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 s and i.e., the twistors of dS4. These can be constructed as pairs of carry weight s 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; δϕ2 ¼ du d rδϕ2 δϕ1 ; ð6Þ S ∂u where the i and complex conjugation on the second spinor h¼ s 2 are for later convenience. The SOð1; 4Þ spinor index a is α ð sÞ 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 Cð sÞðu; r; mÞ¼ ϕð sÞðu; r; mÞ; ð8Þ s h ∂us symmetric, double-traceless spin- gauge potential μ1…μs dS in 4 [13], ∂s ˜ ð sÞ ˜ ð sÞ 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 . Using the complex basis final horizon point is determined by the intersection of that (2) for the S2 tangent space, we can reduce this traceless point’s past lightcone with the initial horizon. Thus, for e.g., part to a pair of scalars, the scalar field, we have
045005-2 SPINOR-HELICITY VARIABLES FOR COSMOLOGICAL HORIZONS … PHYS. REV. D 100, 045005 (2019) Z 1 ∂ϕ 0;v;r0 These modes are waves with frequency p =2 with respect ϕ u; 0; r d2r0 ð Þ ; j j ð Þ¼πu ∂v ð10Þ to the null time u, supported on an antipodal pair of light S2 v¼2ðr·r0−1Þ=u rays r ¼ p=jpj. ¯ which can be obtained from the general formula, Let us now define spinor-helicity variables λα, λα as the spinor square root of p, such that p ¼ λ¯σλ. The Z Z ⟷ ∂ corresponding positive-frequency mode will be a wave ϕ u; 0; r dv d2r0ϕ 0;v;r0 G u; 0; r;0;v;r0 ; supported at r ¼ðλ¯σλÞ=ðλλ¯ Þ, with frequency λλ¯ =2 with ð Þ¼ ð Þ∂v ð Þ pffiffiffi S2 respect to u. Moreover, we can use m ¼ iðλσλÞ=ð 2λλ¯ Þ m¯ μ 0μ 0μ 0 and its complex conjugate as the complex null basis (2) in which Gðx ; x Þ¼ð−1=4πÞδðxμx − 1Þθðu − u Þ is a for the S2 tangent space. Thus, we package the positive- causal Green’s function in dS4. For the analogous general frequency part of the horizon field data (5) into spinor treatment of nonzero spin, see [17]. In fact, the end result fð sÞ λ ; λ¯ (10) holds not only in the static patch ðu>0;v<0Þ,but functions ð α αÞ as follows: also for the horizons’ entire extent u; v ∈ R, which Z λ¯σλ iλσλ includes the antipodal patch ðu<0;v>0Þ. In the rest fð sÞ λ ; λ¯ dueiðλλ¯ Þu=2ϕð sÞ u; ; ffiffiffi : ð α αÞ¼ ¯ p ¯ ð13Þ of the paper, we will avoid specifying the range of u, v, and λλ 2λλ our formulas will apply equally well to both ðu>0;v<0Þ and u; v ∈ R. While the ðu>0;v<0Þ case is linked more These spinor functions have the following manifest directly to observable physics, our formulas “live more symmetries: λ¯σλ naturally” in the more global context u; v ∈ R. (i) By construction, they have momentum under the u → u − 2a r Despite its simplicity, Eq. (10) is not quite satisfactory. Poincare-patch translations · . Since it does not make explicit contact with the horizons’ (ii) Under Poincare-patch dilatations, i.e., static-patch u; v → etu; e−tv symmetries, it is unlikely as a useful starting point for time translations ð Þ ð Þ, they scale ð sÞ ¯ t ð sÞ t=2 t=2 ¯ interacting calculations. as f ðλ; λÞ → e f ðe λ;e λÞ. What, then, are the relevant symmetries? Naively, they (iii) The field’s helicity s is encoded in the scaling fð sÞ eiθ=2λ;e−iθ=2λ¯ e isθfð sÞ λ; λ¯ are the subgroup of dS4 isometries that preserves both relation ð Þ¼ ð Þ. horizons. These are the static-patch time translations In fact, along with the more obvious SOð3Þ rotations, these ðu; vÞ → ðetu; e−tvÞ and the SOð3Þ rotations of r. These symmetries uniquely determine the encoding (13) of symmetries encourage one to work in terms of frequencies horizon data into spinor functions, up to a prefactor that and spherical harmonics. The S-matrix for the free scalar in can only depend on helicity. As a corollary, we conclude this basis was found in [18,19]. However, spherical that this spinor-helicity formalism must coincide with the harmonics are rather unpleasant, so the generalization to one constructed from a different point of view in [7]. interacting theories again does not seem promising. Below, As is frequently useful in the spinor-helicity formalism we will describe a different basis for the S-matrix, which [20], we can analytically continue to complex momenta by ¯ replaces spherical harmonics with plane waves, by fixing making the two spinors ðλ; λÞ independent. As a special ¯ ¯ only one horizon at a time. case, by analytically continuing ðλ; λÞ → ðiλ;iλÞ, we obtain the negative-frequency modes,
V. POINCARE MOMENTUM s ∓s fð Þðiλ;iλ¯Þ¼ðfð Þðλ; λ¯ÞÞ : ð14Þ AND SPINOR-HELICITY Instead of fixing both horizons, let us consider the In these variables, the field’s symplectic form (6) reads residual symmetry from fixing just e.g., the final one. simply This is the symmetry of the Poincare patch: the translations, Z R3 1 X d2λd2λ¯ rotations, and dilatations of . On the horizon, the Ω δf ;δf ½ 1 2 ¼ 2 rotations act on r ∈ S2 in the obvious way, the dilatations i 2πi h s ð Þ rescale u, while a translation by a vector a shifts the light ¼ ðhÞ ¯ ð−hÞ ¯ rays according to u → u − 2a · r. A fixed momentum p × ðδf1 ðλ;λÞδf2 ðiλ;iλÞ − ð1 ↔ 2ÞÞ; ð15Þ with respect to these translations describes two modes on 2 α β the horizon, where d λ is the spinor measure ϵαβdλ dλ =2. The field data (7) on the initial horizon can be treated in 2 p −i p u=2 the same way. Thus, the positive-frequency modes are Positive frequency∶ δ r; þ e j j ; ð11Þ jpj encoded as Z μ¯σμ iμσμ 2 p i p u=2 ˜ s α α i μμ¯ v=2 ˜ s Negative frequency∶ δ r; − eþ j j : ð12Þ fð Þðμ ; μ¯ Þ¼ dve ð Þ ϕð Þ v; ; pffiffiffi : ð16Þ jpj μμ¯ 2μμ¯
045005-3 ADRIAN DAVID, NICO FISCHER, and YASHA NEIMAN PHYS. REV. D 100, 045005 (2019) Z ¯ The negative-frequency modes can again be obtained by ¯ YγuγY YγuγY f Y;Y¯ dueiðYγuYÞu=2ϕ u; ; ffiffiffi ; analytic continuation, as in (14). Note that μα is now the ð Þ¼ ¯ p ¯ iYγuY 2YγuY square root of momentum in a different Poincare coordinate patch—the one associated with the initial horizon. where we removed the helicity superscripts to save space. The initial horizon’s spinor functions (16) are now given by VI. THE FREE S-MATRIX the exact same formulas, but with u replaced everywhere by v In the framework of the previous section, finding the . Of course, this replacement amounts to interchanging the R1;4 S-matrix means relating the spinor functions fð sÞðλ; λ¯Þ and null axes in that define the two horizons. s For the second step of our proof, recall that free massless f˜ ð Þðμ; μ¯Þ. Our paper’s main result (to be derived in the fields in dS4 (or in any conformally flat four-dimensional next section) is that the free-field answer is simply a Fourier [4d] spacetime) can be encoded, via the Penrose transform transform, Fð sÞ Ya Z [21,22],asholomorphic twistor functions ð Þ, with 2 2 Y¯ a d μd μ¯ α ¯ α no dependence on the complex conjugate . The de Sitter ð sÞ ¯ ˜ ð sÞ α α iðλαμ þλαμ¯ Þ f ðλα; λαÞ¼ f ðμ ; μ¯ Þe : ð17Þ SO 1; 4 ð2πiÞ2 group ð Þ [and, in fact, the entire conformal group SOð2; 4Þ] is realized on these functions by linear trans- a This formula can be unpacked explicitly in terms of the formations of the twistor argument Y . In particular, the horizon field data (8) and (9), giving Poincare-patch translations u → u − 2a · r and dilatations ðu; vÞ → ðetu; e−tvÞ are realized, respectively, as 2s Cð sÞðu;r;mÞ¼ πu2sþ1 Fð sÞ λ ;iμ¯ α → Fð sÞ λ ;iμ¯ α i a σ αβλ ; ð α Þ ð α þ ð · Þ βÞ Z s ∂C˜ ð Þ v;r0;m0 ð sÞ α ð sÞ t=2 −t=2 α d2r0 1 − r r0 s ð Þ ; F ðλα;iμ¯ Þ → F ðe λα;ie μ¯ Þ; ð20Þ × ð · Þ ∂v ð18Þ S2 v¼2ðr·r0−1Þ=u along with the obvious action of SOð3Þ rotations. Finally, which reproduces the spin-0 result (10) as a special case. the field’s helicity is encoded in the twistor function’s The phase of the null tangent vector m0 in (18) is fixed via degree of homogeneity,
0 0 2m · m ¼ 1 − r · r : ð19Þ ∂Fð sÞ ∂Fð sÞ ∂Fð sÞ λ μ¯ α Ya −2 2s: α ∂λ þ ∂μ¯ α ¼ ∂Ya ¼ ð21Þ The explicit S-matrix (18) can be derived from the main α result (17) and the definitions (8), (9), (13), (16), via mostly Fourier transform straightforward integrals. The two nontrivial “tricks” are as We can now see that the of the twistor Fð sÞ λ;iμ¯ μ¯ α follows: function ð Þ with respect to has the defining s ¯ (i) When integrating over λλ¯ to invert the transform symmetries of the spinor-helicity function fð Þðλ; λÞ on the (13), it helps to also average over the phase of λα, final horizon! Therefore, up to a prefactor that can be s using the known weight of fð sÞ under such phase absorbed into the definition of Fð Þðλ;iμ¯Þ, we identify rotations. We then have an integral over both Z d2μ¯ magnitudes and phases, which becomes an integral s s iλ¯ μ¯ fð Þðλ; λ¯Þ¼ Fð Þðλ;iμ¯Þe ; ð22Þ dzdz¯ over the complex plane. 2π (ii) Conversely, when integrating over μα, μ¯ α in (17),it helps to localize the integral on values of μα with an or, in SOð1; 4Þ-covariant notation 0 overall phase given byffiffiffi (19), where r ≡ ðμ¯σμÞ=ðμμ¯ Þ Z p dZγvdZ ¯ and m0 ≡ ðiμσμÞ=ð 2μμ¯ Þ. f Y;Y¯ F γ γ Y γ γ Z eiYγuγvZ; ð Þ¼ 4π ð v u þ u v Þ VII. TWISTORIAL DERIVATION where the helicity superscripts are again omitted. But now, To prove our S-matrix formula (17), we will relate it to a by SOð1; 4Þ covariance, upon interchanging u ↔ v,we picture that is covariant under the full SOð1; 4Þ de Sitter must get the spinor-helicity functions f˜ ð sÞðY;Y¯ Þ for the group. First, we will rewrite our transforms (13), (16) initial horizon! Back in SOð3Þ spinor notation, this last between spinor-helicity functions and horizon data in statement reads SO 1; 4 λ ð Þ-covariant language. For this, we combine α Z and iμ¯ α into a twistor, i.e., an SOð1; 4Þ spinor Ya,asin d2λ f˜ ð sÞ μ; μ¯ − Fð sÞ λ;iμ¯ e−iλμ: (3). The final horizon’s spinor functions (13) can now be ð Þ¼ 2π ð Þ ð23Þ treated as functions of the twistor variables ðY;Y¯ Þ, which ¯ just happen to depend only on the ðγuY;γuYÞ components, Combining the two “half”-Fourier transforms (22) and i.e., the ones containing ðλ; λ¯Þ (23), we obtain the free S-matrix (17).
045005-4 SPINOR-HELICITY VARIABLES FOR COSMOLOGICAL HORIZONS … PHYS. REV. D 100, 045005 (2019)
VIII. CONCLUSION Having found the free S-matrix in the spinor-helicity basis, we obtained an explicit expression (18) in terms of In this paper, we applied a spinor-helicity formalism to horizon field data for any spin. More importantly, there is the horizon data of massless fields in de Sitter space and now hope that perturbation theory for interacting fields in derived an elegant expression for the free S-matrix of the de Sitter causal patch can be constructed with relative massless fields with any spin. We extended the known ease. In the future, we would like to explore this possibility, analogies [7] with the spinor-helicity formalism in flat in particular for Yang-Mills and GR. In addition, we expect spacetime, in particular establishing a “half”-Fourier trans- that the language developed here can be fruitfully applied to form relation between spinor-helicity and twistor functions. higher-spin gravity [23,24], the only known working model We also went beyond [7] by working with two different [25] for dS/CFT holography in 3 þ 1 dimensions. horizons, each of which has a different notion of momen- tum. Despite this seeming complication, we saw that the ACKNOWLEDGMENTS momenta on the two horizons are related by a simple Fourier transform (17), once we consider their spinor square root. This work was supported by the Quantum Gravity and This Fourier-transform relationship can also be understood Mathematical and Theoretical Physics Units of the Okinawa as a change of basis in the Hilbert space of a massless Institute of Science and Technology Graduate University. particle on the (complexified) 3d conformal boundary [12]. N. F.’s work was carried out during an internship at OIST.
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