Quantum Supersymmetric Cosmological Billiards and their Hidden Kac-Moody Structure
QQuantum Supersymmetric Cosmological Billiardsand their Hidden Kac-Moody Structure
Thibault Damour ∗ Institut des Hautes Etudes Scientifiques,35 route de Chartres, 91440 Bures-sur-Yvette, France
Philippe Spindel † Unité de Mécanique et Gravitation, Université de Mons,Faculté des Sciences,20, Place du Parc, B-7000 Mons, Belgium (Dated: November 10, 2018)We study the quantum fermionic billiard defined by the dynamics of a quantized supersymmetricsquashed three-sphere (Bianchi IX cosmological model within D = 4 simple supergravity). Thequantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space.We focus on the 15- and 20-dimensional subspaces (with fermion numbers N F = 2 and N F =3 ) where there exist propagating solutions of the supersymmetry constraints that carry (in thesmall-wavelength limit) a chaotic spinorial dynamics generalizing the Belinskii-Khalatnikov-Lifshitzclassical “oscillatory" dynamics. By exactly solving the supersymmetry constraints near each one ofthe three dominant potential walls underlying the latter chaotic billiard dynamics, we compute thethree operators that describe the corresponding three potential-wall reflections of the spinorial statedescribing, in supergravity, the quantum evolution of the universe. It is remarkably found that thelatter, purely dynamically-defined, reflection operators satisfy generalized Coxeter relations whichdefine a type of spinorial extension of the Weyl group of the rank-3 hyperbolic Kac-Moody algebra AE . I. INTRODUCTION
One of the challenges of gravitational physics is to describe the fate of spacetime at spacelikesingularities (such as the cosmological big bang, or big crunches within black holes). A newavenue for attacking this problem has been suggested a few years ago via a conjectured corre-spondence between various supergravity theories and the dynamics of a spinning massless particleon an infinite-dimensional Kac-Moody coset space [1–4]. Evidence for such a supergravity/Kac-Moody link emerged through the study à la Belinskii-Khalatnikov-Lifshitz (BKL) [5] of thestructure of cosmological singularities in string theory and supergravity, in spacetime dimensions ≤ D ≤ [6–8]. [For a different approach to such a conjectured supergravity/Kac-Moody linksee [9, 10].] For instance, the well-known BKL oscillatory behavior [5] of the diagonal componentsof a generic, inhomogeneous Einsteinian metric in D = 4 was found to be equivalent to a billiardmotion within the Weyl chamber of the rank-3 hyperbolic Kac-Moody algebra AE [7]. Similarly,the generic BKL-like dynamics of the bosonic sector of maximal supergravity (considered eitherin D = 11 , or, after dimensional reduction, in ≤ D ≤ ) leads to a chaotic billiard motionwithin the Weyl chamber of the rank-10 hyperbolic Kac-Moody algebra E [6]. The hiddenrôle of E in the dynamics of maximal supergravity was confirmed to higher-approximations(up to the third level) in the gradient expansion ∂ x (cid:28) ∂ T of its bosonic sector [1]. In addi-tion, the study of the fermionic sector of supergravity theories has exhibited a related rôle of ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] A p r Kac-Moody algebras. At leading order in the gradient expansion of the gravitino field ψ µ , thedynamics of ψ µ at each spatial point was found to be given by parallel transport with respect toa (bosonic-induced) connection Q taking values within the “compact” sub-algebra of the corre-sponding bosonic Kac-Moody algebra: say K ( AE ) for D = 4 simple supergravity and K ( E ) for maximal supergravity [2–4]. This led to the study of fermionic cosmological billiards [11, 12].[For definitions, and basic mathematical results on Kac-Moody algebras see Ref. [13]; see alsoRef. [14] for a detailed study of the specific hyperbolic Kac-Moody algebra AE ≡ F that enters4-dimensional gravity and supergravity.]The works cited above considered only the terms linear in the gravitino, and, moreover, treated ψ µ as a “classical” (i.e. Grassman-valued) fermionic field. It is only recently [15, 16] that the fullquantum supergravity dynamics of simple cosmological models has been tackled in a way whichdisplayed their hidden Kac-Moody structures. [For previous work on supersymmetric quantumcosmology, see Refs. [17–24], as well as the books [25, 26].]The work [15, 16] studied the quantum supersymmetric Bianchi IX cosmological model. Thismodel is obtained by the (consistent) dimensional reduction of the simple N = 1 , D = 4 super-gravity to one (timelike) dimension on a triaxially-squashed ( SU (2) -homogeneous) three-sphere.This work allowed to decipher the quantum dynamics of this supersymmetric (mini-superspace)model. The quantum state | Ψ( β ) (cid:105) of this model depends (after a symmetry reduction) on threecontinuous bosonic parameters β a , a = 1 , , (measuring the triaxial squashing of the three-sphere), and on sixty-four spinor indices (which describe the representation space of the anti-commutation relations of the gravitino field displayed below). It was shown that the structureof the solutions of the supersymmetry (susy) constraints depended very much on the eigenvalue N F (going from 0 to 6) of the fermion-number operator: (cid:98) N F = 3 + 12 G ab (cid:98) Φ a γ ˆ1ˆ2ˆ3 (cid:98) Φ b (1.1)Here, (cid:98) Φ aA (with a spatial vector index a = 1 , , , and with a Majorana spinor index A = 1 , , , that we generally suppress) denote the twelve, quantized homogeneous modes of the spatialcomponents of the gravitino field ψ µ (written in a special way that makes more manifest someof their Kac-Moody properties). They satisfy the anticommutation relations (cid:98) Φ aA (cid:98) Φ bB + (cid:98) Φ bB (cid:98) Φ aA = G ab δ AB (1.2)where G ab = 12 − − − − − − . (1.3)defines a contravariant, Lorentzian-signature [ ( − , + , +) ] metric in the three-dimensional spacespanned by the bosonic variables β a . [See Ref. [16] for more details on our notation.]The quantum state | Ψ( β ) (cid:105) must be annihilated by the susy constraints, i.e. (cid:98) S (0) A | Ψ( β ) (cid:105) = 0 , (1.4)where the structure of the susy constraints is (cid:98) S (0) A = i (cid:98) Φ aA ∂ β a + (cid:98) V A ( β, (cid:98) Φ) . (1.5)Here the potential-like term (cid:98) V A ( β, (cid:98) Φ) is a complicated operator which is cubic in the gravitinooperators (cid:98) Φ aA , and involves various potential walls that will be discussed below. Figure 1: Sketchy representation of the propagation in 3-dimensional, Lorentzian β space of the cos-mological quantum supergravity wave function | Ψ( β ) (cid:105) . When considered within our canonical chamber β < β < β , this wave function undergoes successive reflections on the three potential walls thatare present in the supersymmetry constraints (1.4). Two of the potential walls are singular on the hy-perplanes α ( β ) = 0 and α ( β ) = 0 , while the third potential wall grows exponentially when α ( β ) becomes negative. As the twelve (cid:98) Φ aA ’s satisfy the Clifford-algebra anticommutation law (1.2), and as the (cid:98) Φ aA ’s enterthe first term of (cid:98) S (0) A , Eq. (1.5), as coefficients of the partial derivatives ∂ β a , we can view, for eachgiven value of the index A , the susy constraint (1.4) as being a Diraclike [ iγ µ ∂ x µ ψ ( x ) = (cid:98) V ( x ) ψ ( x ) ]equation for the propagation of the wavefunction | Ψ( β ) (cid:105) in the 3-dimensional Lorentzian β space.However, as the Majorana-spinor index A in Eqs. (1.4), takes four values, we see that the statemust simultaneously solve four different Diraclike equations. This represents a huge constrainton possible solutions.The structure of the solution space of these susy constraints has been thoroughly analyzed in[16]. It was found that the structure and generality of the solutions drastically depend on thefermionic level N F , Eq. (1.1). Here, we shall study the cosmological dynamics of the solutionsat levels N F = 2 and N F = 3 that contain two arbitrary real functions of two variables as freeCauchy data, i.e. that have as much freedom as the solutions of the usual, purely bosonic BianchiIX mini-superspace Wheeler-DeWitt equation. More precisely, we are interested in quantumsolutions which, in the WKB approximation, can be viewed as describing the chaotic billiardmotion of the cosmological squashing parameters β , β , β near a big-crunch-type singularity.[This chaotic behavior is a quantum, and spinorial, generalization of the classic BKL oscillatorybehavior of the three Bianchi IX scale factors, a = e − β , b = e − β , c == e − β . The quantum(scalar) version of the Bianchi IX chaos was first studied in Ref. [27].] The type of solution wehave in mind, and will study in detail below, is illustrated in Fig. 1.As illustrated on Fig. 1, we can view these solutions as wave packets bouncing betweenpotential walls. In Fig. 1, these potential walls are drawn as sharp walls located on some(timelike) hyperplanes in β -space. [Note, however, that our analysis will not make any sharp-wall approximation, as was made, e.g., in Ref. [12]. We will compute the reflection of the wavefunction against each exact potential wall; see below.] In particular, we highlighted the three There are similar solutions at level N F = 4 , and in the mirror part of the N F = 2 level that we shall notconsider, which can be obtained by a simple involution acting on fermionic generators. wall hyperplanes defined by the equations α ( β ) = 0 , α ( β ) = 0 , α ( β ) = 0 , (1.6)corresponding to the following three linear forms in the β ’s: α ( β ) ≡ β ; α ( β ) ≡ β − β ; α ( β ) ≡ β − β . (1.7)The three hyperplane equations (1.6) constitute a conventional way of describing the fact thatthe basic equations of the supersymmetric Bianchi IX model, i.e. the susy constraints (1.4),contain operatorial, spin-dependent and β -dependent potentiallike terms that grow when the β ’sapproach these hyperplanes. More precisely, as can be seen on the explicit expressions given inEqs. (6.1)–(6.4) of [16], the potential-like contribution (cid:98) V A ( β, (cid:98) Φ) to the susy constraints operators,Eq. (1.5), contains the following terms (cid:98) V gA = 12 (cid:88) a e − β a (cid:16) γ (cid:98) Φ a (cid:17) A , (1.8)and (cid:98) V sym A = −
18 coth[ β − β ] (cid:104) (cid:98) S (cid:16) γ ˆ1ˆ2 (cid:16)(cid:98) Φ − (cid:98) Φ (cid:17)(cid:17) A + (cid:16) γ ˆ1ˆ2 (cid:16)(cid:98) Φ − (cid:98) Φ (cid:17)(cid:17) A (cid:98) S (cid:105) + cyclic (1.9)where (cid:98) S = 12 (cid:18)(cid:98) Φ γ ˆ0ˆ1ˆ2 ( (cid:98) Φ + (cid:98) Φ ) + (cid:98) Φ γ ˆ0ˆ1ˆ2 (cid:98) Φ + (cid:98) Φ γ ˆ0ˆ1ˆ2 (cid:98) Φ − (cid:98) Φ γ ˆ0ˆ1ˆ2 (cid:98) Φ (cid:19) . (1.10)The operator (cid:98) S , together with similarly defined operators (cid:98) S , (cid:98) S , are spin-like operatorssatisfying the usual su (2) commutation relations : [ (cid:98) S , (cid:98) S ] = + i (cid:98) S , etc .The “gravitational-wall" potential term (1.8) is exponentially small when β , β , and β , arelargish and positive. It starts becoming exponentially large (and confining) when, on the contrary,either β , β , or β , become negative. It is in that sense that the three gravitational-wall hyper-planes α ( β ) = 0 , α ( β ) = 0 and α ( β ) = 0 (where α ≡ β and α ≡ β ) define (softlyconfining) potential walls. The “symmetry-wall" potential (1.9) is similarly made of three differ-ent terms (differing by a (123) cyclic permutation). For instance, the term explicitly displayedin Eq. (1.9), which involves coth[ β − β ] , is singular on the symmetry hyperplane α ( β ) = 0 ,and tends towards a β -independent contribution far from it. [The various β -independent con-tributions coming from the asymptotic ± values of the various coth α ab ’s combine with other β -independent, (cid:98) Φ a -cubic terms to define an effective mass term in the above Diraclike equations.The effect of these mass-like, (cid:98) Φ a -cubic terms will be fully taken into account in our discussionbelow.]It has been shown in [16] that it is enough to consider the evolution of the universe wavefunction | Ψ( β ) (cid:105) within only one of the six different chambers defined by considering the twopossible sides associated with the three symmetry-wall one forms α ( β ) , α ( β ) , α ( β ) (i.e.the two possible signs for, e.g., β − β ). Each such chamber corresponds to some ordering ofthe three β ’s. Here, we shall work within the canonical chamber β < β < β . (1.11)The gravitational wall belonging to this chamber [namely the term e − β (cid:16) γ (cid:98) Φ (cid:17) A in (1.8)]further confines the evolution of the wave packet to stay essentially on the positive side of α = 2 β , so that we can think of the wave function | Ψ( β ) (cid:105) as evolving in the (approximate)billiard chamber (cid:46) β < β < β . (1.12)It is this (approximate) billiard chamber, within which α ( β ) ≥ , α ( β ) ≥ , α ( β ) ≥ ,which is represented in Fig. 1.In the present work, we shall complete the results of Refs. [15, 16] by studying the quantum reflection operators of the universe wave function | Ψ( β ) (cid:105) on the three potential walls (1.6) con-straining its propagation (see Fig. 1). Our present study will thereby represent the quantumgeneralization of Ref. [11], which studied similar reflection operators when treating the gravitinoas a classical, i.e. Grassmann variable. We will study, in turn, the evolution of | Ψ( β ) (cid:105) at thefermionic level N F = 2 (Sec. II) and at the fermionic level N F = 3 (Sec. III). After the com-pletion of this purely dynamical problem, we shall show (in Sec. IV) that our results provide anew evidence for the hidden role of the hyperbolic Kac-Moody algebra AE (and of its compactsubalgebra K [ AE ] ) in supergravity. II. QUANTUM FERMIONIC BILLIARD AT LEVEL N F = 2 The susy constraints, Eqs. (1.4), admit solutions depending on arbitrary functions at level N F = 2 only in a 6-dimensional subspace of the total 15-dimensional N F = 2 space, namely(with p, q = 1 , , ): | Ψ (cid:105) ,N F =2 = k pq ( β )˜ b ( p + ˜ b q ) − | (cid:105) − . (2.1)Here, the amplitude k pq ( β ) parametrizing these solutions is symmetric in the two indices p, q =1 , , , and the two triplets of operators ˜ b a ± = ( b a ± ) † denote the Hermitian conjugates of thefollowing combinations of the basic (Hermitian) gravitino operators (cid:98) Φ aA b a + = (cid:98) Φ a + i (cid:98) Φ a ; b a − = (cid:98) Φ a − i (cid:98) Φ a . (2.2)The vacuum state | (cid:105) − is the unique state annihilated by the six fermionic annihilation operators b a ± . The total 15-dimensional N F = 2 space is generated by acting on | (cid:105) − with two among thesix (anticommuting) creation operators ˜ b a ± . The generic propagating state (2.1) lives in the6-dimensional subspace H (1 , S spanned by the symmetrized products ˜ b ( p + ˜ b q ) − | (cid:105) − .In this Section we shall discuss the reflection law of the N F = 2 spinorial solutions (2.1)against the three different potential walls bounding the chamber within which these solutionspropagate. We are interested in an asymptotic regime (large β ’s, and small wavelengths) wherethe quantum solutions can be approximated (away from the turning points, i.e. sufficiently awayfrom the potential walls) by quasi-classical WKB solutions (see Fig. 1). Like in the usual WKBapproximation, we will obtain the reflection laws against the potential walls by matching theWKB form (away from the walls) to (exact) solutions valid near the walls.Far from all the walls (in our canonical Weyl chamber (cid:46) β ≤ β ≤ β ), the effect of the β -dependent potential terms is negligible, so that the amplitude k pq ( β ) of the general WKB-like N F = 2 spinorial solution can be written as superposition of (rescaled) plane waves: k far − wall pq ( β ) = F ( β ) (cid:88) K pq e iπ (cid:48) a β a . (2.3)Here, the rescaling factor F ( β ) is generally defined (see Eq. (8.4) in Ref. [16], here modified bythe numerical factor − / ) as F ( β ) = e β (8 | sinh β sinh β sinh β | ) − / , (2.4)where we introduced the convenient short-hands β ≡ β + β + β , β ≡ β − β , etc . (2.5)Far from all the walls of the canonical chamber, the rescaling factor F ( β ) is a (real) exponentialof the β ’s, namely F ( β ) ≈ e β e − ( | β | + | β | + | β | ) = e β + β + β . (2.6)As explained in Refs. [15, 16], the rescaling factor F ( β ) is such that the mass-shell condition forthe plane wave factor e iπ (cid:48) a β a takes the simple, special-relativistic-like, form π (cid:48) = − µ , namely G ab π (cid:48) a π (cid:48) b = − µ N F =2 = + 38 , (2.7)where G ab denotes the Lorentzian-signature (inverse) metric in β -space. Note that the N F = 2 mass-shell is tachyonic ( µ N F =2 = − , i.e. π (cid:48) a is a spacelike momentum). As was discussed inRef. [16], this tachyonic character (which holds for all fermionic levels, except N F = 3 ) suggeststhe possibility of a cosmological bounce. In the present study, we are, however, focussing onan intermediate asymptotic regime where the wavepacket is centered, most of the time, aroundcoordinates β a that are large compared to 1, so that many wavelengths separate the successivewall reflections.The amplitude K pq (a “tensor" in β -space) of each plane wave in Eq. (2.3) was found in Ref.[16] to have (for a given momentum vector π (cid:48) a ) only one (complex) degree of freedom, containedin an overall factor, say C N F =2 , i.e. to be of the form K pq = C N F =2 (cid:0) π (cid:48) p π (cid:48) q + L kpq π (cid:48) k + m pq (cid:1) , (2.8)where L kpq and m pq are some fixed numerical coefficients (see Eqs. (19.17) and (19.18) in [16],which are reproduced in Appendix A for the reader’s convenience).A first way of describing the law of reflection of a plane wave (2.3) on a potential wall isto compute the transformation between the incident values of the overall amplitude and of themomentum, say C in N F =2 , π (cid:48) in a , and their reflected (or outgoing) values, say C out N F =2 , π (cid:48) out a . Inorder to derive the scattering map C in N F =2 → C out N F =2 , π (cid:48) in a → π (cid:48) out a we need to go beyond thefar-wall approximation, and study the behaviour of a generic wave packet (2.3) near each typeof potential wall.Anticipating on the results of the computations given in the following subsections, let us alreadyexhibit the simple structure of the scattering maps. The transformation of the momentum π (cid:48) a upon reflection on a potential wall associated with a root α ( β ) is simply given (as expected fromthe classical billiard approximation) by specular reflection (with respect to the β -space geometrydefined by the (contravariant) metric G ab ), i.e. by π (cid:48) out a = π (cid:48) in a − π (cid:48) in · αα · α α a . (2.9)Here, the scalar product between two covariant vectors is defined by π (cid:48) · α ≡ G ab π (cid:48) a α b . [ α a isthe covariant normal to the considered potential wall, which is “located" on the hypersurface α ( β ) ≡ α a β a .]As for the transformation of the overall scalar amplitude C N F =2 , it will be found to be encodedin a global phase, δ global α (which will depend on the considered type of potential wall): C out N F =2 = e iδ global α C in N F =2 . (2.10)A second way of describing the law of reflection of a plane wave (2.3) on a potential wall α is tocompute the “reflection operator" R α , acting on the Hilbert space where the considered quantumspinorial state lives, and transforming the incident state | Ψ (cid:105) in into the corresponding reflectedstate | Ψ (cid:105) out . In the present case, the considered Hilbert space is the 6-dimensional subspace H (1 , S of the 15-dimensional N F = 2 level, and the incident state is the ingoing part of (2.1), i.e.a plane-wave state of the type F ( β ) K pq e iπ (cid:48) in a β a ˜ b ( p + ˜ b q ) − | (cid:105) − . The corresponding reflection operatorthen acts on H (1 , S and is such that | Ψ (cid:105) out ,N F =2 = R ,N F =2 α | Ψ (cid:105) in ,N F =2 . (2.11)[When considering the action of R α we strip | Ψ (cid:105) in and | Ψ (cid:105) out of their corresponding phasefactors e iπ (cid:48) in / out a β a .] As the fundamental billiard chamber of the supersymmetric Bianchi IXmodel is bounded by three walls, described by three linear forms in β -space, namely α ( β ) = β − β , α ( β ) = β − β and α ( β ) = 2 β , the quantum supersymmetric Bianchi IX billiardwill define (at each fermionic level where there exists propagating states) three different reflectionoperators. For instance, at the N F = 2 level, supergravity will define three spinorial reflectionoperators R ,N F =2 α , R ,N F =2 α , R ,N F =2 α , (2.12)all acting in the same 6-dimensional space H (1 , S . We shall compute these (dynamically defined)operators below, and find that they have a remarkable Kac-Moody meaning.In order to derive the reflection laws (2.9), (2.10), (2.11), and, in particular, to compute thevalues of the global phases δ global α , and of the reflection operators, (2.12), we will use a “one-wall"approximation, i.e. we shall separately solve the problems where an asymptotically planar wave F ( β ) K pq e iπ (cid:48) a β a impinges on one of the three possible walls of our canonical chamber, (cid:46) β ≤ β ≤ β , i.e. either on one of the two symmetry walls α ( β ) = β − β , or α ( β ) = β − β ; or onthe gravitational wall α ( β ) = 2 β . In this one-wall approximation, the spinorial wavefunction k pq ( β ) in (2.1) will essentially depend only on one variable (measuring the orthogonal distanceto the wall), which will make the problem of exactly solving the complicated supersymmetryconstraints (1.4) tractable. A. Scattering on the symmetry wall α ( β ) = β − β . In this subsection we study (in the one-wall approximation) the reflection of the N F = 2 spinorial state (2.1) on the symmetry wall α ( β ) = β − β . This study is simplified by usingan adapted basis in β -space. In doing so, we shall treat the building blocks entering (2.1) astensors, with the indicated variance, in β -space. Namely, each creation operator ˜ b p ± is consideredas a (contravariant) vector, while the amplitude k pq is viewed as a (symmetric) covariant 2-tensor. Given a basis of 1-forms (i.e. a set of three independent linear forms in β -space), say α (cid:98) ( β ) = α (cid:98) p β p , α (cid:98) ( β ) = α (cid:98) p β p , α (cid:98) ( β ) = α (cid:98) p β p , we shall then work with the corresponding basis(or dual basis) components ˜ b (cid:98) a ± ≡ α (cid:98) ap ˜ b p ± and k (cid:98) a (cid:98) b ≡ α p (cid:98) a α q (cid:98) b k pq , where we defined α p (cid:98) a α (cid:98) bp ≡ δ (cid:98) b (cid:98) a .It is very useful to use a basis of the type (cid:8) α ⊥ , α u , α v (cid:9) ≡ { α ( β ) , u ( β ) , v ( β ) } , (2.13)where α ( β ) is the reflecting wall form we are considering, i.e., in the present subsection α ( β ) ≡ α ⊥ ( β ) ≡ α ( β ) = β − β , (2.14)while u ( β ) , v ( β ) are two one-forms whose corresponding contravariant vectors (with u (cid:93) p ≡ G pq u (cid:93)q ), are parallel to the wall hyperplane α ( β ) = 0 , i.e. α ⊥ ( u (cid:93) ) = 0 = α ⊥ ( v (cid:93) ) . Geometrically,the (contravariant) vector α (cid:93) is perpendicular to the wall hyperplane α ( β ) = 0 , while the nu-merical function β → α ( β ) measures (modulo a factor √ ) the orthogonal distance away fromthe wall hyperplane. [The squared norms of the wall forms we shall consider here are all equalto 2: α · α = G pq α p α q = 2 . This normalization is adapted to the Kac-Moody interpretation ofthe (dominant) wall forms as simple roots of a Kac-Moody Lie algebra.]It was further found to be convenient to align the two basis elements which are parallel to thewall to the two (intrinsically defined) null directions tangent to the wall. [The wall hyperplane isspacelike in Lorentzian β -space, so that it intersects the lightcone G pq β p β q = 0 along two lines.]Specifically, we use u ( β ) ≡ − (2 β + 12 β + 12 β ) , (2.15) v ( β ) ≡ β + β . (2.16)The only nonzero scalar products among the three basis one-forms (cid:8) α ⊥ , α u , α v (cid:9) ≡{ α ( β ) , u ( β ) , v ( β ) } are α · α = 2 and u · v = 2 , so that the only nonzero components of theinverse metric G (cid:98) a (cid:98) b are G ⊥⊥ = G uv = G vu = 2 . (2.17)Equivalently, the dual (vectorial) basis { α ⊥ , α u , α v } = (cid:110) α p ⊥ ∂∂β p , α pu ∂∂β p , α pv ∂∂β p (cid:111) of (cid:8) α ⊥ , α u , α v (cid:9) is equal to { α ⊥ , α u , α v } = (cid:8) α (cid:93) , v (cid:93) , u (cid:93) (cid:9) , and the nonzero basis components of the covariantmetric G (cid:98) a (cid:98) b are G ⊥⊥ = G uv = G vu = 12 . (2.18)When considering the one-wall approximation, the potential terms (cid:98) V A ( β, Φ) entering the susyconstraints (1.5), (1.4), are easily seen to depend on the β ’s only through the single combination α ( β ) . This immediately implies that the two wall-parallel components π u = − i ∂∂u , π v = − i ∂∂v ofthe momentum are conserved. Actually, it is better to consider the parallel components of the shifted momentum operator, i.e. the differentiation operator acting on the rescaled wave function F ( β ) − | Ψ (cid:105) , i.e. π (cid:48) a = π a + i∂ ln F /∂β a . When considered possibly near the wall α , but far fromthe two other walls, the scale factor F ( β ) , (2.4), reads (as a function of α ≡ α , u, v ) F ( α, u, v ) ≈ e − u + v (2 | sinh α | ) − / . (2.19)Hence, the part of ln F ( α, u, v ) that depends on u and v is − u + v , and shifts the conservedparallel momenta according to: π (cid:48) u = π u − i , π (cid:48) v = π v + i . In keeping with the type of wavelikesolutions (bouncing between potential walls) we are interested in, we shall henceforth considerwave packets having real values of the shifted conserved momenta π (cid:48) u , π (cid:48) v (and therefore complexvalues of π u , and π v ).Putting together the ingredients we just discussed (adapted coordinates, adapted basis, con-served shifted parallel momenta), we finally look for solutions of the susy constraints (1.5), inthe one-wall approximation, of the form | Ψ (cid:105) ,N F =2 = e ( iπ (cid:48) u − ) u +( iπ (cid:48) v + ) v |F ( α ) (cid:105) ,N F =2 , (2.20) The triplet of contravariant vectors (cid:8) α (cid:93) , u (cid:93) , v (cid:93) (cid:9) should not be confused with the vectorial basis that is dual tothe basis of one-forms (2.13). As we shall see below the dual basis { α ⊥ , α u , α v } is (cid:8) α (cid:93) , v (cid:93) , u (cid:93) (cid:9) . where |F ( α ) (cid:105) ,N F =2 = K (cid:98) a (cid:98) b ( α )˜ b ( (cid:98) a + ˜ b (cid:98) b ) − | (cid:105) − . (2.21)Inserting this expression in the (one-wall-approximated) susy constraints (1.5), (1.4) leads toconstraints on |F ( α ) (cid:105) ,N F =2 of the form (with π u = π (cid:48) u + i , π v = π (cid:48) v − i ) i (cid:98) Φ ⊥ A ∂ α |F ( α ) (cid:105) + (cid:18) − π u (cid:98) Φ uA − π v (cid:98) Φ vA + (cid:98) V A ( α, Φ) (cid:19) |F ( α ) (cid:105) = 0 . (2.22)We recall that the spinor index A takes four values. For each value of A = 1 , , , , Eq. (2.22) isa Diraclike equation for the quantum spinor state |F ( α ) (cid:105) , with Φ ⊥ A playing the role of a gammamatrix controlling the evolution with respect to α . The anticommutation law (1.2) implies (cid:98) Φ ⊥ A (cid:98) Φ ⊥ B + (cid:98) Φ ⊥ B (cid:98) Φ ⊥ A = δ AB G ⊥⊥ Id = 2 δ AB Id , (2.23)so that we see that each matrix (cid:98) Φ ⊥ A is invertible (with itself as inverse). Multiplying eachone of the four equations (2.22) by i (cid:98) Φ ⊥ A yields an overdetermined system of ordinary (matrix)differential equations in α of the form ∂ α |F ( α ) (cid:105) = (cid:101) Σ A |F ( α ) (cid:105) ( A = 1 , . . . , . (2.24)The unknowns of this system are the six components K (cid:98) a (cid:98) b ( α ) parametrizing the state (2.20),(2.21). Considering the differences between the equations (2.24), we see that the six components K (cid:98) a (cid:98) b ( α ) are subject to the following system of linear equations (cid:16)(cid:101) Σ − (cid:101) Σ A (cid:17) |F(cid:105) = 0 ( A = 2 , , . (2.25)We found that the rank of this linear system is equal to 2. In other words, the six components K (cid:98) a (cid:98) b can be expressed as linear combinations of two of them, chosen for instance as K ⊥⊥ and K ⊥ v . It is then useful to parametrize the α dependence of K ⊥⊥ and K ⊥ v in terms of two otherfunctions F ( α ) , G ( α ) , as follows (we henceforth work on the half-line α > ) K ⊥⊥ ( α ) = C F sinh / ( α ) F ( α ) ,K ⊥ v ( α ) = C G sinh / ( α ) G ( α ) . (2.26)By appropriately choosing the ratio C F /C G between the proportionality constants, we obtain alinear system for the two functions F and G which reads ∂ α F = G , (2.27) [ ∂ α + coth( α )] G = − ( 14 + π (cid:48) ⊥ ) F . (2.28) Eq. (2.24) also leads to six more algebraic constraints, because the operators (cid:101) Σ A map the |F(cid:105) –componentspartially outside the subspace to which they belongs. However these extra conditions are found to be conse-quences of Eqs (2.25); similar dependences also occur when the same analysis is performed at level N F = 3 , aswell as in the other one-gravitational-wall approximations, at levels N F = 2 , or 3 . π (cid:48) ⊥ denotes the function of π (cid:48) u , π (cid:48) v defined by the far-wall N F = 2 mass-shell constraint G (cid:98) a (cid:98) b π (cid:48) (cid:98) a π (cid:48) (cid:98) b = 2 π (cid:48) ⊥ + 4 π (cid:48) u π (cid:48) v = 38 . (2.29)The general solution of the differential system (2.27), (2.28), contains two arbitrary constants,say C P and C Q . The solution parametrized by C Q involves Q -type Legendre functions, andis singular (in a non square-integrable way) on the considered symmetry wall α = 0 . [E.g., K ⊥ v ( α ) ∝ C Q sinh / ( α ) Q ν [cosh( α )] blows up like O [ α − / ] when α → .] In keeping with thegeneral aim of our work, we shall only consider here the solution parametrized by C P whichinvolves P -type Legendre functions, which vanish on the symmetry wall.Here we use Legendre functions defined on the complex plane cut between z = − and z = 1 by analytically continuing the expression P µν [ z ] = 1Γ(1 − µ ) (cid:18) z + 1 z − (cid:19) µ/ F (cid:20) − ν, ν ; 1 − µ ; 1 − z (cid:21) . (2.30) F and G involve Legendre functions of order µ = 0 or µ = 1 , and degree ν = − + i π (cid:48)⊥ . Here,we conventionally define π (cid:48)⊥ as the positive solution of the far-wall mass-shell condition (2.29).More precisely F = C P P − + i π (cid:48)⊥ [cosh( α )] , (2.31) G = C P P − + i π (cid:48)⊥ [cosh( α )] . (2.32)Note that while the definition (2.30) can be used as is when µ = 0 , the case µ = 1 involves a“regularized" hypergeometric function (where the vanishing pre-factor / Γ(1 − µ ) is needed toregularize the singular coefficients / (1 − µ ) entering the hypergeometric series).Finally, the general (square-integrable) solution of the N F = 2 susy constraints is of the form(2.20), (2.21), with adapted-basis components K (cid:98) a (cid:98) b ( α ) given by (with ν = − + i π (cid:48)⊥ ) K ⊥⊥ ( α ) = C P K ⊥⊥ [ π (cid:48) u , π (cid:48) v ] sinh / ( α ) P ν [cosh( α )] , (2.33) K UV ( α ) = C P K UV [ π (cid:48) u , π (cid:48) v ] sinh / ( α ) P ν [cosh( α )] , (2.34) K ⊥ U ( α ) = C P K ⊥ U [ π (cid:48) u , π (cid:48) v ] sinh / ( α ) P ν [cosh( α )] . (2.35)Here, the indices U, V run over the two values u, v parametrizing the parallel components ofthe wave function, and the π (cid:48) U -dependent (but α -independent) polarization tensors K (cid:98) a (cid:98) b ( π (cid:48) U ) aregiven by K ⊥⊥ [ π (cid:48) u , π (cid:48) v ] = ( π (cid:48) u + i π (cid:48) v − i , (2.36) (cid:8) K UV [ π (cid:48) u , π (cid:48) v ] (cid:9) UV = uu,uv,vv = (cid:26) − π (cid:48) v + i π (cid:48) v + i π (cid:48) v − , − π (cid:48) u π (cid:48) v + i π (cid:48) u − π (cid:48) v )) − , − π (cid:48) u + i (cid:18) π (cid:48) v − π (cid:48) u (cid:19) + 1332 (cid:27) , (2.37) (cid:8) K ⊥ U [ π (cid:48) u , π (cid:48) v ] (cid:9) U = u,v = (cid:26) ( i π (cid:48) v + 18 ) , ( i π (cid:48) u −
34 ) (cid:27) . (2.38)We have checked that the values of the various π (cid:48) U -dependent coefficients K (cid:98) a (cid:98) b ( π (cid:48) U ) are in agree-ment with the general, far-wall plane-wave solution (2.8). To perform this check, and to finallyobtain the scattering laws (2.9), (2.10), (2.11), we will need to use the far-wall ( α → + ∞ )asymptotic expression of the Legendre functions, namely: P µν [cosh( α )] ≈ √ π (cid:18) Γ( + ν )Γ(1 − µ + ν ) e ν α + Γ( − − ν )Γ( − µ − ν ) e − ( ν +1) α (cid:19) , α → + ∞ . (2.39)1 B. Reflection laws on the symmetry wall α ( β ) = β − β . Let us now extract from the explicit structure of the one-wall solution (2.33), (2.34), (2.35)the reflection laws (2.9), (2.10), (2.11). In the following, we shall conventionally assume thatthe wavepackets we are considering are “future-directed" in the sense that the (shifted, far-wall)contravariant momentum vector π (cid:48) (cid:93) is directed towards increasing values of the timelike variable β = β + β + β . [Physically, as β = − ln( abc ) , this means that we are considering a contractinguniverse, going towards a Big-Crunch-like singularity where the volume abc → .] With thisconvention, and given the fact that the wavepacket evolves in the half-space α = α > ,the ingoing piece of the asymptotic solution is characterized by having a complex phase factor ∝ e − i π (cid:48)⊥ α , while its reflected piece should have a phase factor ∝ e + i π (cid:48)⊥ α . Here, as above, π (cid:48)⊥ isdefined as being the positive root of the mass-shell condition (2.29).In the case of the α symmetry wall that interest us here, we should insert ν = − + i π (cid:48)⊥ inthe asymptotic expression (2.39). This yields P − + i π (cid:48)⊥ (cid:39) e − α √ π (cid:18) Γ( − i π (cid:48)⊥ )Γ( − i π (cid:48)⊥ ) e − i π (cid:48)⊥ α + Γ(+ i π (cid:48)⊥ )Γ( + i π (cid:48)⊥ ) e + i π (cid:48)⊥ α (cid:19) , (2.40) P − + i π (cid:48)⊥ (cid:39) e − α √ π (cid:18) Γ( − i π (cid:48)⊥ )Γ( − − i π (cid:48)⊥ ) e − i π (cid:48)⊥ α + Γ(+ i π (cid:48)⊥ )Γ( − + i π (cid:48)⊥ ) e + i π (cid:48)⊥ α (cid:19) . (2.41)Let us first note that the combination of the exponentially decaying prefactor e − α with theoverall factor sinh / ( α ) in Eqs. (2.33), (2.34), (2.35), and with the real exponential factorlinked to the imaginary additions to π (cid:48) u , π (cid:48) v in Eq. (2.20), reproduces (in the limit α (cid:29) )the real exponential factor e β + β + β = e − α − u + v present in the general far-wall solution(2.8). Then, the presence of the two complex-conjugated phase factors e ± i π (cid:48)⊥ α (in addition tothe conserved phase factors e i ( π (cid:48) u u + π (cid:48) v v ) ) shows that the reflection law for the shifted momentumreads π (cid:48) in (cid:98) a = ( − π (cid:48)⊥ , π (cid:48) u , π (cid:48) v ) → π (cid:48) out (cid:98) a = (+ π (cid:48)⊥ , π (cid:48) u , π (cid:48) v ) . (2.42)The rewriting of this adapted-basis reflection law, precisely yields the specular reflection law(2.9).In order to extract the global reflection phase-factor e iδ global α , Eq. (2.10), connecting the incidentfar-wall amplitude to the reflected one, one needs to compare both the incident and the reflectedpieces of the solution Eqs. (2.33), (2.34), (2.35) to the generic far-wall solution (2.8). Whendoing so, one can first factor out the amplitude of, say, the incident P − + i π (cid:48)⊥ -type modes (in K ⊥⊥ and K UV ). This yields a π (cid:48)⊥ -dependent factor in the corresponding incident P − + i π (cid:48)⊥ -typemodes (in K ⊥ U ) given by Γ( − i π (cid:48)⊥ )Γ( − − i π (cid:48)⊥ ) = − − i π (cid:48)⊥ , (2.43)where we used the basic identity Γ( z + 1) = z Γ( z ) . Combining this additional π (cid:48)⊥ -linear factor(2.43) in K ⊥ U with the π (cid:48) U -linear factors displayed in Eq. (2.38), we found that all the K (cid:98) a (cid:98) b incident amplitudes of Eqs. (2.33), (2.34), (2.35) nicely agree with the π (cid:48) a -quadratic dependenceof the generic far-wall amplitude (2.8) derived in our previous work [16], and recalled in AppendixA. [The same check holds for the reflected amplitude.]As additional result of this asymptotic analysis, one gets the relation between the overallscalar amplitude C N F =2 of a far-wall wave packet and the overall coefficient C P parametrizing2the amplitude of the P -type solution, namely C ± N F =2 = − C P √ π Γ( ± i π (cid:48)⊥ )Γ( ± i π (cid:48)⊥ ) . (2.44)where the upper sign on C N F =2 refers to the outgoing wave (having α · π (cid:48) > ), while the lowersign refers to the ingoing wave. Taking the ratio between C + N F =2 and C − N F =2 yields the globalphase factor e iδ global α = C + N F =2 C − N F =2 = Γ[+ iπ (cid:48)⊥ ] Γ[ − i π (cid:48)⊥ ]Γ[ − iπ (cid:48)⊥ ] Γ[ + i π (cid:48)⊥ ] . (2.45)In the small wavelength (WKB) limit (large values for the components π (cid:48) a ), this yields, using Γ( z + a )Γ( z + b ) ≈ z a − b as z → ∞ , (2.46) e iδ global , WKB α ≈ e − i π . (2.47)The latter asymptotic value of the global phase is also easily obtained by considering the K ⊥⊥ component of the N F = 2 solution (which is given, for large values of π (cid:48) a , by K ⊥⊥ ≈ C N F =2 π (cid:48)⊥ π (cid:48)⊥ where the factor π (cid:48)⊥ does not change sign upon reflection).Finally, let us extract from our results above the reflection operator (in Hilbert space) R ,N F =2 α mapping the incident state | Ψ (cid:105) in ,N F =2 to the reflected one. We can compute this operator byrelating the various basis spinor states ˜ b ( (cid:98) a + ˜ b (cid:98) b ) − | (cid:105) − making up the one-wall solution (2.21) to eigen-states of various operators defined in terms of the basic gravitino operators (cid:98) Φ aA . Let us recall thatour previous work had emphasized that the building blocks of the susy Hamiltonian operatorwere some operators quadratic in the (cid:98) Φ aA ’s that generated a representation of the compact subal-gebra K [ AE ] of AE . There were two types of such operators: the three spin operators (cid:98) S , (cid:98) S , (cid:98) S , associated with symmetry walls, and three operators (cid:98) J , (cid:98) J , (cid:98) J , associated with the threedominant gravitational walls α = 2 β , α = 2 β , α = 2 β . [See Eq. (8.10) in Ref. [16].]Here, we are considering the reflection by the symmetry wall α , so that one might expect thatthe corresponding reflection operator R ,N F =2 α might be directly related to the correspondingspin operator (cid:98) S . There is, however, a subtlety. Indeed, while the considered dynamical states | Ψ (cid:105) ,N F =2 live in a 6-dimensional subspace H (1 , S of the 15-dimensional N F = 2 level (so that R ,N F =2 α is an endomorphism of H (1 , S ), the spin operator (cid:98) S happens not to leave invariant H (1 , S , but to map it to other sectors within the 15-dimensional N F = 2 state space. How-ever, if one considers, instead of (cid:98) S , its square, namely (cid:98) S , one checks that the latter operatorleaves invariant (and thereby defines an endomorphism of) H (1 , S . [We recall that it is indeedthe squared spin operator which enters each symmetry wall α ab in the Hamiltonian operator,as per ∼ ( (cid:98) S ab − Id ) / (4 sinh ( α ab )) .] In addition, we have shown that the basis of spinor states ˜ b ( (cid:98) a + ˜ b (cid:98) b ) − | (cid:105) − entering (2.21), and which were crucial for finding and simplifying the solution of thesusy constraints happen to be eigenstates of (cid:98) S . More precisely, we have shown that four of ourbasis states are eigenstates of (cid:98) S with zero eigenvalues, (cid:98) S ˜ b ⊥ + ˜ b ⊥− | (cid:105) − = 0 ; (cid:98) S ˜ b ( U + ˜ b V ) − | (cid:105) − = 0 , (2.48)while the other two basis states are eigenstates of (cid:98) S with eigenvalue equal to 4: (cid:98) S ˜ b ( ⊥ + ˜ b U ) − | (cid:105) − = 4 ˜ b ( ⊥ + ˜ b U ) − | (cid:105) − . (2.49)3We then note that these eigenvalues of (cid:98) S are correlated to the Legendre order µ of the corre-sponding wavefunction K (cid:98) a (cid:98) b ( α ) by the simple rule µ = 12 | (cid:98) S | ,N F =2 , (2.50)where we introduced the operator | (cid:98) S | ,N F =2 defined as being the (unique) positive square root of (cid:98) S , considered as an endomorphism of H (1 , S .When comparing the phases of the incident and reflected pieces in the one-wall solution above,one easily sees that they only differ by a phase factor, and that the latter phase factor, say e iδ µ only depends on the value of the Legendre order µ , and can be written as e i δ µ = Γ( − µ − i π (cid:48)⊥ ) Γ( i π (cid:48)⊥ )Γ( − µ + i π (cid:48)⊥ ) Γ( − i π (cid:48)⊥ ) . (2.51)In view of the strict correlation (2.50), we conclude that the reflection operator R ,N F =2 α is anoperatorial function of | (cid:98) S | ,N F =2 , which is given by R ,N F =2 α = Γ[+ iπ (cid:48)⊥ ] Γ[ − i π (cid:48)⊥ − | (cid:98) S | ,N F =2 ]Γ[ − iπ (cid:48)⊥ ] Γ[ + i π (cid:48)⊥ − | (cid:98) S | ,N F =2 ] . (2.52)In the small wavelength (or WKB) limit ( π (cid:48)⊥ (cid:29) ), we have (cid:2) e i δ µ (cid:3) WKB = e iπ ( µ − ) , (2.53)so that the reflection operator depends only on the spin operator, namely R ,N F =2 ,W KBα = e + iπ ( | (cid:98) S | ,NF =2 − = e iδ global α e + iπ | (cid:98) S | ,NF =2 . (2.54)In the second form, we have factored out the (WKB limit of the) global phase factor (2.47)(which corresponds to the (cid:98) S = 0 eigenvalues). Note that this result can also be written as R ,N F =2 ,W KBα = e − iπ e − iπ | (cid:98) S | ,NF =2 , (2.55)because the eigenvalues of | (cid:98) S | ,N F =2 are 0 and 2. C. Scattering and reflection laws on the symmetry wall α ( β ) = β − β . We shall be briefer in our discussion of the scattering of a N F = 2 wave packet (2.1) on theother symmetry wall of our canonical chamber, i.e. the wall form α ( β ) = β − β . Thoughthere are some differences in intermediate expressions (because of the dissymetric role of the twosymmetry walls bounding one given billiard chamber) the final results are obtained by applyingthe cyclic permutation (231) → (123) to the previous final results concerning the scattering onthe α ( β ) = β − β wall. By definition, we require this square root to have the same eigenstates as (cid:98) S ,N F =2 . (cid:8) ˜ α ⊥ , ˜ α u , ˜ α v (cid:9) ≡ { ˜ α ( β ) , ˜ u ( β ) , ˜ v ( β ) } , (2.56)with ˜ α ( β ) = β − β , ˜ u ( β ) ≡ − (2 β + 12 β + 12 β ) , ˜ v ( β ) ≡ ( β + β ) . (2.57)The metric components in this adapted basis are the same as the previous ones, Eq. (2.17), sothat the far-wall mass-shell condition reads as before, namely G (cid:98) a (cid:98) b π (cid:48) (cid:98) a π (cid:48) (cid:98) b = 2 π (cid:48) ⊥ + 4 π (cid:48) ˜ u π (cid:48) ˜ v = 38 . (2.58)(2.59)The state reflecting on the α wall is looked for in the form | Ψ (cid:105) ,N F =2 = e ( iπ (cid:48) ˜ u − )˜ u +( iπ (cid:48) ˜ v + )˜ v | ˜ F ( α ) (cid:105) ,N F =2 , (2.60)where the real-exponential contributions are slightly modified (because of the non cylic invarianceof the original scale factor F ( β ) , Eq. (2.4)), and where | ˜ F ( α ) (cid:105) ,N F =2 = K (cid:98) a (cid:98) b ( α )˜ b ( (cid:98) a + ˜ b (cid:98) b ) − | (cid:105) − . (2.61)Here it is now understood that the basis indices (cid:98) a = ( ⊥ , u, v ) must be replaced by their tildedavatars, corresponding to the new basis (2.56), (2.57).As above, we find Q µν -type and P µν -type Legendre solutions, with the order µ related to (cid:98) S via µ = 12 | (cid:98) S | ,N F =2 , (2.62)so that µ = 0 or . The degree ν is again given by ν = − + i π (cid:48)⊥ . The Q -type solutions aresingular and we discard them. On the other hand, the P -type solutions are regular and areexpressed by formulas similar to Eqs. (2.33), (2.34), (2.35), when using projections on the tildedbasis (2.56).The final reflection laws are the same, mutatis mutandis , as before. Namely, the standardspecular reflection law (2.9) (on the new wall α ), and (defining as before π (cid:48)⊥ as the positiveroot of the mass-shell condition (2.58)) e iδ global α = Γ[+ iπ (cid:48)⊥ ] Γ[ − i π (cid:48)⊥ ]Γ[ − iπ (cid:48)⊥ ] Γ[ + i π (cid:48)⊥ ] ≈ e − iπ (2.63)(where the last approximation corresponds to the WKB limit) and the (23) → (12) version ofthe reflection operator (2.52), which yields, in the WKB limit R ,N F =2 ,W KBα = e − iπ e ± iπ | (cid:98) S | ,NF =2 . (2.64)As before, we can indifferently choose here the ± sign because the eigenvalues of | (cid:98) S | ,N F =2 are,as before, and .5 D. Scattering on the gravitational wall α ( β ) = 2 β . We shall also be brief in discussing the scattering of a N F = 2 wave packet (2.1) on a gravita-tional wall. Gravitational walls correspond to terms in the Hamiltonian that are proportional to e − α = e − β , e − α = e − β , or e − α = e − β . The main differences between a gravitationalwall and a symmetry wall are that: (i) a gravitational wall is softer than a symmetry wall inthat it does not become singular on the corresponding wall hyperplane α aa = 0 ; and (ii) theoperator (cid:98) J coupled (in the Hamiltonian) to the wall factor e − α is quadratic in the gravitinooperators (cid:98) Φ aA , while we had quartic-in-fermions operators, such as (cid:98) S , for symmetry walls (seeEq. (8.11) in Ref. [16]). Similarly to the (sharper) symmetry wall case, we shall impose theboundary condition that the wave function exponentially decreases as one penetrates within theconsidered gravitational wall (i.e. when, say, α ( β ) = 2 β becomes negative).As in the symmetry-wall case, we shall solve the susy constraints in the one-wall approximation.It is again very useful to introduce an adapted basis of one-forms, namely g α (cid:98) a = ( g α ⊥ , g α u , g α v ) with g α ⊥ ( β ) = 2 β ; g α u ( β ) = g u = β + β ; g α v ( β ) = g v = β + β (2.65)[Below, we simplify the notation by deleting the pre-subscript g .] Again, we have chosen a direc-tion normal to the considered wall, and two null directions parallel to the wall. The normalizationof this co-frame is now slightly different from before, with G ⊥⊥ = 2 ; G uv = G vu = − , (2.66)so that the far-wall mass-shell condition reads π (cid:48) ⊥ − π (cid:48) u π (cid:48) v = 38 . (2.67)In the following, we shall define π (cid:48)⊥ as the positive root of the latter mass-shell condition, i.e. π (cid:48)⊥ = (cid:113) π (cid:48) u π (cid:48) v − µ , where µ = − is the squared mass at level N F = 2 . The dual (vectorial)basis is equal to { α ⊥ , α u , α v } = (cid:8) α (cid:93) , − v (cid:93) , − u (cid:93) (cid:9) .Let us introduce the shorthand notation (here generalized, in anticipation of the corresponding N F = 3 discussion, to a mass-shell condition involving a different squared mass µ ) U µ ( β ; j ) ≡ e ( β + β ) e iπ (cid:48) u ( β + β )+ iπ (cid:48) v ( β + β ) e β W − j, i π (cid:48)⊥ [ e − β ] , (2.68)where W κ,µ ( z ) denotes the standard Whittaker function.The solution of the susy constraints near a gravitational wall (and decaying under the wall) isthen of the usual form (see (2.1)) | Ψ (cid:105) ,N F =2 = k (cid:98) a (cid:98) b ( β )˜ b ( (cid:98) a + ˜ b (cid:98) b ) − | (cid:105) − , (2.69)where the frame indices now refer to the gravitational basis (2.65), and where the componentsof the state are given by6 k uu ( β ) = C J (cid:18)
54 + i ( π (cid:48) u + 2 π (cid:48) v ) − π (cid:48) v (cid:19) U − [ β, −
32 ] , (2.70) k uv ( β ) = C J
14 (1 + 2 i π (cid:48) u )(1 + 2 i π (cid:48) v ) U − [ β, −
32 ] , (2.71) k vv ( β ) = C J
14 (1 + 2 i π (cid:48) u ) U − [ β, −
32 ] , (2.72) k ⊥ u ( β ) = − C J
14 (3 + 2 i π (cid:48) v )(1 + 4 π (cid:48) u π (cid:48) v ) U − [ β,
12 ] , (2.73) k ⊥ v ( β ) = − C J
14 (1 + 2 i π (cid:48) u )(1 + 4 π (cid:48) u π (cid:48) v ) U − [ β,
12 ] , (2.74) k ⊥⊥ ( β ) = − C J (1 + 4 π (cid:48) u π (cid:48) v ) (cid:18) U − [ β, −
32 ] − (cid:18) e − β + 14 (cid:19) U − [ β,
12 ] (cid:19) . (2.75)The last component can be rewritten as k ⊥⊥ ( β ) = C J
18 (1 + 4 π (cid:48) u π (cid:48) v )(3 + 4 π (cid:48) u π (cid:48) v ) U − [ β,
52 ] . (2.76)The latter form displays the role of the value j = in the second argument of the function U µ ( β ; j ) describing the behavior of the basis state ˜ b ⊥ + ˜ b ⊥− | (cid:105) − (in correspondence with the factthat the latter state is an eigenstate of the operator (cid:98) J with the eigenvalue j = ).The behaviour of the Whittaker function near the origin e − β → yields the far-wall limit ofthe wave function: U µ [ β, j ] ∼ β →∞ e ( β + β + β ) e i ( π (cid:48) v ( β + β )+ π (cid:48) u ( β + β ) ) × Γ [ − i π (cid:48)⊥ ]Γ (cid:104) (1+ j )2 − i π (cid:48)⊥ (cid:105) e − i π (cid:48)⊥ β + Γ [ i π (cid:48)⊥ ]Γ (cid:104) (1+ j )2 + i π (cid:48)⊥ (cid:105) e i π (cid:48)⊥ β . (2.77)We have checked that the π (cid:48)⊥ -dependence of the successive ratios between the incident andreflected amplitudes exhibited in (2.77), which follow from the Euler-gamma function identity Γ (cid:20) (1 + j + 2)2 − i π (cid:48)⊥ (cid:21) = (cid:18) (1 + j )2 − i π (cid:48)⊥ (cid:19) Γ (cid:20) (1 + j )2 − i π (cid:48)⊥ (cid:21) , (2.78)agree with the general far-wall solution (2.8), obtained in Ref. [16].From Eq. (2.77), we also immediately get the phase shifts, for each component of the wavefunction, between the incident ( ∝ e − i π (cid:48)⊥ β ) and reflected ( ∝ e + i π (cid:48)⊥ β ) amplitudes, uponscattering on the α = 2 β gravitational wall: e iδ α ( j ; π (cid:48)⊥ ) = Γ (cid:2) j − i π (cid:48)⊥ (cid:3) Γ [ i π (cid:48)⊥ ]Γ (cid:2) j + i π (cid:48)⊥ (cid:3) Γ [ − i π (cid:48)⊥ ] . (2.79)In the cases of the reflections upon symmetry walls discussed above, the global phase fac-tor, entering Eq. (2.10), could be read off from the reflection behavior of the perpendicular-perpendicular amplitude, k ⊥⊥ ( β ) . The reason for this fact was that, in those cases, theperpendicular-perpendicular projection of the farwall amplitude (2.8), i.e. the quantity K ⊥⊥ ( π (cid:48) a ) ,happened to be independent of the sign of the (corresponding) perpendicular component π (cid:48)⊥ of7 π (cid:48) a (which is the only adapted-basis component of π (cid:48) a which changes upon reflection). [The firstcontribution ∝ π (cid:48) ⊥ in K ⊥⊥ ( π (cid:48) a ) is always invariant under the sign flip π (cid:48)⊥ → − π (cid:48)⊥ , but we aretalking also here about the second contribution ∝ L k ⊥⊥ π (cid:48) k , which is linear in π (cid:48)⊥ and could, apriori, change under reflection. It does not in the case of symmetry-wall reflections because L ⊥⊥⊥ happens to vanish.] By contrast, in the case of reflection upon the gravitational wall α , wefound that the corresponding coefficient L ⊥⊥⊥ does not vanish, so that the (linear in π (cid:48) a ) contribu-tion ∝ L k ⊥⊥ π (cid:48) k changes upon reflection. On the other hand, we found that all the parallel-parallelcoefficients L ⊥ UV measuring the dependence on ± π (cid:48)⊥ vanish in the case of the gravitational wall α . As a consequence, in that case, the global phase can be read off from the reflection behaviorof the parallel-parallel amplitudes, k UV ( β ) . The latter amplitudes correspond to the eigenvalue j = − , so that e iδ global α = Γ (cid:2) − − i π (cid:48)⊥ (cid:3) Γ [ i π (cid:48)⊥ ]Γ (cid:2) − + i π (cid:48)⊥ (cid:3) Γ [ − i π (cid:48)⊥ ] . (2.80)Contrary to the symmetry-wall cases, the dependence of the global reflection phase δ global α on π (cid:48)⊥ does not admit a limit when π (cid:48)⊥ → + ∞ , rather one has δ global α ∼ π (cid:48)⊥ ln π (cid:48)⊥ . This divergence is eas-ily understood classically: because of the energy conservation law π (cid:48)⊥ + k e − β = π (cid:48)⊥ − wall , arelativistic particle impinging on a gravitational wall with incident normal momentum π (cid:48)⊥ far − wall will penetrate within the wall up to the turning point π (cid:48)⊥ = 0 , i.e. up to the energy-dependent position β = − ln( π (cid:48)⊥ − wall /k ) . The shift β in the effective locationof the wall then leads to an additional (energy-dependent) phase shift ∼ − π (cid:48)⊥ far − wall β ∼ π (cid:48)⊥ far − wall ln π (cid:48)⊥ far − wall .Of more importance for our purpose is the dependence of the phase factors (2.79) on the secondargument j of the mode function U µ ( β ; j ) , Eq. (2.68). Indeed, we have shown that our adaptedbasis was such that each corresponding spinor state ˜ b ( (cid:98) a + ˜ b (cid:98) b ) − | (cid:105) − was an eigenspinor of the operator (cid:98) J associated with the α gravitational wall. More precisely, the perpendicular-perpendicular, ⊥⊥ , state has eigenvalue j = , the two perpendicular-parallel states, ⊥ u , ⊥ v have eigenvalues j = , and the three parallel-parallel states uu, uv, vv have eigenvalues − . Note that thesevalues are precisely the j -values entering the corresponding mode functions (2.68). We cantherefore re-express the result above by saying that the reflection operator, in Hilbert space,against the α gravitational wall is given by the following operatorial expression R ,N F =2 α = Γ (cid:104) (cid:98) J − i π (cid:48)⊥ (cid:105) Γ [ i π (cid:48)⊥ ]Γ (cid:104) (cid:98) J + i π (cid:48)⊥ (cid:105) Γ [ − i π (cid:48)⊥ ] . (2.81)In the large momentum (WKB) limit, this yields R ,N F =2 α ≈ e iδ ( π (cid:48)⊥ ) e − i π e − i π (cid:98) J , (2.82)where we defined δ ( π (cid:48)⊥ ) ≡ π (cid:48)⊥ ln(4 π (cid:48)⊥ /e ) . III. QUANTUM FERMIONIC BILLIARD AT LEVEL N F = 3 The analysis done in the previous section of the various reflection laws at the fermionic level N F = 2 can also be performed at the fermionic level N F = 3 . This level corresponds to a 20-dimensional subspace of the total spinorial state space. Actually, there is a natural decomposition8of the N F = 3 space into two 10-dimensional subspaces. The latter two subspaces are mappedonto each other via the involution b a ± → b a ∓ , ˜ b a ± → ˜ b a ∓ , between the basic fermionic annihilationand creation operators. Here, we shall work in only one of these equivalent 10-dimensionalsubspaces.As found in our previous work the general structure of the propagating solution of the susyconstraints can then be written as | Ψ (cid:105) ,N F =3 = f ( β ) | η (cid:105) + h pq ( β ) | B pq (cid:105) , (3.1)where | η (cid:105) = 13! η klm (cid:101) b k − (cid:101) b l − (cid:101) b m − | (cid:105) − ; | B pq (cid:105) = 12 (cid:88) k,l η pkl (cid:101) b k + (cid:101) b l + (cid:101) b q − | (cid:105) − . (3.2)Here, η klm = √− G(cid:15) klm denotes the Levi-Civita tensor in β -space, and the first index on η pkl ismoved by the Lorentzian metric G pq in β -space. The general solution (3.1) is parametrized bythe (pseudo-)scalar f and the (dualized) tensor h pq . The latter tensor is not symmetric in itstwo indices and has, in general, nine independent components. With the additional degree offreedom described by the scalar f , this means that the general N F = 3 solution a priori containsten independent components (as befits its belonging to a 10-dimensional subspace of the N F = 3 level).It was found in Ref. [16] that, far from all the walls, the general propagating solution at level N F = 3 simplifies because several irreducible components among the ten generic ones eithervanish or become related to each other. Specifically, in our canonical chamber both the scalar f , and the antisymmetric part of the tensor h pq , vanish far from the walls. In addition, theremaining components, namely the six components of the symmetric part h ( pq ) of h pq , can all bepolynomially expressed in terms of the shifted momenta π (cid:48) a according to a formula of the sametype as for the N F = 2 solution, i.e. h far − wall( pq ) = C N F =3 e iπ (cid:48) a β a e (cid:36) a β a (cid:0) π (cid:48) p π (cid:48) q + L kpq π (cid:48) k + m pq (cid:1) , (3.3)where L kpq and m pq are some fixed numerical coefficients (see Eqs. (19.32) and (19.33) in [16],which are reproduced in Appendix A for the reader’s convenience). [The coefficients L kpq and m pq describing the N F = 3 far-wall solutions are different from their N F = 2 analogs.] Here theshifted (far-wall) momenta π (cid:48) a satisfy the (non-tachyonic) mass-shell condition G ab π (cid:48) a π (cid:48) b = − µ N F =3 = − . (3.4)The real exponential factor e (cid:36) a β a = e β + β + β is the far-wall asymptotic form of the rescalingfactor F ( β ) , Eq. (2.4).We wish to generalize to the N F = 3 level the reflection laws (2.9), (2.10), (2.11), discussedabove for the N F = 2 level. As before, these reflection laws will be obtained by matchingthe general far-wall solution (3.3) to three separate approximate susy solutions, obtained byconsidering, in turn, the various one-wall cases where the solution propagates near each one of thethree walls of our canonical chamber, i.e., the symmetric walls α or α , and the gravitationalwall α . We will again find that the first reflection law (2.9) is always satisfied, and we willcompute the values of the other scattering data, namely the N F = 3 global phase factor C out N F =3 = e iδ global α C in N F =3 , (3.5)and the reflection operator acting in the considered 10-dimensional subspace of the N F = 3 level,such that | Ψ (cid:105) out ,N F =3 = R ,N F =3 α | Ψ (cid:105) in ,N F =3 . (3.6)9As in the N F = 2 case, we found that it is very useful to use (for each wall) the same wall-adapted basis as above to be able both to solve the corresponding one-wall susy constraints, andto compute the scattering data. When working in some basis of one forms, say α (cid:98) a ( β ) = α (cid:98) ap β p ,we shall write the general solution (3.1) in the form | Ψ (cid:105) ,N F =3 = f ( β ) | η (cid:105) + h (cid:98) a (cid:98) b ( β ) | α (cid:98) a α (cid:98) b (cid:105) , (3.7)where h (cid:98) a (cid:98) b ≡ α p (cid:98) a α q (cid:98) b h pq are the components of the tensor h pq in the dual basis ( α p (cid:98) a α (cid:98) aq = δ pq or,equivalently α p (cid:98) a α (cid:98) bp = δ (cid:98) b (cid:98) a ), and where we introduced the short-hand notation | α (cid:98) a α (cid:98) b (cid:105) ≡ α (cid:98) ap α (cid:98) bq | B pq (cid:105) . (3.8)[One must also tensorially transform the Levi-Civita tensor, and the metric.]In the following subsections, we shall briefly summarize the main results of our analysis at the N F = 3 level. A. N F = 3 reflection on the symmetry wall α ( β ) = β − β . We use the same basis of one forms as above, namely (2.13), with (2.14) and (2.15). Onedecomposes the wavefunctions f ( β ) and h (cid:98) a (cid:98) b ( β ) entering the general N F = 3 solution (3.7) in theproducts of the factor e ( iπ (cid:48) u − ) u +( iπ (cid:48) v + ) v , (3.9)and of functions of α = α ( β ) . Here, the conserved shifted momenta π (cid:48) u , π (cid:48) v (which measure themomentum parallel to the wall plane) must satisfy (when receding far from the wall) the N F = 3 mass-shell condition (3.4) which explicitly yields π (cid:48) ⊥ + 4 π (cid:48) u π (cid:48) v = − µ N F =3 = − . (3.10)One then writes down the N F = 3 analogs of equations (2.22), (2.24) and (2.25) (written interms of adapted-basis objects). The rank of the latter linear system is again found to be equal to2. This means that the ten components of f, h (cid:98) a (cid:98) b can be expressed as linear combinations of onlytwo of them. One also finds that the three antisymmetric components of h (cid:98) a (cid:98) b must separatelyvanish. We could then express the seven remaining components, i.e. f, h ( (cid:98) a (cid:98) b ) , in terms of twofunctions of α = α ( β ) = β − β , say F ( α ) and G ( α ) . For instance, h ⊥⊥ ∝ sinh ( α ) G ( α ) , (3.11) h ( ⊥ u ) ∝ sinh ( α ) F ( α ) . (3.12)The F and G have to satisfy the differential system ∂ α F + 14 coth( α ) F + G = 0 , (3.13) ∂ α G + 34 coth( α ) G − (cid:18)
116 + π (cid:48) ⊥ (cid:19) F = 0 , (3.14)from which follows ∂ α F + coth( α ) ∂ α F + (cid:18)
14 + π (cid:48) ⊥ −
116 sinh ( α ) (cid:19) F = 0 . (3.15)0The general solution of this system is F = c + P + − + i π (cid:48)⊥ [cosh( α )] + c − P − − + i π (cid:48)⊥ [cosh( α )] , (3.16) G = c + (cid:18)
116 + π (cid:48) ⊥ (cid:19) P − − + i π (cid:48)⊥ [cosh( α )] − c − P + − + i π (cid:48)⊥ [cosh( α )] . (3.17)The solution for f and h ( (cid:98) a (cid:98) b ) also involves the combination G + coth( α ) F which can be shownto be equal to G + 12 coth( α ) F = c + P + − + i π (cid:48)⊥ [cosh( α )] − c − (cid:18)
916 + π (cid:48) ⊥ (cid:19) P − − + i π (cid:48)⊥ [cosh( α )] . (3.18)As we see, there is a two-parameter family of solutions: (i) the c + family involving P + − + i π (cid:48)⊥ , P + − + i π (cid:48)⊥ and P − − + i π (cid:48)⊥ ; and (ii) the c − family involving P − − + i π (cid:48)⊥ , P − − + i π (cid:48)⊥ and P + − + i π (cid:48)⊥ . Nearthe wall ( α → ), the Legendre functions behave like (we recall that α > ) : P µν (cosh α ) ∼ − µ ) (cid:16) α (cid:17) − µ . (3.19)Though the above Legendre functions enter the solution after being multiplied by sinh ( α ) , the c + family of solutions will be singular at α = 0 in a non square integrable way. We thereforeexclude it, and retain only the c − family of solutions. [This family is mildly singular at α = 0 because of the presence of sinh ( α ) P + − + i π (cid:48)⊥ [cosh( α )] . But the latter mode is square integrable.]Finally, defining (for µ = − , − , + ) the mode functions h µ ( β ) ≡ e ( iπ (cid:48) u − ) u +( iπ (cid:48) v + ) v sinh ( α ) P µ − + i π (cid:48)⊥ [cosh( α )] , (3.20)we have been able to write the only regular solution of the susy constraints near the α symmetrywall as a sum of the type | Ψ (cid:105) ,N F =3 = C (cid:88) µ = − , − , + N iµ ( π (cid:48) u , π (cid:48) v ) h µ ( β ) | µ, i (cid:105) . (3.21)Here, i is a degeneracy index, which labels, for each value of µ various states associated with thesame value of the order µ of the corresponding Legendre mode h µ ( β ) .Parallely to the N F = 2 analysis above, there is again a direct link between the various modestates | µ, i (cid:105) and the spinorial operator (cid:98) S . Namely, the states | µ, i (cid:105) span, for each value of µ (when the degeneracy index i varies), the eigenspace of (cid:104) (cid:98) S (cid:105) ,N F =3 with eigenvalue (2 µ ) .More precisely, we have (cid:104) (cid:98) S (cid:105) ,N F =3 | µ, i (cid:105) = (2 µ ) | µ, i (cid:105) for i = 1 , . . . g ( µ ) , (3.22)where the various degeneracies (which sum, as needed, to ten) are g (cid:18) − (cid:19) = 1 ; g (cid:18) − (cid:19) = 5 ; g (cid:18) + 34 (cid:19) = 4 . (3.23)1Let us briefly indicate the structure of the various eigenstates, and how they are intimately linkedto the basis adapted to the considered wall α ⊥ = α | − (cid:105) = | u v (cid:105) + | v u (cid:105) − | ⊥⊥(cid:105) − | η (cid:105) , (3.24) | − , i (cid:105) i =1 ,..., = | ⊥ u (cid:105) , | u ⊥(cid:105) , | ⊥ v (cid:105) , | v ⊥(cid:105) , | η (cid:105) + | G (cid:105) , (3.25) | + 34 , i (cid:105) i =1 ,..., = | ⊥⊥(cid:105) − | η (cid:105) , | uu (cid:105) , | vv (cid:105) , | uv (cid:105) − | vu (cid:105) , (3.26)where we used the notation (3.8), together with the following shorthand for the trace state | G (cid:105) ≡ G pq | B pq (cid:105) = G (cid:98) a (cid:98) b | α (cid:98) a α (cid:98) b (cid:105) = ( | ⊥⊥(cid:105) + | uv (cid:105) + | vu (cid:105) ) .The possibility of expressing the solution of the susy constraints near a symmetry wall α S asa combination of modes of the type (3.20) (involving Legendre functions P µν (cosh( α )) ) can bedirectly seen when considering the second-order equation (Hamiltonian constraint) which mustbe satisfied as a consequence of the first-order susy constraints. Indeed, the near-wall form ofthe Hamiltonian constraint reads (with ˆ π a = − i∂/∂β a , and | Ψ (cid:48) ( β ) (cid:105) = F ( β ) − | Ψ( β ) (cid:105) ) (cid:32) G ab ˆ π a ˆ π b + µ N F + 12 (cid:98) S α S − Id sinh ( α S ) (cid:33) | Ψ (cid:48) ( β ) (cid:105) = 0 . (3.27)Decomposing the solution of this near-symmetry-wall second-order equation in eigenspinors ofthe squared spin operator (cid:98) S α S , one finds that the general solution pertaining to an eigenvalue S of (cid:98) S α S is expressible in terms of the Legendre modes (3.20) for µ = ± | S | . (3.28)As the eigenvalues S (with multiplicities) of the squared spin operators at level N F = 3 are(for all symmetry walls) (cid:16)(cid:0) (cid:1) (cid:12)(cid:12) , (cid:0) (cid:1) (cid:12)(cid:12) , (cid:0) (cid:1) (cid:12)(cid:12) (cid:17) , we recover the fact that the Legendre order µ can take the values ± , ± , ± . However, such an analysis based on the second-order equationalone cannot determine which subset of indices µ belong to a given solution of the first-order susyconstraints. Nor can they determine the subset of indices belonging to a square-integrable solu-tion, by contrast to a non square-integrable one. To determine that the one-parameter family ofsquare-integrable solutions of the susy constraints were associated with the set µ = (cid:8) − , − , + (cid:9) of indices we had to go through the more complicated analysis of the susy constraints sketchedabove.Finally, we can extract from our analysis the scattering data for the α symmetry-wall re-flection. The basic fact to be used is the asymptotic decomposition of the Legendre function P µν given in Eq. (2.39) above. To determine the global phase relating the incident far-wall amplitude C N F =3 to the reflected one, it is enough (as in the N F = 2 case) to consider the ⊥⊥ componentof the wave amplitude h pq . [Indeed, we have checked that, for the α symmetry-wall reflection,the N F = 3 coefficient L ⊥⊥⊥ measuring the sensitivity of h farwall ⊥⊥ , Eq. (3.3), to the sign of π (cid:48) perp vanishes]. We have exhibited in Eq. (3.11), the fact that h ⊥⊥ is proportional to G ( α ) , andtherefore (for the square-integrable solution) to P + − + i π (cid:48)⊥ [cosh( α )] . This shows that the globalphase factor is the one belonging to the value µ = + . Using, the general result (2.51), we thenget (cid:2) e i δ global (cid:3) α = Γ( − − i π (cid:48)⊥ ) Γ( i π (cid:48)⊥ )Γ( − + i π (cid:48)⊥ ) Γ( − i π (cid:48)⊥ ) . (3.29)2In the WKB limit this yields (cid:2) e i δ global (cid:3) W KBα = e i π . (3.30)Moreover, the map between the incident spinor state and the reflected one is obtained by thereflection operator R ,N F =3 α = Γ[+ iπ (cid:48)⊥ ] Γ[ − i π (cid:48)⊥ − (cid:113) (cid:98) S ,N F =3 ]Γ[ − iπ (cid:48)⊥ ] Γ[ + i π (cid:48)⊥ − (cid:113) (cid:98) S ,N F =3 ] , (3.31)which yields in the WKB limit R ,N F =3 ,W KBα = e − i π e i π √ (cid:98) S ,NF =3 . (3.32)Here, (cid:113) (cid:98) S ,N F =3 denotes an operator squareroot of (cid:98) S ,N F =3 which is not equal to itspositive squareroot, but which is defined as (cid:113) (cid:98) S ,N F =3 ≡ (cid:98) µ , (3.33)by which we mean the following squareroot version of Eq. (3.22) (cid:113) (cid:98) S ,N F =3 | µ, i (cid:105) = 2 µ | µ, i (cid:105) for 2 µ = (cid:26) − , − , + 32 (cid:27) and i = 1 , . . . g ( µ ) . (3.34) B. N F = 3 reflection on the symmetry wall α ( β ) = β − β . Concerning the reflection on the second symmetry wall of our canonical chamber, namely α ( β ) = β − β , the needed computations are very similar to the ones above, with, however,some significant differences. Though one would have expected that a simple cyclic permutationwould suffice to translate the results of the α wall into results for the α wall, there are somesubtleties in intermediate results, linked to the fact that the explicit form of the susy constraintsis not manifestly cyclically symmetric. However, the end results are correctly obtained from apermutation (231) → (123) .We have already introduced above the basis adapted to the α ( β ) = β − β symmetry wall,namely Eqs. (2.56), (2.57). In terms of the frame components of the state (3.7), there are somesimplifications because we found that the algebraic constraints on the state imply the vanishingnot only (as before) of the antisymmetric components of h (cid:98) a (cid:98) b , but also the vanishing of thescalar f . This leaves us with only six propagating components: h ( (cid:98) a (cid:98) b ) . Again the adapted-framedecomposition of these components is directly linked with eigenstates of the relevant squared spinoperator, namely (cid:104) (cid:98) S (cid:105) ,N F =3 . The good (square-integrable) modes are again of the form (3.20)with the corresponding µ -decomposition (3.21) of the solution. However, there is a differencein the link between each Legendre P µν mode and eigenspinors of (cid:104) (cid:98) S (cid:105) ,N F =3 , with eigenvalues (2 µ ) , as in Eq. (3.22) above. We have now, when considering a full basis of the 10-dimensional3space, even if some coefficient modes vanish | µ = − (cid:105) = | ⊥⊥(cid:105) , (3.35) | µ = − , i (cid:105) i =1 ,..., = | η (cid:105) , | u ⊥(cid:105) , | ⊥ u (cid:105) , | v ⊥(cid:105) , | ⊥ v (cid:105) , (3.36) | + 34 , i (cid:105) i =1 ,..., = | uu (cid:105) , | vv (cid:105) , | uv (cid:105) , | vu (cid:105) . (3.37)Let us only exhibit here, for illustration, the form of the ⊥⊥ mode: h ⊥⊥ ∝ sinh ( α ) P − − + i π (cid:48)⊥ [cosh( α )] . (3.38)Contrary to the N F = 2 case, h farwall ⊥⊥ is sensitive to the sign of π (cid:48)⊥ (i.e. the projected coefficient L ⊥⊥⊥ does not vanish). However, the other projections L ⊥ UV of L kpq do vanish, so that the farwallparallel-parallel components of h pq are insensitive to the sign of π (cid:48)⊥ . Finally, the global phasefactor for the N F = 3 reflection on the α symmetry wall is given by the behavior of the µ = mode, i.e. (cid:2) e i δ global (cid:3) α = Γ( − − i π (cid:48)⊥ ) Γ( i π (cid:48)⊥ )Γ( − + i π (cid:48)⊥ ) Γ( − i π (cid:48)⊥ ) , (3.39)with WKB limit: (cid:2) e i δ global (cid:3) W KBα = e i π . (3.40)The corresponding reflection operator reads R ,N F =3 α = Γ[+ iπ (cid:48)⊥ ] Γ[ − i π (cid:48)⊥ − (cid:113) (cid:98) S ,N F =3 ]Γ[ − iπ (cid:48)⊥ ] Γ[ + i π (cid:48)⊥ − (cid:113) (cid:98) S ,N F =3 ] , (3.41)(where (cid:113) (cid:98) S ,N F =3 is again defined as being (cid:98) µ ) which yields in the WKB limit R ,N F =3 ,W KBα = e − i π e i π √ (cid:98) S ,NF =3 . (3.42) C. Reflection at level N F = 3 on the gravitational wall α ( β ) = 2 β . When considering the reflection on the gravitational wall α ( β ) = 2 β of a N F = 3 solution weuse the same adapted basis as in our corresponding N F = 2 analysis, namely (2.65). Instead ofthe Legendre-like mode functions (3.20), we will have Whittaker-like mode functions, U µ ( β ; j ) ,as defined in Eq. (2.68). The only difference is that the squared mass value µ labelling thesemodes must now be taken to be µ N F =3 = + (instead of µ N F =2 = − ). Actually, the value of The vanishing of such or such component depends on the choice of basis. What is important is that we wereable to describe the exact solution of the susy constraints within the ten-dimensional (half) N F = 3 state space. µ only enters indirectly in the expression of U µ ( β ; j ) via a modified link between the shiftedparallel momenta π (cid:48) u , π (cid:48) v and π (cid:48)⊥ . In the present case, this explicit link reads: π (cid:48)⊥ = (cid:113) π (cid:48) u π (cid:48) v − .In the case of symmetry walls, we were decomposing the state | Ψ (cid:105) ,N F =3 into eigenstates ofthe squared spin operator (cid:98) S ,N F =3 (labelled by µ with (2 µ ) = S measuring the eigenvaluesof (cid:98) S ,N F =3 ), as in Eq. (3.21). Here, we shall decompose | Ψ (cid:105) ,N F =3 into eigenstates of theoperator (cid:98) J (with eigenvalues denoted j ), according to | Ψ (cid:105) ,N F =3 = C (cid:88) j = − , , N ij ( π (cid:48) u , π (cid:48) v ) U ( β ; j ) | j, i (cid:105) . (3.43)At level 3, the eigenvalues j (with their degeneracies labelled above by i ) of (cid:104) (cid:98) J (cid:105) ,N F =3 are (+2 | , | , − | ) . More importantly, the eigenspinors corresponding to these eigenvalues aredirectly linked with objects naturally constructed within our present adapted basis. Namely, wehave (using the notation (3.8), now applied to our new adapted basis) | j = 2 , i (cid:105) i =1 , = | u ⊥(cid:105) , | v ⊥(cid:105) , (3.44) | j = 0 , i (cid:105) i =1 ,..., = | ⊥⊥(cid:105) , | uu (cid:105) , | vv (cid:105) , | uv (cid:105) , | vu (cid:105) , | η (cid:105) , (3.45) | j = − , i (cid:105) i =1 , = | ⊥ , u (cid:105) , | ⊥ , v (cid:105) . (3.46)Let us only cite the form of our final result, namely the expression of all the components h (cid:98) a (cid:98) b ( β ) (modulo an overall factor that we omit) of the main N F = 3 polarization tensor h pq along ouradapted basis. [The scalar polarization f happens to vanish, as well as h uv − h vu .] h u ⊥ = − i π (cid:48) u U ( β, +2) ,h v ⊥ = − i π (cid:48) u π (cid:48) v ( π (cid:48) u − i/ U ( β, +2) ,h ⊥ u = iπ (cid:48) v U ( β, − ,h ⊥ v = i ( π (cid:48) u − i/ U ( β, − ,h ⊥⊥ = 12 (2 π (cid:48) u π (cid:48) v − i π (cid:48) u − / π (cid:48) v ( π (cid:48) u − i/ U ( β, ,h uu = ( π (cid:48) u − i/ π (cid:48) v U ( β, ,h vv = π (cid:48) v + i ( π (cid:48) u − π (cid:48) v ) + 1 / π (cid:48) v ( π (cid:48) u − i/ U ( β, ,
12 ( h uv + h vu ) = 12 U ( β, . (3.47)Using the asymptotic behaviour of the Whittaker modes [see Eq. (2.77)], we deduce thereflection laws on the gravitational wall β = 0 . We checked that (because, for the presentcase, L ⊥⊥⊥ = 0 ) the global phase is read off the h ⊥⊥ expression (involving j = 0 ) and reads e iδ global α = Γ (cid:2) − i π (cid:48)⊥ (cid:3) Γ [ i π (cid:48)⊥ ]Γ (cid:2) + i π (cid:48)⊥ (cid:3) Γ [ − i π (cid:48)⊥ ] . (3.48)As before it is energy-dependent, and has no limit as π (cid:48)⊥ → + ∞ .5The reflection operator against the α gravitational wall (acting in Hilbert space and trans-forming the incident state into the reflected one) is given by the following operatorial expression R ,N F =3 α = Γ (cid:104) (cid:98) J − i π (cid:48)⊥ (cid:105) Γ [ i π (cid:48)⊥ ]Γ (cid:104) (cid:98) J + i π (cid:48)⊥ (cid:105) Γ [ − i π (cid:48)⊥ ] . (3.49)In the large momentum (WKB) limit, this yields R ,N F =3 α = e iδ ( π (cid:48)⊥ ) e − i π e − i π (cid:98) J , (3.50)where δ ( π (cid:48)⊥ ) = 2 π (cid:48)⊥ ln(4 π (cid:48)⊥ /e ) . These are formally the same expressions as at level N F = 2 ,but, here, (cid:98) J denotes the endomorphism of the 10-dimensional N F = 3 subspace in which weare working.Let us note that the solution (3.47) contains more excited components than the previoussymmetry-wall N F = 3 solutions. In particular, the antisymmetric components h u ⊥ − h ⊥ u and h v ⊥ − h ⊥ v do not vanish, while they vanished before. However, using the asymptotic behavior,Eq. (2.77), of the relevant functions U ( β, ± , one finds that their leading-order asymptoticapproximations (as β → + ∞ ) are exactly proportional to each other: U asympt ( β, +2) = − π (cid:48) u π (cid:48) v U asympt ( β, − . (3.51)Inserting this asymptotic relation in Eqs. (3.47) one finds that the antisymmetric components h u ⊥ − h ⊥ u and h v ⊥ − h ⊥ v vanish far from the gravitational wall (in keeping with the far-wallanalysis of Ref. [16]). IV. HIDDEN KAC-MOODY STRUCTURE OF THE SPINOR REFLECTIONOPERATORS
Let us consider the WKB limit of the reflection operators R rep α that map the incident spinorstates | Ψ (cid:105) in to the reflected ones | Ψ (cid:105) out . These spinor reflection operators depend both on theconsidered reflection wall form α ( β ) and on the representation space, say V rep , in which livesthe considered incident and reflected quantum states. More precisely, we derived above twodifferent triplets of such reflection operators: (i) one triplet associated with the reflection (onthe three potential walls of our canonical billiard chamber) of the propagating quantum susystates at level N F = 2 , which live in a 6-dimensional representation; and (ii) a second tripletassociated with the reflection (on the same three bounding walls) of the propagating quantumsusy states at level N F = 3 , which live in a 10-dimensional representation. In the WKB limit(and after factorization of the classical, energy-dependent part of the gravitational-wall reflection, δ ( π (cid:48)⊥ ) = 2 π (cid:48)⊥ ln(4 π (cid:48)⊥ /e ) ), we found the following operatorial expressions for these two tripletsof reflection operators: R ,N F =2 ,W KBα = e − iπ e ± iπ | (cid:98) S | ,NF =2 , R ,N F =2 ,W KBα = e − iπ e ± iπ | (cid:98) S | ,NF =2 , R ,N F =2 α = e − i π e − i π (cid:98) J ,NF =2 , (4.1)where we recall that | (cid:98) S | ,N F =2 and | (cid:98) S | ,N F =2 were defined as the positive squareroots ofthe corresponding squared spin operators (cid:98) S , (cid:98) S , which are both endomorphisms of the 6-dimensional subspace H (1 , S of the N F = 2 level. The “gravitational" operator (cid:98) J is also anendomorphism of H (1 , S . [See, e.g., the second Table in Appendix B of [16].]6The corresponding results for the reflection operators in the 10-dimensional subspace of the N F = 3 level where live the propagating quantum states read: R ,N F =3 ,W KBα = e − i π e i π √ (cid:98) S ,NF =3 , R ,N F =3 ,W KBα = e − i π e i π √ (cid:98) S ,NF =3 , R ,N F =3 α = e − i π e − i π (cid:98) J ,NF =3 . (4.2)Here, there is a crucial difference in the way the squareroots of the squared-spin operators aredefined. We recall that both squared-spin operators have eigenvalues (2 µ ) = (cid:8) ( ) , ( ) , ( ) (cid:9) .The squareroot operators (cid:113) (cid:98) S ab ,N F =3 are defined as having the eigenvalues µ = (cid:8) − , − , + (cid:9) on the corresponding eigenspaces of (cid:98) S ab ,N F =3 . This sign pattern is such that the corresponding,successive values of the Legendre order µ , namely (cid:8) − , − , + (cid:9) differ by 1 (so as to correspondto the regular solution of the first-order susy constraints).Let us emphasize that the results above for the reflection operators have resulted from a purely dynamical computation within supergravity. However, a remarkable fact is that the end resultsof these supergravity calculations can be expressed in terms of mathematical objects having a(hyperbolic) Kac-Moody meaning. More precisely, we are going to show that the two tripletsof spinorial reflection operators satisfy some relations that are related to a spin-extension of theWeyl group of the rank-3 hyperbolic Kac-Moody algebra AE . The notion of spin-extendedWeyl group was introduced, within the use of specific representations of the maximally compactsubalgebra K [ AE ] of AE (and K [ E ] ⊂ E ), in Ref. [11]. More precisely, Ref. [11] studiedthe one-wall reflection laws of the classical, Grassmann-valued gravitino field ψ , in the case where,near each potential wall (with bosonic potential ∝ e − α ( β ) ), the coupling of the gravitino is alsoToda-like and ∝ e − α ( β ) , so that the law of evolution of ψ near each separate wall reads ∂ t ψ ≈ i e − α ( β ) Π α J α ψ , (4.3)where Π α is a conserved momentum.Under these assumptions, Ref. [11] found that the transformation linking the incident valueof the Grassmann-valued ψ to its reflected value was given by a classical, fermionic reflectionoperator of the form R classical α = e i π ε α J α , (4.4)where ε α = ± denotes the sign of the momentum Π α . Here, J α is a matrix acting on the repre-sentation space defined by a classical homogeneous gravitino. In the case of Ref. [11], this was(when considering 4-dimensional supergravity) a 12-dimensional space in which live the twelvecomponents of a Majorana (spatial) gravitino ψ iA , with i = 1 , , (spatial index) and A = 1 , , , (Majorana spinor index). This 12-dimensional representation is (essentially) equivalent to thedirect sum of the two (complex-conjugated) -dimensional complex representations that live atlevels N F = 1 and N F = 5 within our 64-dimensional quantized-gravitino Hilbert space. [In viewof the hidden, but crucial, importance of the existence of such finite-dimensional representations,we briefly discuss in Appendix B the structure of some of the low-dimensional representations of K [ AE ] .]Motivated by these physical findings, a mathematical definition of spin-extended Weyl groups(for general Kac-Moody algebras) was then implemented (as part of the definition of spin-coversof maximal compact Kac-Moody subgroups of the K [ AE ] type) and studied in Ref. [29].Ref. [11] showed that the reflection operators, say r Gi = R classical α i , describing the Grassmanianscattering on the dominant potential walls (labelled by the index i = 1 , · · · , rank ) of the cosmo-logical supergravity billiards (both in dimension D = 11 and in D = 4 ) satisfied some spinorial7generalization of the usual Coxeter relations satisfied by the corresponding Weyl-group gen-erators. [We recall that a basic finding of cosmological billiards [8] is that the gravity-definedbilliard chamber coincides with the Weyl chamber of some corresponding Kac-Moody algebra.]The (Grassman-supergravity-based) spin-extended Weyl group was then defined as the infinite,discrete matrix group generated by the r Gi ’s. [Here, the index i labels the nodes of the Dynkindiagram, corresponding to the simple roots of a Kac-Moody algebra, and to the dominant wallsof the supergravity dynamics.] The generalized Coxeter relations satisfied by the Grassmanianreflection operators r Gi can be written as r i = 1; (4.5) r i r j r i · · · = r j r i r j · · · with m ij factors on each side . (4.6)Here, i , and j , with i (cid:54) = j (which includes both i < j and i > j ), are labels for the nodes ofthe Dynkin diagram of the considered Kac-Moody group. The positive integers m ij entering the“braid relations" (4.6) are defined from the corresponding values of the nondiagonal elements ofthe Cartan matrix a ij (which are supposed to be negative integers, while a ii = 2 ). Namely (see[28]) m ij = { , , , , } if a ij a ji = { , , , , ≥ } (respectively) . (4.7)In addition to the generalized Coxeter relations, (4.5), (4.6), Ref. [11] had found that thesquared Grassman reflection operators ( r Gi ) had simple properties. Namely, they generated afinite-dimensional, normal subgroup of the corresponding (Grassman-based) spin-extended Weylgroup.According to the mathematical definition of Ref.[29], the spin-extended Weyl group of a Kac-Moody algebra with Dynkin diagram Π is a discrete subgroup of a spin cover of the maximallycompact Kac-Moody subgroup K [Π] that is generated by elements of order eight (involving thepolar angle π ). This mathematically-so-defined spin-extended Weyl group can also be chara-terized by generators and relations. Namely, its (abstract) generators r i satisfy not only thegeneralized Coxeter relations above (4.5), (4.6), but also the following ones: r − j r i r j = r i r n ij j , (4.8)where, as above i (cid:54) = j , and where the positive integers n ij are defined from the correspondingvalues of the nondiagonal elements of the Cartan matrix a ij via n ij = 0 (respectively = 1) if a ij is even (resp . odd) . (4.9)The additional (non-Coxeter-like) relations (4.8) are the same as those that enter the Tits-Kac-Peterson [28] extension of the Weyl group (generated by elements of order four: t i = 1 ). Theirorigin is not clear to us, and we shall see below that the quantum-motivated reflection operatorsthat have appeared in our dynamical study above, namely (4.1) and (4.1), satisfy the generalized In the notation of Eqs. (4.5), (4.6) below, the usual Coxeter relations defining the Weyl group, i.e. the groupgenerated by geometrical reflections in the simple-root hyperplanes in Cartan space are: r i = 1 and the braidrelations (4.6). Here, following standard mathematical lore [28], we rewrite the relations written in Ref. [11] in a form thatonly involves the multiplicative identity, rather than the “minus identity operator" used there when dealingwith concrete, matrix forms of the r Gi ’s. AE , and its Dynkin diagram is • J (cid:107) (cid:115) (cid:43) (cid:51) • S • S (4.10)Here, we use the labelling: (1 , ,
3) = ( J , S , S ) . The two arrows and the double line betweennodes 1 and 2 mean that a = a = − , while the single line between nodes 2 and 3 mean that a = a = − . Finally, a = a = 0 . As a consequence, the relevant values of the integers n ij and m ij to be used in Eqs. (4.5), (4.6), and (4.8), are: n = n = 0 ; m = m = 0 ; n = n = 1 ; m = m = 3 ; n = n = 0 ; m = m = 2 . (4.11)The three relations r i = 1 , Eq. (4.5), are satisfied for each one of our triplets of reflection oper-ators (4.1), (4.2). [This is clear without calculation because the eigenvalues of all our reflectionoperators are e ik π for some integer k .] By explicit (matrix) calculations, we have verified thatthe AE braid relations (4.6), namely r r r = r r r ; r r = r r (4.12)(note that m = m = 0 so that there are no braid relations between the nodes J and S )are also satisfied by our two triplets of reflection operators (4.1), (4.2).Concerning the non-Coxeterlike relations (4.8), let us first emphasize that we view them asexpressing constraints on the sub-group generated by the squared operators r i . As in Ref.[11], we looked directly at the values taken (within the two matrix representations that we areconsidering) by the squares of our two triplets of generators (4.1), (4.2). We found that theyhave extremely simple values; namely they only differ from the identity matrix by some simplephase factors, namely (cid:0) R ,N F =2 ,W KBα (cid:1) = − Id = e iπ Id , (cid:0) R ,N F =2 ,W KBα (cid:1) = − Id = e iπ Id , (cid:0) R ,N F =2 ,W KBα (cid:1) = e − i π Id , (4.13)and (cid:0) R ,N F =3 ,W KBα (cid:1) = e i π Id , (cid:0) R ,N F =3 ,W KBα (cid:1) = e i π Id , (cid:0) R ,N F =3 ,W KBα (cid:1) = − Id = e iπ Id . (4.14)In both cases the subgroup generated by the squared reflection operators is central (i.e. com-mutes with everything else) and isomorphic to the multiplicative group of order four generatedby e i π .Finally, in view of the simple results (4.13), (4.14), it is a simple matter to see whether thenon-Coxeterlike relations (4.8) are satisfied or not. One can easily see that, with the precisedefinitions (4.1), (4.1), they are not satisfied as written. However, they are satisfied modulothe inclusion of additional phase factors in the relations (4.8). The latter phase factors can beeasily reabsorbed in suitable redefinitions of the basic reflection operators. For instance, if we9had defined, at level N F = 2 (with an arbitrary integer n in the third line) R ,N F =2 ,W KB,newα = e ± iπ | (cid:98) S | ,NF =2 , R ,N F =2 ,W KB,newα = e ± iπ | (cid:98) S | ,NF =2 , R ,N F =2 ,newα = e in π e − i π (cid:98) J ,NF =2 , (4.15)and, at level N F = 3 , R ,N F =3 ,W KBα = e − i π e i π √ (cid:98) S ,NF =3 , R ,N F =3 ,W KBα = e − i π e i π √ (cid:98) S ,NF =3 , R ,N F =3 α = e in π e − i π (cid:98) J ,NF =3 , (4.16)these two new triplets of operators would satisfy all the relations (4.5), (4.6), and (4.8). In thatcase, the corresponding squared operators are simply equal to unity (for an appropriate choiceof n ).Let us also mention in passing that if we define, within the full sixty-four-dimensional spinorialspace which gathers all the fermionic levels (from N F = 0 to N F = 6 ) the quantum analogs ofthe Grassmann-motivated operators defined in Ref. [11], namely R α = e − i π (cid:98) S , R α = e − i π (cid:98) S , R α = e − i π (cid:98) J , (4.17)the latter reflection operators satisfy all the relations (4.5), (4.6), and (4.8). V. CONCLUSIONS
We solved the susy constraints (1.4) of the supersymmetric Bianchi IX model in the one-wallapproximation, i.e. taking into account one potential wall at a time. This allowed us to derive thequantum laws of reflection of the wave function of the universe | Ψ( β ) (cid:105) during its chaotic evolutionnear a big crunch singularity, i.e. in the domain of large (positive) values of the three squashingparameters β , β , β (considered in the symmetry chamber β < β < β ). Our analysis couldlimit itself to two subspaces of the total 64-dimensional fermionic state space because we hadshown in previous work that propagating states only exist in subspaces of the fermion levels N F = 2 , N F = 3 and N F = 4 . In addition, given the symmetry between the N F = 2 and the N F = 4 levels, and the self-symmetry of the N F = 3 level, and in view of the special structureof the propagating states, it was enough to work (separately) in a 6-dimensional subspace of the N F = 2 level, and in a 10-dimensional half of the N F = 3 level.Our main results are contained in Eqs. (2.52), (2.81), (3.31), (3.41), (3.49), and are summarized(in the small-wavelength limit, which allows one to highlight their structure) in the reflection op-erators (4.1), (4.2). We remarkably found that the latter, purely dynamically-defined, reflectionoperators satisfy generalized Coxeter relations which define a type of spinorial extension of theWeyl group of the rank-3 hyperbolic Kac-Moody algebra AE . More precisely, we found thatour dynamical reflection operators satisfy the generalized Coxeter relations (4.5) and (4.6) asso-ciated with the Dynkin diagram (4.10) of AE , and selected in Ref. [11] (in a slightly differentform) as characteristic of a spin-extension of the Weyl group. We also found that the squares ofour dynamical reflection operators commute with all the reflection operators. In addition, some0phase-modified versions of the reflection operators, see Eqs. (4.15), (4.16) satisfy the relations(4.8) that are part of the defining relations of the mathematically-defined spin-extended Weylgroup of Ref. [29]. The fact that our dynamically-defined spinorial reflection operators satisfyrelations that appear as being partly more general than those of Ref. [29] (though only modulosome extra phase factors, Eqs. (4.15), (4.16)) might suggest the need to define more general spin-covers than those mathematically constructed in Ref. [29]. Anyway, independently of such aneventual generalization, let us repeat that our findings provide a new evidence for the existenceof hidden Kac-Moody structures in supergravity. In particular, our results have gone beyondprevious related evidence for Kac-Moody structures in two directions: (i) we quantized the grav-itino degrees of freedom instead of treating ψ µ as a classical, Grassmann-valued object, and (ii)in our quantum treatment the symmetry walls necessarily involved operators quartic in fermions(through the squared spin operators (cid:98) S , (cid:98) S ), while the previous (Grassmann) treatment of Ref.[11] had assumed a linear coupling to the quadratic spin operators. Let us also note that the linkbetween our present dynamical reflection operators, Eqs. (4.15), (4.16), and representations of K [ AE ] is more indirect than what was suggested by the Grassmann-based work of Ref. [11]. Inparticular, the 6-dimensional subspace in which live the N F = 2 reflection operators is strictlysmaller than the full 15-dimensional N F = 2 space within which live the operators (cid:98) J , (cid:98) S , (cid:98) S that carry a representation of K [ AE ] . Moreover, the operators that appear in exponentiatedform in Eqs. (4.15), (4.16), do not define a representation of K [ AE ] .In view of our results, we can associate with the evolution of the supergravity state of theuniverse | Ψ( β ) (cid:105) (considered at each fermion level) a word in the group generated by the threereflection operators Eqs. (4.15), (4.16), i.e. a product of the form · · · r i n r i n − · · · r i r i . Thematrix group generated by such products is infinite. However, we must recall that our study wasassuming a type of intermediate asymptotic behavior with a sparse sequence of wall collisions,separated by large enough distances in β space to be able to treat each collision of the wave packetas a separated one-wall reflection. Such an approximation is not expected to maintain itself for aninfinite number of collisions. Indeed, on the one hand, at level N F = 3 the (shifted) momentum π (cid:48) a is timelike ( G ab π (cid:48) a π (cid:48) b = − ) so that, after a finite number of reflections, one expects the trajectoryof the wave packet to end up in a direction which does not meet anymore a potential wall. Onthe other hand, at level N F = 2 the (shifted) momentum π (cid:48) a is spacelike ( G ab π (cid:48) a π (cid:48) b = + ) so that,after a finite number of reflections, one expects π (cid:48) a to tip over, i.e. to migrate from the upper half[where π (cid:48) + π (cid:48) + π (cid:48) > , corresponding to decreasing spatial volume V = abc = e − ( β + β + β ) ]of its (one-sheeted) hyperboloidal mass-shell , to its lower half (corresponding to increasingspatial volumes). Such a cosmological bounce (further discussed in Ref. [16]) is then expectedto generate a finite number of reflections during the re-expansion regime, before driving thewavefunction in the (non-billiard-like) Friedman-type expansion regime. We leave to future worka discussion of the global evolution of the quantum state of such a universe, which is classicallyexpected to bounce back and forth, indefinitely, between large volumes and small volumes (seeFig. 3 in Ref. [16], and discussion in Sec. XX there). Acknowledgments
We thank Igor Frenkel and Christophe Soulé for informative discussions. Ph. S. thanksIHES for its kind hospitality; his work has been partially supported by the PDR “Gravity andextensions” from the F.R.S.-FNRS (Belgium) (convention T.1025.14).1
Appendix A: Asymptotic plane-wave solutions
We display hereafter the explicit values of the numerical constants entering the linear ( L kpq π (cid:48) k )and constant ( m pq ) contributions entering the amplitudes K pq ∝ π (cid:48) p π (cid:48) q + L kpq π (cid:48) k + m pq (A1)of the N F = 2 and N F = 3 asymptotic plane-wave solutions to which we referred in Eqs. (2.3),(3.3). For each wall form α ⊥ , with adapted basis α (cid:98) a = (cid:8) α ⊥ , α u , α v (cid:9) , the values of the projectedcomponents L ⊥ (cid:98) a (cid:98) b that vanish determine the global reflection phase factor (see text). • Level N F = 2 : L kpq π (cid:48) k = − i π (cid:48) + π (cid:48) + π (cid:48) ( π (cid:48) + π (cid:48) ) ( π (cid:48) + 3 π (cid:48) ) ( π (cid:48) + π (cid:48) ) 2 π (cid:48) + π (cid:48) ( π (cid:48) + π (cid:48) ) ( π (cid:48) + 3 π (cid:48) ) ( π (cid:48) + π (cid:48) ) π (cid:48) , m pq = −
13 9 39 5 13 1 1 . (A2) • Level N F = 3 : L kpq π (cid:48) k = i − π (cid:48) + π (cid:48) + π (cid:48) − π (cid:48) + π (cid:48) − ( π (cid:48) − π (cid:48) ) − π (cid:48) + π (cid:48) − π (cid:48) + π (cid:48) − π (cid:48) − ( π (cid:48) − π (cid:48) ) − π (cid:48) − π (cid:48) , m pq = + 14 − . (A3)The above expressions correspond to the canonical billiard chamber β < β < β . See Ref. [16]for a discussion of the other chambers. Appendix B: On finite-dimensional representations of K [ AE ] The finite-dimensional representations of the “maximally compact” subalgebra K [ AE ] thatnaturally enter our supergravity study constitute special ones. We have investigated more generalfinite-dimensional representations of K [ AE ] , and briefly report here some of our findings.The algebra K [ AE ] is defined as the subalgebra of the hyperbolic Kac–Moody AE algebra[13, 14] that is fixed by the Chevalley involution ω . We recall that the latter is defined by itsaction on the Kac-Moody generators ( e i , f i , h i ) : ω ( e i ) = − f i , ω ( f i ) = − e i and ω ( h i ) = − h i ; sothat, for any Kac-Moody algebra A , its maximally compact subalgebra K [ A ] is generated bythe differences x i ≡ e i − f i . In the case of AE , with Dynkin diagram (4.10), this yields thethree generators x , x , x , which are respectively equivalent (modulo a factor i ) to the threegenerators (cid:98) J , (cid:98) S , and (cid:98) S .Any three generators (cid:98) J , (cid:98) S , (cid:98) S satisfying the following five relations [30] ad (cid:98) S (cid:98) S = (cid:98) S , ad (cid:98) S (cid:98) S = (cid:98) S , (B1) ad (cid:98) S (cid:98) J = 4 ad (cid:98) S (cid:98) J , ad (cid:98) J (cid:98) S = 4 ad (cid:98) J (cid:98) S , (B2)2 ad (cid:98) J (cid:98) S = 0 , (B3)define a representation of K [ AE ] . As in the text, we use here hermitian-type generators, corre-sponding to − ix , − ix , − ix rather than antihermitian-type ones x i = e i − f i , as generally usedin mathematical works.We are looking for finite-dimensional representations of the three generators (cid:98) J , (cid:98) S , (cid:98) S (i.e.three matrices, say J , S , S ), with emphasis on finding low-dimensional representations.[For a related study (oriented, however, towards finding high-dimension representations) in thecase of K [ E ] see Ref. [31].] Conditions (B1) show that S and S may be interpreted as usual su (2) generators. Note that if, given S and S , a matrix J satisfies relations (B2), (B3),so does the matrix − J . Moreover, the complex conjugate of any solution triplet of matrices J , S , S will also be a solution. In addition, if J , S , S is a n -dimensional solution, thetriplet J + kId n , S , S is also a solution for an arbitrary value of k .One can first look for representations that are irreducible with respect to su (2) , i.e. with S and S given, modulo conjugation, by the standard (2 j + 1) -dimensional, spin- j matrices, say(with m, m (cid:48) varying by steps of 1 between − j and + j ) (cid:16) S ( j )12 (cid:17) m, m (cid:48) = m δ m, m (cid:48) , (cid:16) S ( j )23 (cid:17) m, m (cid:48) = 12 (cid:16)(cid:112) ( j − m )( j + m + 1) δ m − , m (cid:48) + (cid:112) ( j + m )( j − m + 1) δ m +1 , m (cid:48) (cid:17) . (B4)The lowest-dimensional case would be the 2-dimensional spin- su (2) representation. However,we found that, in this case, the only possible solutions of Eqs. (B2), (B3) for J are J ∝ Id .In the present study, we consider such solutions as being “trivial" .The only su (2) -irreducible [cf. (B4)] representations with non-trivial J we found (up to j = 13 / ) correspond to j = 1 and j = , i.e. to 3 and 4 dimensional representation spaces. Weconjecture that these are the only such ones.The lowest-dimensional nontrivial representation of K [ AE ] is 3-dimensional. Its generatorsare given by (B4) (for j = 1 ), together with J ( ± = ± k k + 1 01 0 k , (B5)whose eigenvalues are ± ( k + 1 , k + 1 , k − . Here, k , which corresponds to the shift kId mentioned above, is arbitrary. One can choose k = − if one wishes to normalize the trace of J ( ± to zero.There is a similar 4-dimensional representation with generators given by (B4) (for j = ),together with (modulo a kId shift) J ( ± / = ± − √ − − √ − √ − − √ , (B6) We note, however, that the (real) 4-dimensional Dirac-spinor-type representation of K [ AE ] discussed in Eqs.(4.14), (4.17) of Ref. [11] is equal to a direct sum of two such (complex-conjugated) 2-dimensional representa-tions with “trivial" J ’s, and that the tensor product of the “trivial" 2-dimensional representation, and of thenon-trivial 3-dimensional one discussed next, leads to a non-trivial 6-dimensional representation (see below). ± (1 , , − , − .Other kinds of representations exist, in which the spin generators are not irreducible. Actually,this is the case for the first-found finite-dimensional representation of K [ AE ] , namely the 6-dimensional representation defined by the gravitino operators in -dimensional spacetime[11]. More precisely, Ref. [11] dealt with a real kId shift of J ) to the complex, 6-dimensional representation of K [ AE ] appearing at the N F = 1 level of our total quantum, 64-dimensional space. [The 6-dimensional N F = 1 representationwe are talking about is the representation spanned by the six states ˜ b a ± | (cid:105) − .] In the latterrepresentation, the spin operators are the direct sum of irreducible representations with spins and [i.e. with eigenvalues of S and S equal to + , − , (cid:0) + (cid:1) , (cid:0) − (cid:1) ].Starting from 6-dimensional spin generators given by such a direct sum ( j = ) (cid:76) ( j = ) ,we looked for the most general J satisfying the additional relations (B2), (B3). We found, inabsence of additional requirements, multi-parameter families of solutions. On the other hand,we can require that a non-degenerate sesquilinear form H be left invariant by all the generators,i.e. J † H − H J = 0 , (B7)and similar equations with the spin generators. [The relative minus sign in Eq. (B7) comes fromthe fact that, in our conventions, the one-dimensional group generated by J is e iθ J .] Theinvariance of H under the spin generators restricts it (in the basis of Eqs. (B4)) to the form H = p Id ⊕ q Id , where (in the nondegenerate case) only the sign of the ratio p/q matters. We thenfound that (besides isolated solutions) there exists four different one-parameter families of such6-dimensional representations. Parametrizing the elements of the 6-dimensional representationspace as vector-spinors v A a , with A = 1 , and a = 1 , , , the generators S , S , J ( w ) canbe written in the factorized way discussed in Eqs. (3.11) and (4.16) of Ref. [11], namely, ( S ) AaBb = 12 ( σ ) AB (cid:18) α a α b α · α − δ ab (cid:19) , ( S ) AaBb = 12 ( σ ) AB (cid:18) α a α b α · α − δ ab (cid:19) , (cid:16) J E,L ( w ) (cid:17) AaBb = 12 ( σ ) AB (cid:18) α a E,Lw α w b α E,Lw · α w − δ ab (cid:19) . (B8)Here: σ ≡ σ z = diag(1 , − and σ ≡ σ x are the usual (real) Pauli matrices; σ ≡ Id ; α a and α a are the same linear forms as in Eq. (1.7); their contravariant versions α a , α a aredefined by raising the index either by means of G ab or, equivalently, by means of δ ab ; whilethe third, gravitational-like linear form α w a is the following one-parameter deformation of theusual gravitational linear form: α w ( β ) = α w a β a = 2 β + w ( β + β ) . On the other hand, thethird Eq. (B8) involves, depending on the value E (for Euclidean) or L (for Lorentzian) ofthe superscript, two different contravariant versions of α w , namely, either α a Ew ≡ δ ab α w b , or, α a Lw ≡ G ab α w b , where G ab is the Lorentzian (contravariant) metric defined in Eq. (1.3). Theparameter w runs over the real line (except, in the Lorentzian case, for the singular value w = Here, spinors mean two-component su (2) spinors ξ A . As in Ref. [11], the presence of vectorial projectors α a α b /α in Eqs. 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