ ffiffiffiffikβ jδ ffiffiffiffik ; pffiffiffiffij α α α þ p pffiffiffiffi p fhi;M βg¼0; fρij;M βg¼0; fΦ;M βg¼0; sk sj s˜k s˜j X ˜ α ¯ ˜ γ ¯ fhi;Ug¼0; fρij;Ug¼0; fΦ;Ug¼2: i ZkZkβZjZjδ ¼ 2 sksj The commutation with the second-class constraints means k −1 −1 −1 −1 that the Dirac bracket for the algebra generators coincides × ðsk − sk þ s˜k − s˜k Þ ≡ 0; ð26Þ with the Poisson bracket. For the group elements, we have Z˜ αZ¯ Mab;Gα i GΓab α ; M˜ ab;Gα −i ΓabG α : α ffiffiffiffiffii iβffiffiffiffiffi f βg¼ ð Þ β f βg¼ ð Þ β (with shorthand notation Gi β ≔ p p ) 2si 2s˜i ð27Þ α fhi;G βg¼0; As for the brackets of the algebra generators M, they follow 1 X Z˜ αZi Z˜¯ iαZ¯ Φ;Gα ffiffiffiffiffiffii βffiffiffiffiffiffi ffiffiffiffiffiffi iffiffiffiffiffiffiβ immediately by linearity from the ones with a single f βg¼2 p2s p2s˜ þ p2s p2s˜ i i i i i twistor. We remark that no constraints were used: the Poisson brackets reproduce the right algebra on the whole α 1 1 I;h 8 − Gi β þ ¼0; of T ⋆. The role of the constraints is truly to restrict the 2si 2s˜i matrix to be unitary and special unitary. Z˜ αZ¯ Gα Gα Z˜ αZ¯ TSU 2; 2 α j kβ kβ j β I;h j kβ For the final step leading to ð Þ, note the Poisson fρkj;G βg¼i pffiffiffiffiffiffiffipffiffiffiffiffiffiffi þ iρkj − ¼i ; U; G 2sk 2s˜k 2si 2sj 2sk algebra we obtained is separable, since ð det Þ form a canonical pair with brackets (17), as can be easily verified ð24Þ using Z˜ αZ¯ Gα Gα Z˜ αZ¯ X α j kβ kβ j β I;h j kβ ab k abα β ab fρ˜kj;G βg¼i pffiffiffiffiffiffiffipffiffiffiffiffiffiffi þ iρkj − ¼i : fZ;M g¼iZ ZαΓ βZk ¼ iZTrðΓ Þ ≡ 0 2sk 2s˜k 2sk 2sj 2sk k ð25Þ [except of course if M ¼ U is the Uð1Þ generator, in which case we get correctly 2iZ]. Then, to reduce to TSU 2; 2 , Even though the group element does not commute with the ð Þ we simply impose det G 1 as a (first-class, real) con- incidence constraints, its Dirac bracket with itself coincides ¼ P straint, which modules out U si as a gauge orbit. Since with the Poisson bracket, thanks to opposite contributions ¼ i we already know that two helicities need to have opposite from the two sets: signs, we can fix U ¼ 0 without loss of generality. ▪ X α −1 γ Three remarks are in order. Firstly, when two or fG β; ρijgfρij; ρ¯ijg fρ¯ij;G δg i≠j more twistors in the same set have the same helicity, some or all of the incidence constraints become first class. α ¯ −1 ¯ γ I;h þfG β; ρ˜ijgfρ˜ij; ρ˜ijg fρ˜ij;G δg¼0: The symplectic reduction describes a smaller phase space not parametrized by a unitary group element, because α Therefore, the Dirac bracket of all reduced variables fρij;G βg ≉ 0. Secondly, since the helicities can always coincides with the Poisson bracket. Furthermore, this be made to match in projective twistor space, this shows the

104064-6 OCTAHEDRON OF COMPLEX NULL RAYS AND CONFORMAL … PHYS. REV. D 99, 104064 (2019) importance of using the full twistor space for our sym- dilatation constraint, forms a pair of first-class constraints, plectic reduction to work. and we recover the symplectic reduction to TSLð2; CÞ Finally, instead of working with eight copies of twistor already established in [6–8]. The final reduction to space, we could have picked a pair of self-dual tetrahedra T TSUð2Þ relevant to LQG is done introducing a timelike and T˜ in CP3 from Sec. II. By construction of these direction, which identifies an SUð2Þ subgroup of SLð2; CÞ 2 tetrahedra, the incidence constraints ρij ¼ ρ˜ij ¼ 0 have and a Hermitian structure k · k . From the twistorial view- already been imposed. To impose the helicity constraints point, the constraint achieving this reduction is the inci- hi ¼ 0 we assign four different colors to vertices of each dence of two twistors on the same chosen timelike tetrahedron and define G as a Σ-unitary matrix acting direction. See [9] for a review.2 As a side comment of on a configuration space of self-dual tetrahedra and mathematical interest, it is known [22] that TSUð2Þ ≅ preserving colors of vertices. If we interpret Z ¼ ZðTÞ C4=C obtained in this way is the maximal coadjoint as a holomorphic volume of the tetrahedron T, then orbit of SUð2; 2Þ and that SUð2; 2Þ and Uð1Þ form a the final constraint Φ ¼ 0 is that G preserves the phase Howe pair. It may be interesting to establish a precise of this holomorphic volume, which can also be put in a relation between the Howe pairs and the reduction pre- form sented in Sec. III. Coming back to our physical motivations, the work 1 X Z ϵijklI I ZαZβZγ ZδD ; presented has two applications. First, the twistorial para- ¼ 6 αβ γδ i j k l ijkl ð28Þ TSU 2; 2 i;j;k;l metrization of ð Þ obtained provides a convenient starting point to construct SUð2; 2Þ spin networks and their 2 holonomy-flux algebra through a generalized Schwinger where Iαβ is the infinity twistor and Dijkl ≡ jXij − Xklj are representation. The flux operators will be the standard squared distances between the vertices of the octahedron holomorphic algebra operators used in quantum twistor from Fig. 4 in Sec. II taken with respect to the holomorphic M theory, whereas the holonomy operators can be built metric on C. from a suitable operator ordering of (11). Secondly, our The only other context where a volume of a polygon classical results are sufficient to deduce how the geometric in the twistor space plays a role in physics is the interpretation of LQG spin networks should be seen amplituhedron of [20]. It remains to be seen whether from the perspective of SUð2; 2Þ spin networks. The there is any connection between the amplituhedron and reduction discussed above from TSUð2; 2Þ to TSUð2Þ our work. acts trivially on the algebra generators; hence, the spin label j describing LQG’s quantum of area is simply the SUð2Þ IV. BREAKING THE CONFORMAL SYMMETRY Casimir with respect to the canonical timelike direction I In twistor theory it is common to break the conformal N ¼ð1; 0; 0; 0Þ, namely with respect to the canonical symmetry introducing an infinity twistor, which specifies 3-vector (1, 0, 0, 0, 1, 1) in E4;2. The effect on the holonomy the asymptotic structure of the conformally flat metric [2]. is less trivial. In particular, the SLð2; CÞ matrix element is The choice of infinity twistor determines if the remaining given by symmetry is Poincaré or the (anti–)de Sitter. Here we are ffiffiffi ffiffiffi A A p p interested instead in a different reduction that takes us A ω˜ πB −π˜ ωB i s s˜ A AC D_ h B ¼ ffiffiffiffiffiffipffiffiffiffiffiffiffi ≈ pffiffiffiffiffiffipffiffiffiffiffiffiffiðG B þϵ ðG _ ÞϵDBÞ directly to SLð2; CÞ, since this is the local gauge group of pπω π˜ ω˜ 2 πω π˜ ω˜ C general relativity. As shown in [9], this reduction can be achieved without using the infinity twistor but rather requir- ð29Þ ing conservation of the dilatations between the two sets of A D_ twistors. This means that we preserve not only the pseudo- on shell of the constraints, where G B and GC_ are the 2 × 2 A A Hermitian structure Σ, but also γ5.Sinceγ5 is the equivalent diagonal blocks of (11). Here ðω ; π Þ are the spinor α A in the Clifford algebra of the Hodge dual, it is clear that constituents of Z , and πω ≔ πAω . The LQG SUð2Þ preserving this structure fixes scales. And from the suð2; 2Þ holonomy carrying the extrinsic curvature of the quantum algebra we see that this condition breaks translations and space can be recovered from the Lorentz holonomy as conformal boosts, allowing only the Lorentz subalgebra. explained in [6], and the embedding (29) shows how it On the dilatation constraint surface, the description of determines the argument of an SUð2; 2Þ spin network. From 8 the remaining Lorentz algebra in T ⋆ becomes largely these considerations we can also remark that the LQG area is redundant. Building on the results of [6], we know it is invariant under the SUð2; 2Þ dilatations, whereas the enough to work with a pair of twistors only. To eliminate extrinsic geometry is affected, in agreement with [9]. the redundancy, we thus impose the additional constraints Z Z Z Z Z˜ Z˜ Z˜ Z˜ 1 ¼ 2 ¼ 3 ¼ 4 and 1 ¼ 2 ¼ 3 ¼ 4. 2See also [21] for related reductions to the little groups ISOð2Þ On shell of these constraints, I and Φ become trivial, and SUð1; 1Þ stabilizing, respectively, a null and a spacelike and h reduces to a single equation. This, together with direction.

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A suggestion in line with Penrose’s original program is ACKNOWLEDGMENTS to introduce a notion of scale not from the Casimirs but The work of M. D. has been partially supported D directly from the eigenvalues of the dilatation generator . by STFC consolidated Grant No. ST/P000681/1. M. L. Such interpretation is at odds with LQG, and we have acknowledges Grant No. 266101 by the Academy of clarified why. On the other hand, it may be relevant to allow Finland. S. S. is grateful to Sergey Alexandrov and one to extend the spin network construction of the Hilbert Kristina Giesel for discussions on symplectic reductions space of loop quantum gravity to more general theories like and Dirac brackets. M. D. thanks Dmitry Alekseevsky and Poincar´e gauge theory of gravity or conformal gravity. Nick Woodhouse for useful correspondence.

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