PPrepared for submission to JHEP
Massive Celestial Fermions
Sruthi A. Narayanan a, a Center for the Fundamental Laws of Nature, Harvard University,Cambridge, MA 02138, USA
E-mail: [email protected]
Abstract:
In an effort to further the study of amplitudes in the celestial CFT (CCFT),we construct conformal primary wavefunctions for massive fermions. Upon explicitly calcu-lating the wavefunctions for Dirac fermions, we deduce the corresponding transformation ofmomentum space amplitudes to celestial amplitudes. The shadow wavefunctions are shownto have opposite spin and conformal dimension − ∆ . The Dirac conformal primary wave-functions are delta function normalizable with respect to the Dirac inner product providedthey lie on the principal series with conformal dimension ∆ = 1 + iλ for λ ∈ R . It is shownthat there are two choices of a complete basis: single spin J = or J = − and λ ∈ R ormultiple spin J = ± and λ ∈ R + ∪ . The massless limit of the Dirac conformal primarywavefunctions is shown to agree with previous literature. The momentum generators on thecelestial sphere are derived and, along with the Lorentz generators, form a representationof the Poincaré algebra. Finally, we show that the massive spin- conformal primary wave-functions can be constructed from the Dirac conformal primary wavefunctions using thestandard Clebsch-Gordan coefficients. We use this procedure to write the massive spin- ,Rarita-Schwinger, conformal primary wavefunctions. This provides a prescription for con-structing all massive fermionic and bosonic conformal primary wavefunctions starting fromspin- . Corresponding author. a r X i v : . [ h e p - t h ] S e p ontents (Dirac) Conformal Primary Wavefunctions 5 Polarizations 116.2 Rarita-Schwinger Fields 12
A Solving Constraint Equations 13B Bulk Integrals of Propagators 14C Boundary Integrals of Propagators 15
Recently there has been considerable focus on the conformal primary basis. The transfor-mation from a plane wave basis to the conformal primary basis provides a mapping from4-dimensional (4D) momentum space scattering amplitudes to 2-dimensional (2D) "celestialamplitudes". These celestial amplitudes transform as conformal correlators under SL (2 , C ) .The conformal primary basis was introduced in [1, 2] for amplitudes containing masslessand massive scalars, photon and graviton fields. It has since been extended to all masslessfields in [3–5]. This construction was then generalized to massive bosons in [6]. It hasbeen further used to study the analytic structure of amplitudes as well as conformally softtheorems in [7–15]. – 1 –he correspondence between a 4D bulk theory and a 2D boundary conformal fieldtheory has its roots in earlier work. In [16], de Boer and Solodukhin introduced a 3-dimensional (3D) hyperbolic slicing of 4D Minkowski space where the common boundaryof the slices was a 2-sphere, the celestial sphere, CS . This led to an important holographicrelation between bulk Lorentz symmetries and conformal symmetries on CS as demonstratedin [17, 18]. The authors of [19] were then able to recast 4D scattering amplitudes as 2Dconformal correlators which have come to be referred to as celestial amplitudes.To widen the set of amplitudes that we can study using the conformal primary basis,it is a natural next step to understand the transformation of fermionic fields. Whereasit is easier, in principal, to generalize transformations for massless particles because, ina particular gauge, they are Mellin transforms of plane wave solutions, massive fields aretrickier and have a different functional form for each spin. The goal of this paper is toexplicitly construct the spin- (Dirac) conformal primary wavefunctions and to show thatthey can be used to construct the spin-1 primaries. This results in a prescription to constructmassive conformal primary wavefunctions for arbitrary integer and half integer spin via thestandard Clebsch-Gordan coefficients.This paper is organized as follows. In section 2 we detail the conventions used through-out this paper. In particular, we review pertinent recent results for conformal primary wave-functions of massive bosons and the Dirac equation in Minkowski space with ( − , + , + , +) signature. In section 3 we explicitly construct the Dirac conformal primary wavefunctionsusing constraints and discuss how to transform an amplitude containing fermions from mo-mentum space to a celestial amplitude. In section 4 we write the explicit functional form ofthe wavefunctions in terms of the previously found scalar conformal primaries. We computethe shadow field and show that it shifts the conformal dimension from ∆ to − ∆ as wellas flipping the spin. We compute the Dirac inner product of two conformal primary wave-functions to demonstrate that they are delta function normalizable as long as we considerthe principal continuous series ∆ = 1 + iλ for λ ∈ R . We note that there are two choicesfor a complete basis: J = or J = − and λ ∈ R or J = ± and λ ∈ R + ∪ . Finallywe compute the massless limit and show it is in agreement with the spinor solutions in [5].In section 5 we derive the spin − momentum generators in the celestial basis and showthat they are not diagonal and that they form a representation of the Poincaré algebraalong with the Lorentz generators. In section 6 we arrive at a prescription for writing downarbitrary massive half-integer and integer spin conformal primary wavefunctions by first re-lating the spin- to the spin- conformal primary wavefunctions and then constructing theRarita-Schwinger fields, massive spin- , using Clebsch Gordan coefficients. In appendix Awe write the details of solving the differential equations to get the wavefunctions. In ap-pendix B we compute a weighted bulk integral of scalar propagators which is useful for theDirac inner product. In appendix C we compute a boundary integral of scalar propagatorswhich is useful for the completeness relation.After this work was completed, independent and overlapping results were posted in [20].– 2 – Preliminaries
We start with a hyperbolic slicing of 4D Minkowski spacetime with signature ( − , + , + , +) where the metric on each 3D hyperbolic slice, H , is given by ds = dy + dzd ¯ zy . (2.1)The coordinate y varies as we move along H . When y → we approach CS , parametrizedby complex coordinates z, ¯ z . The momentum of a massive particle is parametrized as p µk = m k (cid:18) y k + z k ¯ z k y k , ¯ z k + z k y k , i (¯ z k − z k )2 y k , − y k − z k ¯ z k y k (cid:19) ≡ m k ˆ p µk (2.2)where p k = − m k . Massless (null) momenta are parametrized as q µ = ω (1 + w ¯ w, ¯ w + w, i ( ¯ w − w ) , − w ¯ w ) ≡ ω ˆ q µ . (2.3)Each hyperbolic slice has an SL (2 , C ) isometry given by the coordinate transformations z → z (cid:48) = ( az + b )(¯ c ¯ z + ¯ d ) + a ¯ cy ( cz + d )(¯ c ¯ z + ¯ d ) + c ¯ cy , y → y (cid:48) = y ( cz + d )(¯ c ¯ z + ¯ d ) + c ¯ cy (2.4)where ad − bc = ¯ a ¯ d − ¯ b ¯ c = 1 . The above SL (2 , C ) transformation induces Möbius transfor-mations of the complex coordinates w, ¯ w on CS w → w (cid:48) = aw + bcw + d , ¯ w → ¯ w (cid:48) = ¯ a ¯ w + ¯ b ¯ c ¯ w + ¯ d . (2.5)The appropriate Lorentz transformation matrix is given explicitly [21] by Λ µν = 12 a ¯ a + b ¯ b + c ¯ c + d ¯ d b ¯ a + a ¯ b + d ¯ c + c ¯ d i ( − b ¯ a + a ¯ b − d ¯ c + c ¯ d ) − a ¯ a + b ¯ b − c ¯ c + d ¯ dc ¯ a + a ¯ c + d ¯ b + b ¯ d d ¯ a + a ¯ d + c ¯ b + b ¯ c i ( − d ¯ a + a ¯ d + c ¯ b − b ¯ c ) − c ¯ a − a ¯ c + d ¯ b + b ¯ di ( c ¯ a − a ¯ c + d ¯ b − b ¯ d ) i ( d ¯ a − a ¯ d + c ¯ b − b ¯ c ) d ¯ a + a ¯ d − c ¯ b − b ¯ c i ( − c ¯ a + a ¯ c + d ¯ b − b ¯ d ) − a ¯ a − b ¯ b + c ¯ c + d ¯ d − b ¯ a − a ¯ b + d ¯ c + c ¯ d i ( b ¯ a − a ¯ b − d ¯ c + c ¯ d ) a ¯ a − b ¯ b − c ¯ c + d ¯ d . (2.6)We will be primarily concerned with its infinitesimal form. For a particular infinitesimalLorentz transformation a = 1 + αβ − γ , b = α, c = β, d = 1 − γ (2.7)the matrix is given to first order in the infinitesimal parameters by Λ µν = ( β + ¯ β + α + ¯ α ) i ( β − ¯ β + ¯ α − α ) − ( γ + ¯ γ ) ( α + ¯ α + ¯ β + β ) 1 i ( γ − ¯ γ ) ( α + ¯ α − ¯ β − β ) i ( ¯ α − ¯ β + β − α ) i (¯ γ − γ ) 1 i ( ¯ α + ¯ β − β − α ) − ( γ + ¯ γ ) ( β + ¯ β − ¯ α − α ) i ( β − ¯ α − ¯ β + α ) 1 . (2.8)– 3 – .1 Review of Massive Spinning Bosons In this subsection we consolidate some useful results from [6]. The conformal primarywavefunction for a spin- s massive boson was found to be of the form φ µ ··· µ s , ± ∆ ,J,m ( X ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆+ s s (cid:88) b = − s g ( s ) J,b ( (cid:126)w ; y, (cid:126)z ) (cid:15) µ ··· µ s b e ± im ˆ p · X (2.9)where J ∈ {− s, · · · , s } is the spin of the conformal primary, G ∆+ s = ( − q · ˆ p ) − ∆ − s is thescalar bulk to boundary propagator [22], (cid:15) µ ··· µ s b is the spin- s polarization tensor, the g ( s ) J,b are scalar functions for each
J, b and (cid:90) H [ d ˆ p ] ≡ (cid:90) ∞ dyy (cid:90) d z = (cid:90) d ˆ p i ˆ p (2.10)is the integral over a hyperbolic slice. It will be useful to define (cid:15) µ ··· µ s J ≡ s (cid:88) b = − s g ( s ) J,b ( (cid:126)w ; y, (cid:126)z ) (cid:15) µ ··· µ s b (2.11)so that (2.9) becomes φ µ ··· µ s , ± ∆ ,J,m ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆+ s (cid:15) µ ··· µ s J e ± im ˆ p · X . (2.12)Comparing the explicit expressions for spin- and spin- , results in the following expectedrelations (cid:15) µν − = (cid:15) µ − (cid:15) ν − , (cid:15) µν − = i √ (cid:15) µ − (cid:15) ν + (cid:15) µ (cid:15) ν − ) , (cid:15) µν = (cid:114) (cid:18) (cid:15) µ (cid:15) ν − (cid:15) µ − (cid:15) ν − (cid:15) µ (cid:15) ν − (cid:19) (cid:15) µν = − i √ (cid:15) µ (cid:15) ν + (cid:15) µ (cid:15) ν ) , (cid:15) µν = (cid:15) µ (cid:15) ν (2.13)which are consistent with the usual Clebsch-Gordan coefficients up to overall factors result-ing from the wavefunction normalization. In a similar fashion, one can construct the higherinteger spin fields. It is also useful to note that this is consistent with the construction ofthe polarization tensors in momentum space [23]. In this paper we will consider solutions to the Dirac equation in 4D Minkowski space ( γ µ ∂ µ − m ) ψ = 0 . (2.14)The gamma matrices are given by γ = (cid:32) − I I (cid:33) , γ i = (cid:32) σ i σ i (cid:33) , (2.15) We have chosen one particular convention for the gamma matrices. The following construction can bedone with any convention choice but will result in differences in overall factors. – 4 –here σ i are the usual Pauli matrices and { γ µ , γ ν } = 2 η µν I . We will use familiar conven-tions where σ ≡ { I , σ i } , ¯ σ ≡ { I , − σ i } . (2.16)Solutions to the Dirac equation are of the form ψ ± s = u ± s ( p ) e ± im ˆ p · X where ± denote positiveand negative energy solutions and s ∈ { , − } . The spinors satisfy ( ± iγ µ p µ − m ) u ± s ( p ) = 0 . (2.17)The spin representation of the infinitesimal Lorentz transformation (2.8) can be computedto be Λ = − γ β α γ γ − ¯ α − ¯ β − ¯ γ . (2.18) (Dirac) Conformal Primary Wavefunctions In this section we construct the Dirac conformal primary wavefunctions. These wavefunc-tions are solutions to the massive Dirac equation in 4D as well as 2D conformal primaries.Using symmetry constraints and a convenient ansatz, we are able to find an integral formfor the wavefunctions. Once we have the wavefunctions we are able to extract the transfor-mation from a momentum space amplitude to a celestial amplitude.
Inspired by [6] we propose the following ansatz for the conformal primary wavefunctions ψ ± ∆ ,J,m ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ± J (ˆ p ; (cid:126)w ) e ± im ˆ p · X ≡ (cid:90) H [ d ˆ p ] G ∆+ u ± J (ˆ p ; (cid:126)w ) e ± im ˆ p · X (3.1)where u ± J (ˆ p ; (cid:126)w ) is a linear combination of the momentum space spinors u ± s ( p ) and J ∈{− , } labels the spin of the conformal primary. We consider a linear combination ofspins since a general Lorentz transformation transforms a spinor to a linear combinationof spinors. The conformal primary wavefunction should transform as a spin- conformalprimary and as a spinor under Lorentz transformations. Since the measure and the exponentin (3.1) are invariant under these transformations, we require G J (ˆ p ; (cid:126)w ) → G (cid:48) J (Λˆ p ; (cid:126)w (cid:48) ) ≡ ( cw + d ) ∆+ J (¯ c ¯ w + ¯ d ) ∆ − J Λ G J (ˆ p ; (cid:126)w ) . (3.2)Using the coordinate transformations in (2.4) and the explicit form of Λ in (2.18) we canexpand both sides of this expression to first order in infinitesimal parameters and solve the Much of the previous literature uses spin and helicity interchangeably since for massless fields 2D spinand 4D helicity are the same. For massive fields, spin and helicity are not necessarily the same so we willdifferentiate between the two when necessary. – 5 –esulting differential equations. Under such an infinitesimal transformation, we can writean arbitrary function as an expansion f (cid:48) ( z (cid:48) , ¯ z (cid:48) , y, w (cid:48) , ¯ w (cid:48) ) = f + α ( ∂ z + ∂ w ) f + ¯ α ( ∂ ¯ z + ∂ ¯ w ) f + β ( − z ∂ z + y ∂ ¯ z − yz∂ y − w ∂ w ) f + ¯ β ( y ∂ z − ¯ z ∂ ¯ z − y ¯ z∂ y − ¯ w ∂ ¯ w ) f + γ (2 z∂ z + 2 w∂ w + y∂ y ) f + ¯ γ (2¯ z∂ ¯ z + 2 ¯ w∂ ¯ w + y∂ y ) f. (3.3)Using this to expand G (cid:48) J we obtain six matrix-differential equations. Given that we aresolving for a function of five variables, this is generally an overconstrained system. How-ever, one can solve these equations, up to overall constants, using the methods outlined inappendix A. These constants are then fixed by requiring that the primaries solve the Diracequation (2.17). For J = − ψ ± ∆ , − ,m ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆+ √ w ∓ i y +¯ z ( z − w ) y ∓ i w − zy e ± im ˆ p · X (3.4)and for J = ψ ± ∆ , ,m ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆+ √ ± i ¯ z − ¯ wy ± i y + z (¯ z − ¯ w ) y − ¯ w e ± im ˆ p · X . (3.5)To see that these are solutions to the Dirac equation it is important to realize that we canalso write them as ψ ± ∆ , − ,m ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆+ (cid:32) φ ( (cid:126)w ) ± i (ˆ p · σ ) φ ( (cid:126)w ) (cid:33) e ± im ˆ p · X ψ ± ∆ , ,m ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆+ (cid:32) ∓ i (ˆ p · ¯ σ ) φ ( (cid:126)w ) φ ( (cid:126)w ) (cid:33) e ± im ˆ p · X (3.6)where φ ( (cid:126)w ) = √ (cid:32) w (cid:33) and φ ( (cid:126)w ) = √ (cid:32) − ¯ w (cid:33) . We also note that φ † ( (cid:126)w ) φ ( (cid:126)w ) = φ † ( (cid:126)w ) φ ( (cid:126)w ) = 12 ( ¯ w − ¯ w ) (3.7)which ensures the proper orthogonality of the spinors in the integrand as per usual quantumfield theory conventions. One of the primary reasons to study conformal primary wavefunctions is to connect mo-mentum space amplitudes to celestial amplitudes. In the case of massless external particles,for a particular gauge choice, the two are related by a Mellin transform [1–3] ˜ A n = n (cid:89) j =1 (cid:90) ∞ dω j ω ∆ j − j A n . (3.8)– 6 –or massive external scalars the transformation is slightly more complicated and involves aconvolution with a scalar bulk to boundary propagator [2] ˜ A n = n (cid:89) j =1 (cid:90) H [ d ˆ p j ] G ∆ j A n . (3.9)In the case of massive bosons the transformation was found to be [6] ˜ A J ··· J n = n (cid:89) j =1 (cid:90) H [ d ˆ p j ] G ∆ j + s j s j (cid:88) b j = − s j g ( s j ) J j b j A b ··· b n (3.10)where the objects in the integral are the same as those that appear in (2.9). This trans-formation has a more complicated structure since the conformal primary wavefunctioninvolved a sum over polarizations. The amplitude A b ··· b n involves n polarization vectors(or tensors) and in transforming to the celestial amplitudes, we have to consider all possibleconfigurations of polarizations, hence the sum in the integrand.One expects a similar structure for the transformation of fermionic amplitudes. Ageneric amplitude containing fermions is written in terms of the spinors u s . We are able towrite u ± J as a linear combination of the u s as follows u ± J = ¯ u ±− u ± J ¯ u ±− u ±− u ±− + ¯ u ± u ± J ¯ u ± u ± u = (cid:88) s = − U ± s,J u ± s . (3.11)These U ± s,J are the fermionic analogs of the g ( s ) J,b in (2.9). It follows that for an amplitudecontaining n fermions A s ,...,s n with spin indices s i the transformation to the conformalprimary basis is ˜ A J ,...,J n = n (cid:89) i = j (cid:90) H [ d ˆ p j ] G ∆ j +
12 12 (cid:88) s j = − U ± s j ,J j A s ,...,s n . (3.12)Usually a scattering amplitude can contain more than one kind of field. For the sake ofsimplicity we have written the transformation rules for amplitudes containing only onetype of particle. In general one will have a product of transformations, one for each typeof external particle in the amplitude. Although it is often more useful to use the integral expressions for the wavefunctions, in somecontexts it might be useful to have the precise analytic expression. In this section we writethe analytic expression for the wavefunctions by noticing the relation of each component toa scalar conformal primary wavefunction. This allows us to discuss the shadow operators, There are many conventions for writing down the spinor solutions therefore we have chosen to writethis expansion generally to allow the reader to pick their favorite spinors. – 7 –ormalization and massless limit. In order to evaluate the integrals, it is useful to note thatthe spinor components can be written in terms of scalar conformal primaries of dimension ∆ + ψ ± ∆ , − ,m = 1 √ w ∓ i (cid:16) ¯ w∂ ¯ w ∆ − (cid:17) e − ∂ ∆ ± i∂ ¯ w ∆ − e − ∂ ∆ φ ± ∆+ , ψ ± ∆ , ,m = 1 √ ± i∂ w ∆ − e − ∂ ∆ ± i (cid:16) w∂ w ∆ − (cid:17) e − ∂ ∆ − ¯ w φ ± ∆+ . (4.1)Their analytic structure was determined in [2] to be φ ± ∆ ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆ (ˆ p ; (cid:126)w ) e ± im ˆ p · X = 4 πim ( √− X ) ∆ − ( − q ( (cid:126)w ) · X ∓ i(cid:15) ) ∆ K ∆ − ( m √ X ) (4.2)where K α ( x ) is a modified Bessel function of the second kind. Therefore, using the derivativeoperators in (4.1) one can write down a full analytic expression for ψ ± ∆ ,J,m . A two dimensional conformal primary field ϕ ( w, ¯ w ) of integer or half integer spin is labelledby weights h, ¯ h where ∆ = h + ¯ h and J = h − ¯ h . A shadow field for 2D conformal primarieswas defined in [24] to be (cid:101) ϕ ( w, ¯ w ) = Γ(2 − h ) π Γ(2 h − (cid:90) d z w − z ) − h ( ¯ w − ¯ z ) − h ϕ ( z, ¯ z ) . (4.3)The shadow field has weights (1 − h, − ¯ h ) therefore conformal dimension − ∆ and spin − J . It was shown in [2] that the shadow of a dimension ∆ massive scalar is the samewavefunction with conformal dimension − ∆ whereas for massless scalars, photons andgravitons the shadow is a different wavefunction though still a conformal primary. For Diracwavefunctions, the shadow obeys the property (cid:101) ψ ± ∆ , ± ,m = ∓ i Γ( − ∆)Γ(∆ + ) ψ ± − ∆ , ∓ ,m . (4.4)The shadows are conformal primaries of dimension − ∆ and with opposite spin as expected.The shadow as defined in (4.3) is for spin- s conformal primaries where s ∈ Z . Thisshadow transform is not valid for continuous spin wavefunctions as discussed in [25, 26] butis sufficient for our purposes when discussing fermionic fields. It also important to notethat our discussion is specifically in 2D. The shadow for a d -dimensional conformal primaryrequires an uplift to the embedding space as discussed for integer spin in [2, 25] and for halfinteger spin in [20]. We demand that the Dirac conformal primaries are delta function normalizable with respectto the Dirac inner product. Such a normalization was also considered in [5]. We look at– 8 –he X = 0 Cauchy slice where this inner product is given by ( ψ ∆ ,J,m , ψ ∆ ,J (cid:48) ,m ) = 12 (cid:90) d X ( ¯ ψ ∆ ,J γ ψ ∆ ,J (cid:48) + ¯ ψ ∆ ,J (cid:48) γ ψ ∆ ,J ) . (4.5)To compute this inner product we note that the spinors have the property ¯ u ± J ( (cid:126)w ) γ u ± J (cid:48) ( (cid:126)w ) = − ˆ p (cid:20) y + ( ¯ w − ¯ z )( w − z ) y δ J + J (cid:48) , − + y + ( w − z )( ¯ w − ¯ z ) y δ J + J (cid:48) , (cid:21) ∓ i ˆ p (cid:2) ( w − w ) δ J − J (cid:48) , + ( ¯ w − ¯ w ) δ J − J (cid:48) , − (cid:3) (4.6)and that the spatial integral is (cid:90) d Xe i ( p − p (cid:48) ) · X = 2(2 π ) y m ˆ p δ (2) ( z − z (cid:48) ) δ ( y − y (cid:48) ) = (2 π ) m δ (3) (ˆ p − ˆ p (cid:48) ) . (4.7)where δ (3) (ˆ p − ˆ p (cid:48) ) is the Lorentz invariant delta function on H . Using the simplificationsabove and the results in appendix B, the inner product is ( ψ ± iλ ,J,m , ψ ± iλ ,J (cid:48) ,m ) = ∓ π ) m (1 + 4 λ ) δ JJ (cid:48) δ ( λ − λ ) δ (2) ( w − w ) ± i (2 π ) δ ( λ + λ ) m (cid:34) δ J − J (cid:48) , ∂ ¯ w + δ J − J (cid:48) , − ∂ w ( − iλ ) | w | − iλ ) − c.c (cid:35) . (4.8)Just as in [2], convergence of the inner product requires that the wavefunctions be on theprincipal series where ∆ = 1 + iλ for λ ∈ R . We see that the wavefunctions are only deltafunction normalizable if J = J (cid:48) . This gives us two ways to make a basis in order to removeredundancies. We can choose the basis elements to have both spins, J = ± , in whichcase we need to restrict λ ∈ R + ∪ . The wavefunctions with λ ∈ R −∪ will be related by ashadow transform. Alternatively, we can choose the basis elements to have just one spin J = or J = − and let λ ∈ R . Here, the wavefunctions with opposite spin will be relatedby a shadow transform. In what follows we choose the latter.Though we make a particular choice of basis, it is important to note how the two choicesare related to one another. If we start with the set { J = , λ ∈ R } we can write it as theunion (cid:26) J = 12 , λ ∈ R (cid:27) = (cid:26) J = 12 , λ ∈ R −∪ (cid:27) ∪ (cid:26) J = 12 , λ ∈ R + ∪ (cid:27) . (4.9)We can take the first set and take the shadow of those wavefunctions to getShadow (cid:18)(cid:26) J = 12 , λ ∈ R −∪ (cid:27)(cid:19) = (cid:26) J = − , λ ∈ R + ∪ (cid:27) . (4.10)Therefore we see thatShadow (cid:18)(cid:26) J = 12 , λ ∈ R −∪ (cid:27)(cid:19) ∪ (cid:26) J = 12 , λ ∈ R + ∪ (cid:27) = (cid:26) J = ± , λ ∈ R + ∪ (cid:27) (4.11)which is the other choice of basis. – 9 – .3 Completeness We can now show that our choice of basis is complete. One can show for the spinor solutionsthat ∓ i ¯ u ± J (ˆ p ; (cid:126)w ) u ± J (ˆ p ; (cid:126)w ) = − (ˆ p + ˆ p ) · ˆ q ( (cid:126)w ) . (4.12)In appendix C we show that (cid:90) ∞−∞ dλµ ( λ ) (cid:90) d w (cid:104) G − iλ (ˆ p ; (cid:126)w ) G + iλ (ˆ p ; (cid:126)w ) + G − iλ (ˆ p ; (cid:126)w ) G + iλ (ˆ p ; (cid:126)w ) (cid:105) = δ (3) (ˆ p , ˆ p ) (4.13)for µ ( λ ) = 2Γ( + iλ )Γ( − iλ )Γ( + iλ )Γ( − iλ ) . (4.14)Using this, one finds that e ± im ˆ p · X = ∓ i (cid:90) ∞−∞ dλ + iλ )Γ( − iλ )Γ( + iλ )Γ( − iλ ) (cid:90) d wG − iλ (ˆ p ; (cid:126)w ) ¯ u ± J ψ ± iλ,J,m ( X µ ; (cid:126)w ) . (4.15)Therefore, the transformation to celestial amplitudes is invertible and the basis is complete.Since we can transform from one basis choice to another using the shadow, we know theother basis choice is also complete. One can also take the massless limit of the massive wavefunctions. In the massless limit [2]the bulk to boundary propagator is G ∆ ( y, (cid:126)z ; (cid:126)w ) → π Γ(∆ − y − ∆ δ (2) ( z − w ) . (4.16)Substituting this and using the change of integration variable ω = m y , the massless limit ofthe spin- wavefunctions is ψ ± ∆ , − ( X µ ; (cid:126)w ) = (cid:90) ∞ dωω ∆ − (cid:32) φ (cid:33) e ± iq · X , ψ ± ∆ , ( X µ ; (cid:126)w ) = (cid:90) ∞ dωω ∆ − (cid:32) φ (cid:33) e ± iq · X (4.17)This is consistent with the massless wavefunctions in [5], where they were used to studysupersymmetry on the CCFT, up to differences between our conventions for the Diracequation and the conventions in [27]. It was proven useful to constrain celestial amplitudes using Poincaré symmetries in [10, 28].In the massless case, the momentum generators are diagonal and do not mix the spin, We would like to thank Monica Pate and Ana-Maria Raclariu for help with the details of this calculation. – 10 –owever it was found in the massive bosonic case that they are not diagonal. Rather, theymix neighboring spins. In the fermionic case, the eigenvalue equation one wishes to solve is (cid:88) I = − P µ, ± J,I G ± J = m ˆ p µ G ± J . (5.1)We are able to find that the momentum generator is P µ, ± J,I = m (cid:20) (cid:20) ± iδ J,I +1 ∆ − (cid:18) ∂ w q µ + q µ ∂ w ∆ − (cid:19) + ± iδ J +1 ,I ∆ − (cid:18) ∂ ¯ w q µ + q µ ∂ ¯ w ∆ − (cid:19)(cid:21) + δ J,I (cid:20)(cid:20) ∂ w ∂ ¯ w q µ + ( ∂ w q µ ) ∂ ¯ w ∆ − − J + ( ∂ ¯ w q µ ) ∂ w ∆ − J + q µ ∂ w ∂ ¯ w (∆ − )(∆ − ) (cid:21) e − ∂ ∆ + (∆ + ) q µ ∆ − e ∂ ∆ (cid:21) (cid:21) (5.2) Up to normalization, this is the same result as (3.3) in [6] with s = . Therefore, one canshow that this operator properly squares to − m and that along with the Lorentz generatorsforms a representation of the Poincaré algebra. We do not reproduce it here.The momentum generator is not diagonal in this representation because, just as in thebosonic case, the wavefunction is constructed with a sum over all spins. The fact that themomentum generator for massive fermionic fields is structurally the same as that of themassive bosonic fields is not surprising given the relation between the fields shown in thenext section. In this section we first relate the spinor solutions to the spin-1 polarization vectors. Weare then able to extrapolate these results to outline a method of writing down the massiveconformal primary wavefunctions for arbitrary integer and half-integer spin . Polarizations
In this section we relate the spinors u ± J to the linear combination of polarization vectorsthat appear in the massive spin-1 conformal primary wavefunctions in [6]. Arbitrary spinbosonic polarization tensors can be constructed from the spin-1 polarization vectors bytaking appropriate linear combinations of tensor products [23] e.g (cid:15) µν ∝ (cid:15) µ (cid:15) ν . The relationbetween spin-1 and spin-2 was outlined in (2.13). To construct these polarization vectorsfrom the Dirac spinors it is necessary to consider a product that includes a gamma matrixto get the appropriate index structure. We consider the inner product ¯ u ± J γ µ u ± J (cid:48) . We canrelate them to the g J,b appearing in (2.9) ¯ u ∓± γ µ u ±± = − (cid:15) µ , ¯ u ±− γ µ u ∓ = ± i (cid:15) µ , ¯ u ± γ µ u ∓− = ∓ i (cid:15) µ − . (6.1)Therefore we explicitly have the relations φ µ, ± ∆ ,J = − ,m = ∓ i (cid:90) H [ d ˆ p ] G ∆+1 (cid:20) ¯ u ∓ γ µ u ±− (cid:21) e ± im ˆ p · X It is useful to note that it might be even cleaner to express these relations using the massive spinor-helicity formalism [29] however we have not explored that in this paper. – 11 – µ, ± ∆ ,J =0 ,m = − (cid:90) H [ d ˆ p ] G ∆+1 (cid:20) ¯ u ∓− γ µ u ±− + ¯ u ∓ γ µ u ± (cid:21) e ± im ˆ p · X φ µ, ± ∆ ,J =1 ,m = ± i (cid:90) H [ d ˆ p ] G ∆+1 (cid:20) ¯ u ∓− γ µ u ± (cid:21) e ± im ˆ p · X . (6.2)Whereas it may appear odd that we need to pair positive and negative energy spinor solu-tions to construct the spin- wavefunctions, we are just matching helicities. For example,the positive energy || s | = 1 , J = − (cid:105) wavefunction has negative helicity therefore we obtainit from a combination of the negative helicity spinors: positive energy || s | = , J = − (cid:105) and negative energy || s | = , J = (cid:105) . We have successfully shown how to construct the massive spin-1 conformal primary wave-function from the Dirac spinors. It is rather convenient, and expected, that they arerelated by the usual Clebsch-Gordan coefficients. Since we already know how to constructhigher integer spin wavefunctions, the natural question is whether we can construct higherhalf-integer spin wavefunctions. To that end, we can construct massive spin- conformalprimary wavefunctions, the Rarita-Schwinger fields. Constructed this way, a spin- massiveconformal primary wavefunction can be written Ψ µ, ± ∆ ,J,m ( X µ ; (cid:126)w ) = (cid:90) H [ d ˆ p ] G ∆+ u µ, ± J ( p ; (cid:126)w ) e ± im ˆ p · X (6.3)where J ∈ {− , − , , } . The integration functions here are given by u µ, ± = (cid:15) µ u ± , u µ, ± = (cid:114) (cid:15) µ u ±− + i (cid:114) (cid:15) µ u ± u µ, ±− = (cid:114) (cid:15) µ − u ± + i (cid:114) (cid:15) µ u ±− , u µ, ±− = (cid:15) µ − u ±− (6.4)up to overall constants that can be chosen to fit an appropriate normalization scheme. It isalso useful to note that these relations hold true in the massless case so the massless limitof these wavefunctions is consistent with the massless spin- fields in [5]. In this way onecan construct any massive integer or half integer spin wavefunction.Upon constructing higher half integer spin wavefunctions one should be able to showthey are normalizable with respect to an appropriately defined inner product. One wouldexpect that they would be constrained to be on the principal series and that the basis wouldbe complete. We have chosen not to generalize these results with the expectation that itcan be done if necessary. Acknowledgements
I am grateful to Alex Atanasov, Erin Crawley, Alfredo Guevara, Rajamani Narayanan,Aditya Parikh, Monica Pate, Ana-Maria Raclariu, Andrew Strominger, and Evan Zayas for In principle one can derive these wavefunctions using the same methods in subsection 3.1 however asthe spin increases, the equations are more difficult to solve. It is easy to see that by construction, thesewavefunctions satisfy the Lorentz and conformal transformation properties. – 12 –any useful discussions and important insights. This work was supported by DOE grantde-sc/0007870.
A Solving Constraint Equations
In this appendix we discuss a particular caveat in finding the solutions to the differentialequations in section 3. In particular there are six differential equations of the form D i u J = A i u J (A.1)where the D i are differential operators for i = 1 , · · · , given by D = ∂ w + ∂ z , D = ∂ ¯ w + ∂ ¯ z D = − ( w ∂ w + yz∂ y + z ∂ z − y ∂ ¯ z ) D = − ( ¯ w ∂ ¯ w + y ¯ z∂ y − y ∂ z + ¯ z ∂ ¯ z ) D = 2 w∂ w + y∂ y + 2 z∂ z D = 2 ¯ w∂ ¯ w + y∂ y + 2¯ z∂ ¯ z . (A.2)There are two different linear combinations of different subsets of the set of D i that areequal to ∂ y . In particular ∂ y = (¯ z − ¯ w ) y ( y + | z − w | ) (cid:18) wz D − D − ( w + z ) D + y ( D − w D )¯ z − ¯ w (cid:19) ∂ y = ( z − w ) y ( y + | z − w | ) (cid:18) w ¯ z D − D − ( ¯ w + ¯ z ) D + y ( D − w D ) z − w (cid:19) . (A.3)This results in two differential equations ∂ y u J = M u J , ∂ y u J = M u J (A.4)where M , M are two × matrices. These are just differential equations for the y dependence of the u J therefore we can let u J = f J ( (cid:126)z, (cid:126)w ) ˜ u J . Setting the two equationsequal to one another, and focusing on just the y dependence one notices that the ˜ u J mustsatisfy ( M − M ) ˜ u J = 0 . (A.5)This is solved assuming that the nullspace of M − M is non-empty. In this particularcase, there are two independent vectors in the nullspace for each spin ˜ u − = w , y w − z − ¯ z , ˜ u = ¯ z − ¯ wy + z (¯ z − ¯ w ) , − ¯ w (A.6)Therefore we are able to write u J as linear combination of the two vectors in the nullspaceand solve for the coefficients, f J (¯ z, ¯ w ) by asserting that the initial six differential equationsmust be satisfied. – 13 – Bulk Integrals of Propagators
In this appendix we include details required for the computation of the integrals appearingin the Dirac inner product. Similar integrals have been discussed in [22]. We would like tocompute an integral of the form I bulk = I + I = (cid:90) ∞ dyy (cid:90) d zG α + − iλ G α + + iλ (cid:20) y + ( ¯ w − ¯ z )( w − z ) y (cid:21) . (B.1)This integral has two terms which are to be computed separately. Both can be computedusing the Fourier transform y + | z − w i | ) ∆ ≡ (cid:90) d k (2 π ) (cid:90) d x e ik · ( x − ( z − w i )) ( y + x ) ∆ = (cid:90) dkd ¯ k (2 π ) (cid:90) dxd ¯ x e ikx + i ¯ k ¯ x ( y + x ¯ x ) ∆ . (B.2)The first term in (B.1) is I = (cid:90) ∞ dyy (cid:90) d z (cid:18) yy + | z − w | (cid:19) α + − iλ (cid:18) yy + | z − w | (cid:19) α + + iλ = (cid:90) ∞ dyy α − − i ( λ − λ ) (cid:90) d z (cid:90) d k d k (2 π ) (cid:90) d x d x e ik · ( x − ( z − w ))+ ik · ( x − ( z − w )) ( y + x ) α + − iλ ( y + x ) α + + iλ = 2 (cid:90) ∞ dyy α − − i ( λ − λ ) (cid:90) d k (2 π ) (cid:90) d x d x e ik · ( x − x )+ ik · ( w − w ) ( y + x ) α + − iλ ( y + x ) α + + iλ (B.3) where to go to the last line we have integrated over z, ¯ z to get delta functions in the k i andthen relabeled k → k . Next we use Schwinger parameters to write y + x ) ∆ = 1Γ(∆) (cid:90) ∞ dββ ∆ − e − β ( y + x ) (B.4)for the two factors appearing the integrand. Then, we can complete the square in theexponent to easily perform the x i integrals. This leaves us with I = 2 π Γ( α + − iλ )Γ( α + + iλ ) (cid:90) ∞ dyy α − − i ( λ − λ ) (cid:90) d k (2 π ) e ik · ( w − w ) × (cid:90) ∞ dβ dβ β α − − iλ β α − + iλ e − β y − β y − k β − k β . (B.5)Next we rescale β i → | (cid:126)k | β i y then y → y | (cid:126)k | and perform the integral over y to get I = 2 − α + i ( λ − λ ) π Γ( α + − iλ )Γ( α + + iλ ) (cid:90) d k (2 π ) e ik · ( w − w ) | (cid:126)k | α − − i ( λ − λ ) × (cid:90) ∞ dβ dβ β α − − iλ β α − + iλ ( β + β + β + β ) . (B.6)Before evaluating the remaining integrals, it is useful to evaluate the second term in (B.1) I = (cid:90) ∞ dyy (cid:90) d z ( ¯ w − ¯ z )( w − z ) (cid:18) yy + | z − w | (cid:19) α + − iλ (cid:18) yy + | z − w | (cid:19) α + + iλ We have defined k = k − ik so that the terms look nice. – 14 – (cid:90) ∞ dyy α − − i ( λ − λ ) (cid:90) d k (2 π ) (cid:90) d x d x ¯ x x e ik · ( x − x )+ ik · ( w − w ) ( y + x ) α + − iλ ( y + x ) α + + iλ (B.7) which can be done in a similar way using Schwinger parameters to get I = 2 − α + i ( λ − λ ) π Γ( α + − iλ )Γ( α + + iλ ) (cid:90) d k (2 π ) e ik · ( w − w ) | (cid:126)k | α − − i ( λ − λ ) × (cid:90) ∞ dβ dβ β α − − iλ β α − + iλ ( β + β + β + β ) . (B.8)Combining this with the first term (B.6) and performing the coordinate transformation β = e U + V and β = e U − V we obtain I bulk = 2 − α + i ( λ − λ ) π Γ( α + − iλ )Γ( α + + iλ ) (cid:90) d k (2 π ) e ik · ( w − w ) | (cid:126)k | α − − i ( λ − λ ) × (cid:90) ∞−∞ dU dV e (2 α − − i ( λ − λ )) U e − i ( λ + λ ) V V . (B.9)When α = 1 both exponentials become purely imaginary and we can evaluate this exactly π δ ( λ − λ ) δ (2) ( w − w )Γ( − iλ )Γ( + iλ ) (cid:90) ∞−∞ dV e − iλ V cosh V . (B.10)In order to evaluate the entire inner product, we need to consider the second term in (4.5)which is the complex conjugate of this term. Combining the two gives us π δ ( λ − λ ) δ (2) ( w − w )Γ( + iλ )Γ( − iλ ) (cid:90) ∞−∞ dV cos(2 λ V )cosh V = 32 π δ ( λ − λ ) δ (2) ( w − w )1 + 4 λ . (B.11) C Boundary Integrals of Propagators
In this appendix we include details required for the computation of the integrals appearingin the completeness relation. Similar integrals have been discussed in [22, 25, 30] but fordifferent pairs of conformal weights. We would like to compute an integral of the form I = (cid:90) d w (cid:18) y y + | z − w | (cid:19) − iλ (cid:18) y y + | z − w | (cid:19) + iλ . (C.1)We can use the Fourier representation in (B.2), integrate over w, ¯ w in the same way as inthe bulk case and rescale integration parameters β , β to obtain I = y − iλ y + iλ (cid:90) d w (cid:90) d k d k (2 π ) (cid:90) d x d x e ik · ( x − ( z − w )) e ik · ( x − ( z − w )) ( y + x ) − iλ ( y + x ) + iλ = 2 π y y Γ( − iλ )Γ( + iλ ) (cid:90) d k (2 π ) e ik · ( z − z ) × (cid:90) ∞ dβ dβ β − − iλ β − + iλ e − | (cid:126)k | β y − | (cid:126)k | β y − | (cid:126)k | y β − | (cid:126)k | y β . (C.2)– 15 –sing 3.471.9 in [31] we can perform the β i integrals to get Bessel functions I = 8 π y y Γ( − iλ )Γ( + iλ ) (cid:90) d k (2 π ) e ik · ( z − z ) K − iλ ( | (cid:126)k | y ) K − + iλ ( | (cid:126)k | y ) . (C.3)Using the representations of the modified Bessel function of the second kind K ν ( x ) = 1cos νπ (cid:90) ∞ cos( x sinh t ) cosh νtdt = 1sin νπ (cid:90) ∞ sin( x sinh t ) sinh νtdt (C.4)which is valid when − < Re ( ν ) < , I = 16 πy y (cid:90) d k (2 π ) e ik · z (cid:90) ∞ dudt cos( | (cid:126)k | y sinh u ) sin( | (cid:126)k | y sinh t ) × + iλ + λ cosh (cid:20)(cid:18) − iλ (cid:19) u (cid:21) sinh (cid:20)(cid:18) − iλ (cid:19) t (cid:21) . (C.5)To obtain the appropriate completeness relation, we need to consider the other term in 4.13which can be identified with this term where λ → − λ and ↔ . In order to combine theterms conveniently, we write this term as I = 16 πy y (cid:90) d k (2 π ) e ik · z (cid:90) ∞ dudt cos( | (cid:126)k | y sinh u ) sin( | (cid:126)k | y sinh t ) × − iλ + λ cosh (cid:20)(cid:18)
12 + iλ (cid:19) u (cid:21) sinh (cid:20)(cid:18)
12 + iλ (cid:19) t (cid:21) . (C.6)The completeness relation involves a weighted integral over λ of the sum of these two terms I = (cid:90) ∞−∞ dλµ ( λ )[ I + I ]= 16 πy y (cid:90) d k (2 π ) e ik · z (cid:90) ∞ dudt cos( | (cid:126)k | y sinh u ) sin( | (cid:126)k | y sinh t ) I λ . (C.7)We deduce that the integration weight is µ ( λ ) = µ Γ( + iλ )Γ( − iλ )Γ( + iλ )Γ( − iλ ) . (C.8)The λ integral, which is over the terms in the second lines of (C.5) and (C.6), can be doneusing trigonometric sum and difference identities and using that (cid:82) ∞−∞ dλ cos( λx ) = 2 πδ ( x ) .We find that it is I λ = − πµ ∂ t δ ( u − t ) . (C.9)Now we can substitute this into the integral and perform an integration by parts in t toobtain I = 32 µ y y π (cid:90) d k (2 π ) | (cid:126)k | e ik · z (cid:90) ∞ du cos( | (cid:126)k | y sinh u ) cos( | (cid:126)k | y sinh u ) cosh u = 8 µ y y π (cid:90) d k (2 π ) e ik · z (cid:90) ∞−∞ dv [cos( v ( y − y )) + cos( v ( y + y ))] – 16 – µ y y π (cid:90) d k (2 π ) e ik · z [ δ ( y − y ) + δ ( y + y )]= 16 µ y π δ ( y − y ) δ (2) ( z − z ) . (C.10)where we have let v = sinh u and only kept δ ( y − y ) since y , y > . The Lorentz invariantdelta function in hyperbolic coordinates is δ (3) (ˆ p , ˆ p ) = (2 π ) p δ (3) (ˆ p − ˆ p ) = 4 y (2 π ) δ ( y − y ) δ (2) ( z − z ) (C.11)so if µ = 2 then I = δ (3) (ˆ p , ˆ p ) . (C.12) References [1] S. Pasterski, S.-H. Shao and A. Strominger,
Flat Space Amplitudes and Conformal Symmetryof the Celestial Sphere , Phys. Rev.
D96 (2017) 065026 [ ].[2] S. Pasterski and S.-H. Shao,
Conformal basis for flat space amplitudes , Phys. Rev.
D96 (2017) 065022 [ ].[3] S. Pasterski, S.-H. Shao and A. Strominger,
Gluon Amplitudes as 2d Conformal Correlators , Phys. Rev.
D96 (2017) 085006 [ ].[4] H. T. Lam and S.-H. Shao,
Conformal Basis, Optical Theorem, and the Bulk PointSingularity , Phys. Rev. D (2018) 025020 [ ].[5] A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, Extended Super BMS Algebra ofCelestial CFT , .[6] Y. A. Law and M. Zlotnikov, Massive Spinning Bosons on the Celestial Sphere , .[7] L. Donnay, A. Puhm and A. Strominger, Conformally Soft Photons and Gravitons , JHEP (2019) 184 [ ].[8] M. Pate, A.-M. Raclariu, A. Strominger and E. Y. Yuan, Celestial Operator Products ofGluons and Gravitons , .[9] A. Puhm, Conformally Soft Theorem in Gravity , .[10] Y. A. Law and M. Zlotnikov, Poincare constraints on celestial amplitudes , JHEP (2020)085 [ ].[11] M. Pate, A.-M. Raclariu and A. Strominger, Conformally Soft Theorem in Gauge Theory , Phys. Rev. D (2019) 085017 [ ].[12] S. Banerjee, S. Ghosh and P. Paul,
MHV Graviton Scattering Amplitudes and CurrentAlgebra on the Celestial Sphere , .[13] Y. A. Law and M. Zlotnikov, Relativistic partial waves for celestial amplitudes , .[14] E. Casali and A. Puhm, A Double Copy for Celestial Amplitudes , .[15] S. Albayrak, C. Chowdhury and S. Kharel, On loop celestial amplitudes for gauge theory andgravity , .[16] J. de Boer and S. N. Solodukhin, A Holographic reduction of Minkowski space-time , Nucl.Phys.
B665 (2003) 545 [ hep-th/0303006 ]. – 17 –
17] D. Kapec, V. Lysov, S. Pasterski and A. Strominger,
Semiclassical Virasoro symmetry of thequantum gravity S -matrix , JHEP (2014) 058 [ ].[18] D. Kapec and P. Mitra, A d -Dimensional Stress Tensor for Mink d +2 Gravity , JHEP (2018) 186 [ ].[19] C. Cheung, A. de la Fuente and R. Sundrum,
4D scattering amplitudes and asymptoticsymmetries from 2D CFT , JHEP (2017) 112 [ ].[20] L. Iacobacci and W. Mueck, Conformal Primary Basis for Dirac Spinors , .[21] B. Oblak, From the Lorentz Group to the Celestial Sphere , 8, 2015, .[22] M. S. Costa, V. Goncalves and J. Penedones,
Spinning AdS Propagators , JHEP (2014)064 [ ].[23] K. Hinterbichler, A. Joyce and R. A. Rosen, Massive Spin-2 Scattering and AsymptoticSuperluminality , JHEP (2018) 051 [ ].[24] H. Osborn, Conformal Blocks for Arbitrary Spins in Two Dimensions , Phys. Lett. B (2012) 169 [ ].[25] D. Simmons-Duffin,
Projectors, Shadows, and Conformal Blocks , JHEP (2014) 146[ ].[26] P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory , Journalof High Energy Physics (2018) 102 [ ].[27] T. R. Taylor,
A Course in Amplitudes , Phys. Rept. (2017) 1 [ ].[28] S. Stieberger and T. R. Taylor,
Symmetries of Celestial Amplitudes , Phys. Lett. B (2019) 141 [ ].[29] N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang,
Scattering Amplitudes For All Masses andSpins , .[30] M. Pate, Note on Massive Conformal Primary Wavefunctions . unpublished, and privatecommunication, 2020.[31] I. S. Gradshteyn and I. M. Ryzhik,
Table of integrals, series, and products .Elsevier/Academic Press, Amsterdam, seventh ed., 2007..Elsevier/Academic Press, Amsterdam, seventh ed., 2007.