Steinmann Relations and the Wavefunction of the Universe
aa r X i v : . [ h e p - t h ] S e p Steinmann Relations and the Wavefunction of the Universe
Paolo Benincasa,
1, 2, ∗ Andrew J. McLeod, † and Cristian Vergu ‡ Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen,Blegdamsvej 17, 2100 København, Denmark Instituto de F´ısica Te´orica UAM/CSIC, Calle Nicol´as Cabrera 13-15,Cantoblanco, 28049 Madrid, Spain
The physical principles of causality and unitarity put strong constraints on the analytic structureof the flat-space S-matrix. In particular, these principles give rise to the Steinmann relations, whichrequire that the double discontinuities of scattering amplitudes in partially-overlapping momentumchannels vanish. Conversely, at cosmological scales, the imprint of causality and unitarity is in gen-eral less well understood—the wavefunction of the universe lives on the future space-like boundary,and has all time evolution integrated out. In the present work, we show how the flat-space Stein-mann relations emerge from the structure of the wavefunction of the universe, and derive similarrelations that apply to the wavefunction itself. This is done within the context of scalar toy modelswhose perturbative wavefunction has a first-principles definition in terms of cosmological polytopes .In particular, we use the fact that the scattering amplitude is encoded in the scattering facet of cos-mological polytopes, and that cuts of the amplitude are encoded in the codimension-one boundariesof this facet. As we show, the flat-space Steinmann relations are thus implied by the non-existence ofcodimension-two boundaries at the intersection of the boundaries associated with pairs of partially-overlapping channels. Applying the same argument to the full cosmological polytope, we also deriveSteinmann-type constraints that apply to the full wavefunction of the universe. These argumentsshow how the combinatorial properties of cosmological polytopes lead to the emergence of flat-spacecausality in the S-matrix, and provide new insights into the analytic structure of the wavefunctionof the universe.
INTRODUCTION
The physics of the early universe involves processes atultra-high energies, as the Hubble parameter during infla-tion can be as large as 10 GeV. This physics is encodedin correlation functions, or equivalently in the wavefunc-tion of the universe, living at a space-like boundary of aquasi- dS spacetime at the end of inflation. Our ability tounderstand physics at such scales depends in part on ourunderstanding of the analytic structure of these quanti-ties, which we expect to be constrained by basic physicalprinciples such as unitarity and causality.We can draw a parallel with what happens in flatspacetime, where the relevant quantum mechanical ob-servable is the S-matrix. In this case there already existsa good understanding of how physical principles such asLorentz invariance, unitarity, and causality are reflectedin the structure of S-matrix elements. For instance, uni-tarity is encoded in the factorization properties of theS-matrix [1–3], while causality is encoded in its analyticproperties [4]. One particular set of constraints that fol-low from causality and that have recently proven use-ful for bootstrapping amplitudes and integrals in planar N = 4 supersymmetric Yang-Mills theory [5–11] are theSteinmann relations [12–19], which require that doublediscontinuities in partially overlapping channels vanish:Disc s I (Disc s J M ) = 0 , where I * JJ * II ∩ J 6 = ∅ . (1)Here, M denotes a scattering amplitude, while s I = (cid:0)P i ∈I p i (cid:1) is the Mandelstam invariant that de-pends on the external momenta whose labels belong tothe set I . The Steinmann relations apply to any quan-tum field theory and even to individual Feynman inte-grals [20]—however, their implications can be subtle incases involving massless external particles [5, 20].In contrast, it is not clear how these physical principlesmake their appearance in cosmological processes, wherethe relevant observables are cosmological correlators or,equivalently, the wavefunctions of the universe which gen-erate them. Such quantities live on the future space-likeboundary and, consequently, have all past time evolu-tion integrated out. So, in what ways are the imprints ofcausality and unitary visible?A similar question—regarding how flat-space unitar-ity and Lorentz invariance emerge from the wavefunctionof the universe—was discussed in [21] in the context of cosmological polytopes [22, 23]. Cosmological polytopesare geometrical-combinatorial objects which encode all ofthe properties of the wavefunction despite having a first-principles definition that makes no reference to physicalconcepts such as Hilbert space or spacetime. In particu-lar, they constitute a special class of positive geometries,which come equipped with a canonical form that has log-arithmic singularities (only) along the boundaries of thepolytope. In the case of cosmological polytopes, these We emphasize that positive geometries can be defined and stud-ied independently of any physical interpretation [24]. η η = 0 η = −∞ −→ p −→ p −→ p −→ p −→ p x x x x x x FIG. 1: On the left, we show a Feynman graph that contributes to the wavefunction of the universe. On the right, we depictthe associated reduced graph, which is obtained from the Feynman graph by suppressing the external edges. singularities come in one-to-one correspondence with thesingularities of the wavefunction, and each of them givesrise to an associated canonical function that provides thecontribution of a Feynman graph G to the wavefunctionitself [22]. One of these singularities occurs at the van-ishing locus of the total energy E tot = P i | ~p i | , where ~p i is the spatial-momentum of the i th external state (notethat for physical processes | ~p i | is always positive and,hence, such locus can be reached upon analytic contin-uation only). As such a singularity is approached, thewavefunction of the universe reduces to (the high energylimit of) the flat-space scattering amplitude [26–28]. Inthe cosmological polytope picture, the total energy singu-larity identifies the scattering facet , which is a polytopeliving on a codimension-one boundary of the cosmologicalpolytope, whose canonical form returns the relevant scat-tering amplitude [22]. The vertex structure of this facetmakes flat-space unitarity manifest; in particular, thecodimension-one faces of the scattering facet themselvesfactorize into pairs of lower-dimensional scattering facetsand a simplex that encodes the Lorentz-invariant phase-space measure. This provides a combinatorial statementof the cutting rules, which encode unitarity. In a sim-ilar manner, Lorentz invariance on the scattering facetis made manifest by a contour integral representation ofthe canonical form [21].In this work, we adopt a similar strategy for studyinghow the flat-space causality is encoded in the wavefunc-tion of the universe. In particular, we investigate how theSteinmann relations are encoded in the scattering facet,utilizing the correspondence between the boundaries ofthese polytopes and the cuts of Feynman integrals. We When the scattering amplitude vanishes or the states under con-sideration do not have a flat-space counterpart, the total energysingularity is softer and its coefficient is a typical cosmologicaleffect—see [25] for an example. The connection between individual discontinuities and cut inte- also ask whether Steinmann-like relations hold for thewavefunction itself, whose cuts now correspond to facetsof the full cosmological polytope.As we review below, for both the scattering amplitudeand the wavefunction of the universe, these cuts are morespecifically encoded in the residues of the canonical form ω ( P ) on the boundaries of the associated polytope P ,where P is either the full cosmological polytope P G orthe scattering facet S G . These boundaries come in one-to-one correspondence with the subgraphs g ⊂ G , andcan be characterized by hyperplanes W g in dual projec-tive space; thus, we denote the residue of ω ( P ) on theboundary corresponding to g by Res W g ω ( P ).The canonical form has the property that its residuealong the hypersurface W g itself constitutes the canonicalform on this boundary of P : ω ( P ∩ W g ) = Res W g ω ( P ) . (2)Importantly, this new canonical form only has singulari-ties along the hyperplanes in W g which contain its facets.In other words, Res W g Res W g ω ( P ) = 0 (3)if W g ∩ W g does not contain a codimension-two facetof P . Below, we will show that the resemblance betweenequations (1) and (3) is not a coincidence.In what follows, we first review the wavefunction ofthe universe and the salient aspects of cosmological poly-topes. We then investigate the question of how flat-spacecausality emerges on the scattering facet of cosmologi-cal polytopes, where the Steinmann relations are knownto apply. We demonstrate that the Steinmann relations grals has long been understood [29]. For more details on howsequential discontinuities of scattering amplitudes can be com-puted using cut integrals, see [20, 30] (also [31, 32]). emerge naturally from the face structure of the scatteringfacet, insofar as the codimension-one boundaries of thisfacet that correspond to partially-overlapping momen-tum channels never intersect to form codimension-twoboundaries. This elucidates the mechanism by whichflat-space causality emerges from the wavefunction ofthe universe, complementing the existing understandingof how flat-space unitarity and Lorentz invariance alsoemerge [21]. Having uncovered the combinatorial mech-anism by which these relations are encoded, we go onto derive novel constraints on the wavefunction of theuniverse that follow from the same argument. COSMOLOGICAL POLYTOPES
Let us consider a massless scalar theory in ( d + 1)-dimensional flat spacetime with polynomial interactionsthat have time-dependent couplings, S [ φ ] = Z d d x Z −∞ dη
12 ( ∂φ ) − X k ≥ λ k ( η ) k ! φ k . (4)This theory describes a conformally-coupled scalar in anFRW cosmology, where ds = a ( η )[ − dη + δ ij dx i dx j ],provided that the couplings λ k ( η ) are taken to be λ k ( η ) = λ k ϑ ( − η )[ a ( η )] (2 − k )( d − / for some constants λ k .The wavefunction of the universe for this theory isgiven by Ψ[Φ] = φ (0)=Φ Z φ ( −∞ (1 − iε )) = 0 Dφ e iS [ φ ] , (5)where the iε prescription regularizes the path integral atearly times, and Φ corresponds to the boundary state atlate time. We split φ into a classical mode and a quantumfluctuation via φ ( ~p, η ) = Φ( ~p ) e i | ~p | η + ϕ , (6)where the classical part of the solution has the correctBunch-Davies oscillatory behavior at early times, and werequire the fluctuations ϕ to vanish at both early andlate times ( η = −∞ , λ k ( η ) = Z + ∞−∞ dǫ e iǫη ˜ λ k ( ǫ ) , (7)the perturbative wavefunction can be computed in termsof Feynman diagrams, which now include an integral over the energy ǫ associated with each graph site. We herepostpone consideration of these integrals over site ener-gies, as well as the integrals over the spatial loop mo-menta in the case of loop graphs, and focus on the re-maining contribution coming from a graph G with sites V and edges E , ψ G ( x v , y e ) = Z −∞ Y v ∈V (cid:2) dη v e ix v η v (cid:3) Y e ∈E G ( η v e , η v ′ e , y e ) , (8)where x v = P i ∈ v | ~p i | is the sum of external energies en-tering site v ∈ V , y e is the energy flowing through edge e ∈ E , and G ( η v e , η v ′ e , y e ) = (2 y e ) − (cid:2) e − iy e ( η ve − η v ′ e ) ϑ ( η v e − η v ′ e )+ e + iy e ( η ve − η v ′ e ) ϑ ( η v ′ e − η v e ) − e iy e ( η ve + η v ′ e ) (cid:3) (9)is the bulk-to-bulk propagator that satisfies the boundarycondition that all fluctuations vanish at η = 0. Thiscontribution to the Feynman diagram is universal, insofaras the details of the specific theory are entirely encodedin the Fourier coefficients ˜ λ k ( ǫ ), which we have factoredout as part of the energy integrals. For further detailssee [22].Since the integrals over conformal time in equation (8)only depend on the total energy entering each site, theycan be represented by reduced graphs in which the linesconnected to external states have been suppressed. Anexample Feynman graph and the associated reducedgraph are shown in Figure 1. In [22] it was shown thatthese reduced graphs are in one-to-one correspondencewith polytopes whose canonical forms encode the samesingularity structure as ψ G ( x v , y e ). To identify the poly-tope associated with a given graph G , we associate a tri-angle with each of its propagators, where the edges ofthis triangle can be thought of as representing the twosites and the edge that constitute this propagator: x i x ′ i y i x i x ′ i y i ←→ We can situate these triangles in projective space by tak-ing x i , y i , and x ′ i to be generic vectors in P n e − , where n e is the number of propagators in G . The vertices of the i th triangle will then be given by the vectors x i − y i + x ′ i , In order to avoid language clashes, we will refer to the vertices ofthe graphs as sites , reserving the name vertices for the polytopes. For recent work on the integrals over site energies, see [33]. While the energy flowing through each edge will be partially fixedby momentum conservation, we leave these energies general asthe function ψ G ( x v , y e ) doesn’t know about these constraints. x ′ i x i x j x i x ′ i x i x ′ i y i x j y j x i x j y i y j FIG. 2: Examples of cosmological polytopes obtained from the intersection of two triangles. The images in the left columnillustrate the intersection of pairs of triangles at either one or two midpoints, while the corresponding convex hulls are depictedin the middle column. The column on the right shows the associated reduced graphs. x i + y i − x ′ i , and − x i + y i + x ′ i . However, when multi-ple propagators end on the same graph site, the corre-sponding midpoints of these triangles should be identi-fied, which implies that these triangles will be containedwithin a lower-dimensional subspace of P n e − . In par-ticular, the vertices appearing in this collection of trian-gles will only span a subspace of dimension n v + n e − n v and n e represent the number of sites and edgesin G . The cosmological polytope P G associated with G isdefined to be the convex hull of these triangle vertices.Examples are shown in Figure 2.Let us illustrate the construction described above onthe simple case of a triangle ∆ i , whose edge midpointsare at x i , y i , and x ′ i . We can embed the triangle inthe projective plane via the identifications x i = [1 : 0 :0], y i = [0 : 1 : 0] and x ′ i = [0 : 0 : 1]. Then, wecan parametrize a generic point within the correspondingcosmological polytope by Y = x i x i + y i y i + x ′ i x ′ i ∈ ∆ i ,where x i , y i , and x ′ i are real numbers that satisfy x i + y i ≥ x i + x ′ i ≥ y i + x ′ i ≥ Y inside a cosmologicalpolytope P G , we can formulate a unique canonical form ω ( Y , P G ) = Ω( Y , P G ) hY d N Yi (10)in P G . This form has logarithmic singularities on—andonly on—all of its boundaries [22]. Moreover, Ω( Y , P G ) iscalled the associated canonical function , and turns out tobe precisely the universal part of the Feynman diagram Note that we sometimes leave the dependence of the canonicalform and the canonical function on Y implicit. from equation (8), namelyΩ( Y , P G ) = ψ G ( x v , y e ) . (11)One of the general features of so-called canonical func-tions is that they are singular (only) on the boundariesof the associated positive geometry. Thus, by virtue ofequation (11), the boundary structure of P G characterisesthe residues of ψ G ( x v , y e ). Moreover, as shown in [22],the codimension-one boundaries (or facets) of P G are inone-to-one correspondence with the connected subgraphsof G . In particular, each of the codimension-one facets of P G can be characterized by a dual vector W g = X v ∈ g ˜ x v + X e ∈E ext g ˜ y e , (12)where g is one of the connected subgraphs of G , ˜ x v and˜ y e are dual vectors to x v and y e (meaning ˜ x v · x v ′ = δ vv ′ ,˜ y e · y e ′ = δ ee ′ , and ˜ x v · y e = ˜ y e · x v = 0), and E ext g isthe set of edges departing from the subgraph g . It canbe checked that W g · Y ≥ Y ∈ P G . Wealso associate an energy to the subgraph g , namely E g = X v ∈ g x v + X e ∈E ext g y e , (13)and we have that E g → In cases involving internal massless states in a generic FRWcosmology, the function ψ G ( x v , y e ) is instead determined by a covariant form associated to the very same cosmological poly-tope [23, 34]. by introducing a simple marking which identifies thosevertices which are not on the facet: x i x ′ i y e W · ( x i − y e + x ′ i ) > x i x ′ i y e W · ( x i + y e − x ′ i ) > x i x ′ i y e W · ( − x i + y e + x ′ i ) > In particular, the facet of P G identified with a subgraph g ⊆ G is given by marking all internal edges of g in themiddle, and all the edges that depart this subgraph onthe side closest to g . This association fully characterisesthe facets of P G .Finally, let us highlight that the combinatorial struc-ture of cosmological polytopes is such that each of theircodimension- k faces is given by the intersection of apair of codimension-( k −
1) faces. Given that P G has di-mension n v + n e −
1, the intersection of two of thesecodimension-( k −
1) faces can only have codimension k if the intersecting faces share enough vertices to span a( n v + n e − k − necessary but notsufficient condition for such a face to exist. STEINMANN RELATIONS AND THESCATTERING FACET
One facet of particular interest is the so-called scatter-ing facet S G , on which the total external energy vanishesand is consequently conserved. The residue of ψ G ( x v , y e )on this boundary returns the contribution to the scat-tering amplitude coming from the graph G . We begin byinvestigating how the Steinmann relations are encoded inthe canonical form of S G , by studying the correspondencebetween its facet structure and cut integrals.The scattering facet of the cosmological polytope P G is characterized by the dual vector in equation (12) when g is chosen to be the full (reduced) graph G , namely S G = P G ∩ W G . (14)This facet is given by the convex hull of the 2 n e vertices { x i + y ij − x j , − x i + y ij + x j } , where we have switchedto a notation in which y ij represents the edge betweensites x i and x j . Since it corresponds to a codimension-one face of the cosmological polytope, it has dimension n v + n e − Codimension-one Faces and Individual Cuts
Let us first review how the codimension-one faces of S G relate to the individual cuts of ψ G ( x v , y e ) [21]. Each ofthese faces corresponds to a connected subgraph g ⊂ G and is given by the convex hull of the vertices { x j + y ij − x i | x i / ∈ g ∨ x j ∈ g } . This corresponds to deletingjust the vertices of S G whose markings appear closestto g on the edges that depart this subgraph. On thisface, Ω( Y , S G ) factorises into a pair of lower-dimensional x x x x x x x g x x x x x x x g ¯ g FIG. 3: An example of a codimension-one face of the scat-tering facet, which is associated with an individual cut of thescattering amplitude. In the diagram on the left, we mark thevertices that do not appear on the scattering facet S G by ,and the additional vertices that get eliminated when we inter-sect it with the facet corresponding to g by . In the diagramon the right, we depict the vertices which do contribute to P G ∩ W g by open circles. The vertices marked by and areassociated with the scattering facets S g and S ¯ g , respectively.The remaining vertices, marked by , are associated with thesimplex Σ . scattering facets and a simplex that encodes the Lorentz-invariant phase space of the cut propagators. That is,Ω( Y , P G ∩ W g ) = Ω( Y g , S g ) × Ω( Y ¯ g , S ¯ g ) × Ω( Y , Σ ) , (15)where S g and S ¯ g are the scattering facets associated withthe reduced graph g and its complement ¯ g , and Σ isthe simplex formed out of the remaining vertices of thescattering facet on the cut edges [21]. The direction ofenergy flow is encoded by the vertex structure of Σ . Anexample is shown in Figure 3.We highlight that, since we have only computed a sin-gle cut, the vertices of this face must span a space ofdimension n v + n e −
3. Thus, we must have2 n e − n g ≥ n v + n e − n e − L − , (16)where n g is the number of edges departing from g , andthe last equality is obtained using the fact that n v = n e − L + 1, where L indicates the loop order of G . Thiscan be rephrased as1 ≤ n g ≤ L + 1 . (17)This, it turns out, constitutes both a necessary and suf-ficient condition for this codimension-one face to exist,and for the corresponding cut to be nonzero. Codimension-two Faces and Sequential Cuts
We now consider the intersection of pairs ofcodimension-one faces on the scattering facet, which cor-respond to sequential cuts of the scattering amplitude.We will in particular be interested in pairs of subgraphs g and g that satisfy g ∩ g = ∅ , g ∩ ¯ g = ∅ , ¯ g ∩ g = ∅ , ¯ g ∩ ¯ g = ∅ , (18)where ¯ g j denotes the complement of g j . Such a pair ofsubgraphs corresponds to a pair of partially-overlapping x x x x x x x x x x x x x g x x x x x x x x x x x x x g FIG. 4: An example of a pair of subgraphs that correspondto partially overlapping momentum channels, and their real-ization as faces on the scattering facet. momentum channels; sequential cuts in these channelsare therefore expected to vanish by virtue of the Stein-mann relations.Sequential cuts can be nonzero only if the correspond-ing codimension-two facet of S G exists, and this facet canonly exist if there are a sufficiently large number of ver-tices in this intersection to span a space of dimension n v + n e − n e − L −
3. This corresponds to therequirement that2 n e − n g − n g + n g ∩ g ≥ n e − L − , (19)where n g j denotes the number of edges departing from g j and n g ∩ g is the number of edges simultaneously de-parting from g and g . This inequality can be rewrittenas n g + n g − n g ∩ g ≤ L + 2 . (20)However, unlike the inequality that must be satisfied bycodimension-one faces of S G in equation (17), this turnsout to be a necessary but not sufficient condition for acodimension-two facet to exist.To see this, let us consider the general form this in-tersection of codimension-one facets will take. Just asthe canonical function factorized into a pair of scat-tering facets on the intersection of S G with one of itscodimension-one faces, the canonical function will nowfactorize into up to four scattering facets, associated withthe subgraphs g ∩ g , g ∩ ¯ g , g ∩ ¯ g , and ¯ g ∩ ¯ g . Insome cases, one of these four subgraphs will be empty,and the canonical form will factorize into only three scat-tering facets. There will also in general be contribu-tions coming from a simplex Σ , corresponding to thecut edges among g ∩ g , g ∩ ¯ g , g ∩ ¯ g , and ¯ g ∩ ¯ g . To be more precise, n g is the sum of the degrees of the verticesin g with respect to edges not in g . If an edge does not belongto g but both its sites are in g , then it contributes 2 to n g , eventhough it is a single edge. Note that some of these scattering facets may no longer corre-spond to subgraphs that are connected. x x x x x x x x x x x x x g g ¯ g ∩ ¯ g x x x x x x x x x x x x x g ∩ ¯ g g ∩ g g ∩ ¯ g ¯ g ∩ ¯ g FIG. 5: The intersection of the pair of facets depicted in Fig-ure 4. This intersection factorizes into four lower-dimensionalscattering facets S g ∩ g , S g ∩ ¯ g , S ¯ g ∩ g and S ¯ g ∩ ¯ g , whosevertices are respectively depicted by the markings , , , and. The remaining vertices, denoted by , identify the simplexΣ . This represents an example of an intersection that satis-fies condition (20), yet that does not form a codimension-twofacet of S G . Importantly, each of these lower-dimensional polytopeslives in a space whose dimension is determined by thenumber of sites and edges of the associated subgraph.In particular, a scattering facet associated with a graph g that has n g v sites and n g e edges will have dimension n g v + n g e −
2, while Σ will have dimension n −
1, where n denotes the number of cut edges in G associated withthe simplex. Given this, we can simply check whetherthese polytopes jointly give rise to a space of dimension n v + n e −
4, as required. We finddim( S G ∩ W g ∩ W g ) = X S g ( n g v + n g e −
1) + n − , (21)where the sum is over the scattering facets formed by thesubgraphs g ∩ g , g ∩ ¯ g , g ∩ ¯ g , and ¯ g ∩ ¯ g . Given thatall sites in G contribute to a single scattering facet, andthat all of its edges are associated either with a scatteringfacet or the simplex Σ , this formula reduces todim( S G ∩ W g ∩ W g ) = n v + n e − − X S g . (22)Clearly this will reproduce the correct dimension only ifthe sum is over three scattering facets, and not four.We now recall that pairs of graphs g and g that thatcorrespond to partially-overlapping momentum channelssatisfy equation (18). It is easy to see that the intersec-tion S G ∩ W g ∩ W g for such subgraphs will always fac-torize into four nontrivial scattering facets; an example isshown in Figure 5. As such, it follows from equation (22)that the intersection of the codimension-one faces cor-responding to these subgraphs will always constitute acodimension-three facet of S G . This implies that thesecuts must vanish, and thereby encodes the Steinmannrelations in a combinatorial way. Conversely, it is easyto see that pairs of graphs g and g that dot no sat-isfy equation (18) will generically only give rise to threenontrivial scattering facets, and can thus give rise to in-tersections of the proper dimension.Let us pause here to highlight the essential role be-ing played by the numerators of canonical forms in thisargument. Namely, these numerators eliminate any con-tribution coming from the intersection of the hyperplanes W g and W g outside of the polytope by vanishing alongthese intersections [35]. Thus, while the intersection ofany two k -dimensional hyperplanes in projective spacehas dimension k −
1, the intersection of two facets ofthe cosmological polytope can have lower dimension as aresult of these numerators.This then is the mechanism by which the Steinmannrelations are enforced by the structure of the cosmologi-cal polytope. Any two facets corresponding to partially-overlapping momentum channels intersect outside of thecosmological polytope (up to higher-codimension bound-aries); thus, taking sequential residues on these facetsyields zero due to the vanishing of the numerator alongthis intersection. The corresponding sequential cut—andsequential discontinuity—thereby also vanish.As a final remark, it is important to note that theargument above holds for any graph and for any num-ber of external states. In particular, it is well knownthat the double discontinuity of the box graph does notvanish upon analytic continuation outside the physicalregion [16]. However, there is no contradiction with theanalysis just presented, as this nonzero double discon-tinuity is not visible in Lorentzian signature (and, con-sequently, with real energies), while the scattering facet(and indeed the whole cosmological polytope) is intrinsi-cally Lorentzian.
STEINMANN RELATIONS AND THEWAVEFUNCTION OF THE UNIVERSE
Let us now carry out the same analysis on the fullcosmological polytope P G . Namely, we ask whether theintersection of a pair of codimension-one faces P G ∩ W g and P G ∩ W g on the cosmological polytope correspondsto a codimension-two face when the subgraphs g and g satisfy equation (18).On each of the codimension-one faces P G ∩ W g j ,the canonical function factorizes into a pair of lower-dimension polytopes. Namely,Ω( Y , P G ∩ W g j ) = Ω( Y g j , S g j ) × Ω( Y ¯ g j ∪6E , P ¯ g j ∪6E ) (23)where Ω( Y g j , S g j ) is the canonical function of the scat-tering facet S g j , and Ω( Y ¯ g j ∪6E , P ¯ g j ∪6E ) is the canonicalfunction of the polytope P ¯ g j ∪6E , defined to be the con-vex hull of the vertices associated to the complementary x x x x x x x x x x x x x g ¯ g x x x x x x x x x x x x x g ¯ g FIG. 6: A pair of partially-overlapping codimension-one faces P G ∩W g and P G ∩W g of the cosmological polytope P G . Themarkings depict the complementary subgraphs ¯ g j as well asthe vertices on the cut edges. Together, these vertices formthe polytope P g j ∪6E . graph ¯ g j and the cut edges . This factorization isdepicted in Figure 6.In more detail, the canonical function Ω( Y ¯ g j ∪6E , P ¯ g j ∪6E )is related to the lower-point wavefunction ψ ¯ g j associatedwith the subgraph ¯ g j [22, 36]. Specifically, it is given bya sum over the positive and negative energy solutions forthe energy on the cut edges , divided by the product ofthe energies associated with the cut edges [37]:Ω( Y ¯ g j ∪6E , P ¯ g j ∪6E ) = X { σ e } = ± ψ ¯ g j ( x v ( σ e ) , y e ) Q e ∈6E y e (24)where, like in equation (8), the arguments of ψ ¯ g j are theenergies associated with the sites and edges in ¯ g j . How-ever, the site energies have been shifted by the energiesof the cut edges , namely x v ( σ e ) = x v + X e ∈6E∩E v σ e y e , (25)where E v denotes the set of edges that depart from thesite v (thus, the energies associated with sites that aren’tconnected to a cut edge remain unshifted). Importantly,the explicit factors of (2 y e ) − in (25) cancel after carryingout the sum, and therefore do not correspond to realpoles [36].Now let us consider the intersection P G ∩ W g ∩ W g ofa pair of codimension-one faces associated with the sub-graphs g and g . The canonical function on this intersec-tion will in general factorize into three lower-point scat-tering facets associated with the graphs g ∩ g , g ∩ ¯ g ,and g ∩ ¯ g , and a lower-dimensional polytope associ-ated with g c = (¯ g ∩ ¯ g ) ∪ 6E similar to the second fac-tor in equation (23). An example is shown in Figure 7.There will also be a contribution coming from Σ , which Note that we’re abusing notation here, insofar as ¯ g j ∪ 6E is not agraph in the ordinary sense; we include in it the edges in , butnot the sites outside of ¯ g j that these edges are incident with. x x x x x x x x x x x x x g ∩ ¯ g g ∩ g g ∩ ¯ g ¯ g ∩ ¯ g FIG. 7: The intersection of the pair of facets depicted inFigure 6. This intersection factorizes into a product of threelower-dimensional scattering facets associated with the graphs g ∩ g , g ∩ ¯ g , and g ∩ ¯ g , and a lower-dimensional polytopeassociated with the graph (¯ g ∩ ¯ g ) ∪6E . is formed by the vertices on the cut edges among g ∩ g , g ∩ ¯ g , and g ∩ ¯ g .Like on the scattering facet, it is not sufficient to checkthat this intersection has enough vertices to (in princi-ple) span a space of dimension n v + n e −
3. Thus, weagain count the dimension of each of the polytopes thatthe canonical function factorizes into. As before, eachscattering facet has dimension n g v + n g e −
2, where n g v and n g e denote the number of sites and edges in the corre-sponding reduced graph g . The polytope associated with g c will have dimension n g c v + n g c e + n ¯ E −
1, where n ¯ E isthe number of cut edges departing from this subgraph.Finally, the simplex Σ will have dimension n −
1, where n denotes the number of cut edges in G that contributeto this simplex. We thus finddim ( P G ∩ W g ∩ W g ) = X S g ( n g v + n g e −
1) + n ++ ( n g c v + n g c e + n ¯ E ) − , (26)where the sum is over the scattering facets related tothe subgraphs g ∩ g , g ∩ ¯ g and g ∩ ¯ g . As each ofthe sites and edges in G only contribute to one of thesefactors, this straightforwardly reduces todim ( P G ∩ W g ∩ W g ) = n v + n e − − X S g . (27)Thus, when g and g correspond to partially-overlappingchannels and satisfy equation (18), we see that this inter-section has dimension n v + v e −
4, and the correspondingsequential cut vanishes. This implies new Steinmann-likerelations for the wavefunction of the universe; in partic-ular, we haveRes E g (cid:0) Res E g ψ G (cid:1) = 0 , where g * g g * g g ∩ g = ∅ , (28)where the energies E g j were given in equation (13). Attree level, where the integrals over site energies are known to give rise to polylogarithms [22, 33], this can immedi-ately be promoted to a restriction on the double discon-tinuities of Ψ G . CONCLUSION AND OUTLOOK
Causality and unitarity constitute two of the basic pil-lars of our understanding of physical processes. In thiswork, we have explored how these properties emerge inflat-space from the structure of the wavefunction of theuniverse by studying its formulation in terms of cosmo-logical polytopes. In particular, we have seen that theSteinmann relations are transparently encoded in thecombinatorial properties of the canonical form of thescattering facet. Additionally, we have derived novel re-lations that directly restrict the analytic properties of thewavefunction of the universe.While the Steinmann relations for scattering ampli-tudes are understood to be implied by causality [12–19],it is not yet clear whether this is true for the more gen-eral constraints we have derived. It would therefore beinteresting to see these constraints follow directly frombasic physical principles. In this respect, it is worth high-lighting that the perturbative integrands of in–in corre-lation functions involving the operator O can be writtenas R -products between interaction Hamiltonians and O . R -products were introduced to study causality [38], andplay an important role in the formulation of the originalSteinmann relations for correlation functions.As the new relations we have derived constrain thefunctional form of the wavefunction of the universe, theymay prove useful for bootstrapping this quantity (evenat tree level). In particular, it is worth stressing thatour analysis is valid in any FRW cosmology, as theobject of our investigation was the universal integrandin equation (8). The wavefunction for a specific back-ground is obtained by integrating this quantity over ex-ternal energies with appropriate coefficients ˜ λ ( ǫ ). Con-sequently, these constraints should provide valid input forthe recently-inaugurated boostless bootstrap program [40].Eventually, one would also like to find a generaliza-tion of cosmological polytopes that combinatorially en- However, we note that (just like in for the standard Steinmannrelations) one has to be careful how one chooses the analyticcontinuations used to actually compute these double discontinu-ities [20]. Moreover, the wavefunction we have studied here determineswavefunctions involving more general scalar states, which can bethought of as having a time-dependent mass; these latter wave-functions are determined by recursion relations which the formerwavefunctions enter as seeds [23]. Conformally-coupled scalarsalso play the role of seeds for correlation functions involving bothscalars with more general conformal dimension, and states withspin, by applying suitable differential operators on them [37, 39]. code the perturbative wavefunction of the universe in asingle object, rather than just via the graphs that con-tribute to it. However, as the Steinmann-type constraintswe have derived here apply graph-by-graph, these gener-alized cosmological polytopes would have to obey similarconstraints.Finally, while our analysis is fairly general, we highlightthat it remains unclear whether these Steinmann-typerelations for the wavefunction of the universe extend toprocesses involving states that do not have counterpartsin flat space, which aren’t entirely captured by the cosmo-logical polytope. We leave this important open questionfor future work.
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