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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 Th´eorique, Aix Marseille Universit´e, Universit´e de Toulon, CNRS, 13288 Marseille, France (Received 7 February 2019; published 28 May 2019) We show how the manifold TÃSUð2; 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 TÃSLð2; 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 TÃG 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 TÃSU 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 TÃG. We can thus ask how the classical W Z phase space TÃSUð2Þ used in LQG can be embedded in Z TÃSUð2; 2Þ. Our work answers this question. We provide a uniform parametrization of TÃSUð2Þ, TÃSLð2; CÞ and TÃSUð2; 2Þ in terms of twistors. The embedding is iden- tified 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 TÃSUð2; 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 TÃSLð2; CÞ and Robinson congruence in the Minkowski space. The point Z TÃSUð2Þ 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 Ã 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 Ã W≡ Z∈PT; corresponds to a projective plane f ¯ ¯ WðZÞ¼0g⊂PT.
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