PHYSICAL REVIEW D 100, 045005 (2019) Spinor-helicity variables for cosmological horizons in de Sitter space † ‡ Adrian David,1,* Nico Fischer,2, and Yasha Neiman1, 1Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan 2Friedrich Schiller University Jena, 07737 Jena, Germany (Received 10 June 2019; published 7 August 2019) We consider massless fields of arbitrary spin in de Sitter space. We introduce a spinor-helicity formalism, which encodes the field data on a cosmological horizon. These variables reduce the free S-matrix in an observer’s causal patch, i.e., the evolution of free fields from one horizon to another, to a simple Fourier transform. We show how this result arises via twistor theory, by decomposing the horizon ↔ horizon problem into a pair of (more symmetric) horizon ↔ twistor problems. DOI: 10.1103/PhysRevD.100.045005 I. INTRODUCTION Poincare patches) to obtain the free S-matrix in the static patch for massless fields of any spin. Our formalism and In field theory on flat spacetime, the S-matrix between result provide a plausible starting point for efficiently past and future infinity is an object of fundamental including the effects of interactions in future work. importance. For massless theories such as Yang-Mills and general relativity (GR), the spinor-helicity formalism [1] has emerged as the ideal language [2] for studying the II. GEOMETRIC SETUP S-matrix (with the exception of some highly symmetric De Sitter space is best described as a hyperboloid of unit cases, in which twistor language is superior [3–5]). Since spacelike radius embedded in flat 4 þ 1d spacetime, our Universe appears to have a positive cosmological constant, it is of great theoretical interest to study the μ 1;4 μ dS4 x ∈ R xμx 1 : 1 “S-matrix” in a static (i.e., observable) patch of de Sitter ¼f j ¼ g ð Þ space, with an observer’s past and future horizons in the We will use lightcone coordinates xμ ¼ðu; v; rÞ for R1;4, roles of past/future infinity. So far, there has been remark- 3 μ ably little work on this problem. Instead, the main focus of where r is an R vector, and the metric is dxμdx ¼ 2 theoretical attention in de Sitter space has been with −dudv þ dr . These coordinates are adapted to a de Sitter correlations on its conformal boundary [6–8], which are observer, whose initial and final horizons are defined by unobservable in a true asymptotic de Sitter space (but ðu ¼ 0;v<0Þ and ðu>0;v¼ 0Þ, respectively. The hori- become observable in approximate, temporary de Sitter zons’ spatial section is the two-sphere S2 of unit vectors 2 scenarios such as inflation). r ¼ 1. The tangent space of this S2 at a point r can be In this paper, we take some first steps toward the de Sitter spanned by a complex null basis ðm; m¯ Þ, S-matrix. First, we encode the lightlike field data on a cosmological horizon in terms of spinor-helicity variables, m · r ¼ 0; m2 ¼ 0; m × m¯ ¼ −ir: ð2Þ equivalent to those introduced in [7] for the Poincare patch (see also the constructions for anti-de Sitter, in the Poincare This basis is defined up to phase rotations ðm; m¯ Þ → iθ −iθ patch [9] and in stereographic coordinates [10]). Then, in ðe m;e m¯ Þ, which describe SOð2Þ rotations of the S2 our main result, we relate the spinor-helicity variables tangent space. In our setup, these rotations will play the associated with two cosmological horizons (and thus two role of the massless fields’ little group. 1;4 Vectors in dS4 are simply R vectors constrained to the tangent space of the hyperboloid (1). Spinors in dS4 can be *[email protected] † constructed similarly from embedding-space spinors (see [email protected] ‡ [email protected] e.g., [11]), but we will not need that construction here. For the statement of our main result, it will suffice to introduce Published by the American Physical Society under the terms of the two-component spinors ψ α of spatial SOð3Þ rotations. the Creative Commons Attribution 4.0 International license. The antisymmetric metric on SOð3Þ spinors is ϵαβ, with Further distribution of this work must maintain attribution to ϵαγϵ δα the author(s) and the published article’s title, journal citation, inverse βγ ¼ β. We raise and lower indices via 3 β α and DOI. Funded by SCOAP . ψ α ¼ ϵαβψ . We denote the Pauli matrices by σ β. 2470-0010=2019=100(4)=045005(5) 045005-1 Published by the American Physical Society ADRIAN DAVID, NICO FISCHER, and YASHA NEIMAN PHYS. REV. D 100, 045005 (2019) α α α ðsÞ i1 is Spinors have a complex conjugation ψ → ψ¯ → −ψ , ϕ ðu; r; mÞ¼m …m ϕi …i ðu; 0; rÞ; α 1 s under which ϵαβ is real but σ β is imaginary. ð−sÞ i1 is ϕ ðu; r; mÞ¼m¯ …m¯ ϕi …i ðu; 0; rÞ: ð5Þ For the derivation of our main result, we will also need 1 s the four-component spinors of the R1;4 embedding space, These respectively describe fields of helicity Æs and i.e., the twistors of dS4. These can be constructed as pairs of carry weight Æs under the phase rotation ðm; m¯ Þ → SOð3Þ spinors (see e.g., [12]), iθ −iθ ðe m;e m¯ Þ. The symplectic form for the horizon data λ (5) reads Ya α ; ¼ α ð3Þ iμ¯ X Z Z ⟷ 2 ðhÞ ∂ ð−hÞ Ω½δϕ1; δϕ2¼ du d rδϕ2 δϕ1 ; ð6Þ S ∂u where the i and complex conjugation on the second spinor h¼s 2 are for later convenience. The SOð1; 4Þ spinor index a is α ðsÞ lowered via Ya ¼ð−iμ¯ ; λαÞ. Complex conjugation is where we sum over the two helicities ϕ in the spinning inherited directly from that of the SOð3Þ spinors. The case, or over just one helicity ϕð0Þ ≡ ϕ in the scalar case. 1;4 R gamma matrices γμ ¼ðγu; γv; γÞ can be written in The boundary data on the initial horizon can be encoded 2 × 2 block notation as in the same way. Replacing the null time u with v, and noticing that the helicity associated with ðm; m¯ Þ is now 00 0 ϵ −iσ β 0 reversed, we write γ a ; αβ ; α : ð μÞ b ¼ αβ α −ϵ 0 00 0 −iσ s β ϕ˜ ð Þ v; r; m m¯ i1 …m¯ is ϕ 0;v;r ; ð Þ¼ i1…is ð Þ −s ϕ˜ ð Þ v; r; m mi1 …mis ϕ 0;v;r : We will sometimes omit both SOð3Þ and SOð1; 4Þ spinor ð Þ¼ i1…is ð Þ ð7Þ indices. In a product, this will imply bottom-to-top index contraction. Finally, it is useful to define the gauge-invariant field strength data corresponding to the gauge potential data [Eqs. (5) and (7)], III. FIELD DATA ON THE HORIZON ∂s We consider the free massless field equation for a totally CðsÞðu; r; mÞ¼ ϕðsÞðu; r; mÞ; ð8Þ s h ∂us symmetric, double-traceless spin- gauge potential μ1…μs dS in 4 [13], ∂s ˜ ðsÞ ˜ ðsÞ C ðv; r; mÞ¼ s ϕ ðv; r; mÞ: ð9Þ 2 ρ ∂v □ 2 s − 1 ϕμ …μ − s∇ρ∇ μ ϕ ð þ ð ÞÞ 1 s ð 1 μ2…μsÞ sðs − 1Þ ν þ ∇ μ ∇μ ϕ ¼ 0; ð4Þ 2 ð 1 2 μ3…μsÞν IV. THE S-MATRIX PROBLEM For our purposes, the S-matrix problem in de Sitter space with a gauge symmetry δϕμ …μ ¼ ∇ μ Λμ …μ for totally 1 s ð 1 2 sÞ is to relate the gauge-invariant field data (8) on the final symmetric, traceless Λμ …μ . The cases s ¼ 0,1,2 1 s−1 horizon to the corresponding data (9) on the initial one. describe the conformally coupled massless scalar, the This statement of the problem, which will be more Maxwell equations, and linearized GR, respectively. In convenient for us, is slightly more general than what is the scalar case, the field’s value ϕ u; 0; r on e.g., the final ð Þ usually termed the S-matrix. Usually, one would relate the horizon constitutes good boundary data for the field quantum states obtained by acting with the fields on some equation □ − 2 ϕ 0. For nonzero spin, good boundary ð Þ ¼ vacuum; by focusing on the fields themselves, we avoid data consists of one complex scalar component for the committing to a particular vacuum state. We will ignore right-handed helicity, and its complex conjugate for the – here any subtleties related to zero-frequency modes, i.e., to left-handed one; see e.g., [14 16] for the standard con- the horizons’ lower-dimensional boundaries (either at struction in flat spacetime and [17] for a general discussion asymptotic infinity or at the horizons’ S2 intersection). in terms of field strengths. In our present context, we can fix In other words, we will be dealing with the “hard part” of a gauge such that ϕμ …μ on the horizon has only spatial 1 s the S-matrix. ϕ i ’ R3 components i1…is . Here, the k s are indices, which For free fields, one can find the S-matrix by “brute must be tangent to the dS4 hyperboloid, and thus to the S2 force”, using the general technique for linear hyperbolic horizon section. The horizon boundary data are then given equations. Essentially, the value of a massless field at some ϕ by the traceless part of this i1…is .