aa r X i v : . [ h e p - t h ] S e p Quantum theory on Lobatchevski spaces
Ugo Moschella and Richard Schaeffer Universit`a dell’Insubria, 22100 Como, Italia, and INFN, Sez. di Milano, Italia Service de Physique Th´eorique, CEA - Saclay, Gif-sur-Yvette, FranceNovember 28, 2018
Abstract
In this paper we set up a general formalism to deal with quantum theories on a Lobatchevskispace, i.e. a spatial manifold that is homogeneous, isotropic and has negative curvature. The heartof our approach is the construction of a suitable basis of plane waves which are eigenfunctionsof the Laplace-Beltrami operator relative to the geometry of the curved space. These functionswere previously introduced in the mathematical literature in the context of group theory; here werevisit and adapt the formalism in a way specific for quantum mechanics. Our developments renderdealing with Lobatchevski spaces, which used to be quite difficult and source of controversies, easilytractable. Applications to the Milne and de Sitter universes are discussed as examples.
There are several good reasons to study quantum theories on a Lobatchevski space. The first reasonis simply to extend our knowledge and skills in quantization. The negative curvature of this modeltogether with its non compactness may, and indeed do, give rise to new and unforeseen phenomenawhich ordinary (canonical) quantization procedures on curved spacetimes [1] are not prepared to dealwith.Secondly, it has for long been advocated by Callan and Wilczek [2] that a negative curvature may act asa covariant regularizer of the infrared problem in a better way than putting a quantum system in a boxor on a sphere. An intuitive way to understand this viewpoint is as follows: since in this geometry thevolume of a sphere increases exponentially with its radius, the flux created by a central charge decreasesaccordingly and photons behave as if they have a ”mass”. Many quantum field models have indeed beenstudied on this background geometry, in particular the two-dimensional Liouville model [3] and a wholeclass of conformally invariant models.From the viewpoint of astrophysics and cosmology, Friedmann-Robertson-Walker open models (moreproperly, models with negative spatial curvature), first introduced in the inflationary context in the earlyeighties [4], became rather popular [5, 6, 7, 8] in the mid-nineties, when the belief was that a negativecurvature of the space could explain the mass content and the expansion rate of the universe. At thattime it was realized that in some cases [9] there were troubles with canonical quantization in such ageometry, although with adaptation the latter can be used to recover the standard [10, 11, 12] Klein-Gordon quantum field in the special case of the open de Sitter manifold and some indications exist [13]for possible ways out in more general situations. One difficulty was the appearance of supercurvaturemodes for theories of sufficiently low mass. Those modes are not square integrable on the Lobatchevskimanifold itself (even not in generalized sense as the exponentials of the flat case do) and therefore donot fit in the standard formalism of quantum mechanics. Even though the physical meaning of thosesupercurvature modes is yet unclear, they are not expected to provide a sizeable contribution to the1icrowave background fluctuations [14]. They might however be relevant in other domains of physics . . . and even in cosmology that from time to time (and even often) gives rise to surprises.The interest in open inflationary models subsequently dropped with the measurements of the fluctuationsof the Cosmic Microwave Background [15] indicating that the universe is most likely flat. However,with the spectacular and unprecedented precision of the forthcoming measurements scheduled by thePLANCK satellite [16], any small deviation from flatness will have to be mastered securely. This callsfor a revision and a complete solution of the problem of describing quantum fluctuations in a negativelycurved space (while the positively curved case is completely under control).From the mathematical viewpoint, part of the material we are going to present belongs to the chapterof harmonic analysis on symmetric spaces (see e.g. [17]; [18] gives an up-to-date account of the topics).Of course there exists a vast literature already on the more specific subject of Lobatchevski spaces.These mathematical results however are often formulated in a rather abstract way. Also, the square-integrability hypothesis that is a pillar of harmonic analysis may be violated in physically interestingsituations and should only be considered as a starting point. On the other hand, most if not allapproaches in the physical literature involve series expansions in terms of special functions which hideto a large extent the underlying symmetries and ultimately the physics.We have therefore decided to reconsider the subject from the beginning and we have found some newways to handle efficiently, and rather simply, the technical difficulties that arise from the lack of acommutative group of space translations. Our approach is especially aimed for physics; it remainspractical and accessible and has also the advantage to avoid the use of theorems concerning expansionsin bases of special functions (those expansions actually result from our approach). Another valuablepoint is that we work solely within the physical space-time and do not rely on any extension or completionof the latter to regions that are not covered by the open chart, as is the case in previous studies. Ourscheme is sufficiently flexible to allow the study of the general dimension in a single step.We focus on the study of a basis for the the standard Hilbert space of square-integrable functions on a( d − d dimensions). This Hilbert space isnot expected to be sufficient [9, 13] to describe all the interesting physical quantum theories in case ofnegative spatial curvature. Modes that are not square-integrable, however, warrant a separate specificstudy, and will be considered elsewhere [19] along the same line of thought.In Section 2 we give an introductory presentation of the geometry of the Lobatchevski space and of the”absolute” [20] of that space which can be identified with the space of momentum directions. We givealso a quite detailed description of the coordinate systems we are going to use.In Section 3 we display the unnormalized eigenmodes of the Laplace-Beltrami operator following theapproach described in [20]. These solutions are ”plane wavefunctions” in the sense they have constantvalues on hyperplanes. Strangely enough, these solutions have been only rarely used to study quan-tum theories in a Lobatchevski space in the physical literature, but they constitute the most naturalpossibility, and actually a cornerstone, to phrase Lobatcehvskian quantum mechanics in strict analogywith Euclidean quantum mechanics. An important technical point is provided in Section 4 where weconstruct some useful integral representations of the above eigenmodes; these representations are linkedto the parabolic coordinate system introduced in Section 2 and are inspired by our earlier work [21].These representations allow to trivialize ( d −
2) integrations when a square-integrable wavefunction isprojected on a mode. The relation of our modes with the basis constructed in spherical coordinates (seee.g. [5] and references therein) is also discussed.In Sections 5, 6 and 7 we set up the basic ingredients for quantum mechanics by building an orthonormalbasis of modes; in particular we show how to deal with the integral representations by computing the L scalar product of the modes. This naturally leads to the introduction of a Fourier-type transform of anarbitrary L function and to an inversion formula obtained here by means of the Kontorovich-Lebedevtransform. 2e conclude the paper by discussing two physical applications to models of QFT that are playing acentral role in contemporary cosmology.In Section 8 we revisit Milne QFT. There has been a regain of interest in this model in recent timescoming from the very different perspectives of string theory [22, 3] and of observational cosmology [23].Here we give a clean treatment of Milne’s QFT based on an expansion of the Minkowskian exponentialplane waves onto the basis of Lobatchevski modes that we have been constructing.In the following Section 9, we study QFT on the open de Sitter universe at any dimension. This hasproven to be technically very difficult and has been source of controversies, even for the quantificationin L space, which is the aim of the present paper [9, 13]. A special attention is given to the twodimensional case, which shares many features of the general case, but is extremely simple.The anti-de Sitter case is of obvious interest: the Lobatchevski manifold is identical to the Euclideananti-de Sitter universe. The physics however is however very different and therefore we have not includedthe AdS case in the present paper. In this section we describe the relevant geometrical setup for our construction, following the ideas andthe layout of Gel’fand, Graev and Vilenkin [20].Let M d be a d -dimensional Minkowski spacetime; an event x has inertial coordinates x , . . . x d − andthe scalar product of two such events is given by x · x ′ = x x ′ − x x ′ − . . . − x d − x ′ d − = η µν x µ x ′ ν . (1)In M d we consider the manifold (Fig. 1) H d − = { x ∈ M d : x = x · x = 1 , x > } . (2)The manifold H d − models a ( d − . It is a homogeneousspace under the action of the restricted Lorentz group of the ambient spacetime SO (1 , d − dl d − is obtained by restriction of the ambient Lorentzian metric to H d − : dl d − = − n(cid:0) d x (cid:1) − (cid:0) d x (cid:1) − . . . − (cid:0) d x d − (cid:1) o(cid:12)(cid:12)(cid:12) H d − . (3) Many interesting coordinate systems on H d − arise from particular decompositions of the symmetrygroup SO (1 , d − x ( r, x) = x = + r r x i = x i r x d − = − x − r r , < r < ∞ , x ∈ R d − . (4)In these coordinates the metric and the invariant volume form have the following explicit expres-sions: dl d − = dr + d x r , dµ ( x ) = r − ( d − drr d x , (5) We use here the convention mostly adopted by topologists in using the notation H d − . The index d − S d − = { x ∈ R d , x + . . . x d = 1 } denotes a ( d − d . Note however that the conventional notation for the hypersurface of thesphere S d − is ω d = 2 π d/ / Γ ( d/ A space of constant negative curvature embedded in an ambient Minkowski spacetime with onedimension more. The manifold H d − does not intersect the lightcone issued at any of its points (theorigin O in the figure); this shows pictorially that the surface is a spacelike manifold i.e. a Riemannianmodel of space. while the scalar product of two event is written x ( r, x) · x ′ ( r ′ , x ′ ) = (x − x ′ ) + r + r ′ rr ′ . (6)Equations (5) and (6) show that the measure and the scalar product are invariant w.r.t. translations inthe x coordinates and this explain why this system is so important and useful. Dirac’s delta distributionon the hyperboloid H d − is understood w.r.t. the invariant measure Z H d − dµ ( x ′ ) δ ( x, x ′ ) f ( x ′ ) = f ( x ); (7)specifically, in the previous coordinate system it is written as follows: δ ( x, x ′ ) = r d − δ ( r − r ′ ) δ (x − x ′ ) . (8)The future light-cone of the ambient spacetime (Fig. 2) plays a crucial role in our construction: C + = { ξ = ξ · ξ = 0 , ξ > } . (9)A useful parametrization for vectors of C + corresponding to (4) is the following: ξ ( λ, η ) = ξ = λ (1 + η ) ξ i = λη i ξ d − = λ (1 − η ) 0 < λ < ∞ , η ∈ R d − . (10)From both the mathematical and the physical viewpoints what matters is [20] the ”absolute” of theLobatchevski space H d − : this is the set of linear generators of the future light-cone, i.e. the light-cone4igure 2: A sight of the asymptotic future lightcone of the ambient spacetime; the vectors ξ belonging to C + play the role of momentum directions. modulo dilatations. The previous parametrization gives in particular a parabolic parametrization of theabsolute that is visualized (Fig. 3) as the parabolic section λ = 1 of C + : ξ ( η ) ≡ ξ (1 , η ) = ξ = (1 + η ) ξ i = η i ξ d − = (1 − η ) η ∈ R d − . (11)Correspondingly, the measure on the absolute is normalized as follows: dµ ( ξ ) = dη. (12)Figure 3: A view of the parabolic and spherical bases of the absolute.
Dirac’s delta on the cone will actually mean Dirac’s delta on the absolute: δ ( ξ, ξ ′ ) = δ ( η − η ′ ) . (13)By using the coordinate systems (4) and (11) the scalar products x · ξ and ξ · ξ ′ are written as fol-5ows: x ( r, x) · ξ ( η ) = (x − η ) r + r ξ (x) · ξ ( η ) r + r ξ ( η ) · ξ ′ ( η ′ ) = ( η − η ′ ) ξ (x) corresponding to x ( r, x). Note that this correspon-dence is tight to the choice of coordinates (4). Starting from ξ (x) we may recover the point x as follows(Fig. 4): x ( r, x ) = ξ (x) r + r ˆ ξ (16)where ˆ ξ = ξ (0) = (cid:0) , , . . . , − (cid:1) generates the only lightlike direction that the parametrization (11)does not cover; this direction is actually attained in the limit r → ∞ as shown by Eq. (16); one hasthat ˆ ξ · x ( r, x) = 1 / r .Figure 4: The choice of the vector ˆ ξ ,given the point x , uniquely determines the vector ξ (x) . The latteris the vector obtained by intersecting the cone with the plane containing the vector ˆ ξ and the point x (this plane actually contains the whole curve x ( r, x) , r > . ”Infinity” on the curve x ( r, x) is at r = 0 in the direction ξ (x) -which depend on the chosen point x - and at r = ∞ in the direction ˆ ξ . There is the other natural and widely used coordinate system on the manifold H d − where the rotationsymmetry SO ( d −
1) is apparent: x ( ψ, φ ) = x = cosh ψx = sinh ψ cos φ ... x d − = sinh ψ sin φ . . . sin φ d − cos φ d − x d − = sinh ψ sin φ . . . sin φ d − sin φ d − ψ ∈ R , φ = ( φ , ..., φ d − ) , < φ i < π, < φ d − < π. (17)6orrespondingly, there is the important (compact) spherical basis ξ = 1 for the absolute (see Fig.(3)): ξ ( θ ) = (cid:26) ξ = 1 ~ξ = n θ = ξ = 1 ξ = cos θ ... ξ d − = sin θ . . . sin θ d − cos θ d − ξ d − = sin θ . . . sin θ d − sin θ d − (18)with θ = ( θ , ..., θ d − ) and where n θ is a unit vector (n θ · n θ = 1) pointing in the direction identified bythe angles θ . In these coordinates the scalar products are written as follows x ( ψ, φ ) · x ( ψ ′ , φ ′ ) = cosh ψ cosh ψ ′ − sinh ψ sinh ψ ′ n φ · n φ ′ (19) x ( ψ, φ ) · ξ ( θ ) = cosh ψ − sinh ψ n φ · n θ (20) ξ ( θ ) · ξ ( θ ′ ) = 1 − n θ · n θ ′ (21)The measure on the absolute is the rotation invariant measure normalised as follows: dµ ( θ ) = d − Y i =1 (sin θ i ) d − − i dθ i Let us consider the vector space S ( H d − ) of rapidly decreasing complex functions defined on the manifold H d − and let us introduce the natural scalar product w.r.t. the invariant measure dµ ( x ): h f, g i = Z H d − f ∗ ( x ) g ( x ) dµ ( x ) . (22)The space S ( H d − ) can be completed to construct the Hilbert space H = L ( H d − , dµ ). By analogywith the flat case, H is the natural Hilbert space one would consider to study quantum mechanics onthe homogeneous and isotropic hyperbolic space H d − . To pursue this analogy, the first object to beexamined is the free Hamiltonian operator H = − ∆ , (23)where ∆ denotes the Laplace-Beltrami operator associated with the geometry (3). The operator − ∆is self-adjoint on a suitable domain of the Hilbert space H and its spectrum is the set [ ( d − , ∞ ).The eigenfunctions of the operator − ∆ can be labeled by a forward lightlike vector ξ ∈ C + and a realnumber q as follows [20]: ψ iq ( x, ξ ) = const ( x · ξ ) − d − + iq , (24)7nd one easily verifies that − ∆ ψ iq ( x, ξ ) = "(cid:18) d − (cid:19) + q ψ iq ( x, ξ ) = k ψ iq ( x, ξ ) . (28)For fixed q and any λ >
0, the vectors ξ and λξ identify the same eigenfunction because of the homo-geneity properties of the expression ( 24). Therefore the modes (24) corresponding to real values of theparameter q and to vectors ξ on the absolute (i.e. on a basis of the asymptotic lightcone) can be used toconstruct a basis of the Hilbert space H and are the strict analogue of the purely imaginary exponentials e ip · x of the flat case. Indeed, like the exponentials, also the wavefunctions (24) take constant valueson planes, here the hyperplanes x · ξ = const. In this sense the modes ( 24) may be called “planewaves”.As for the physical interpretation of the labels, k may be thought of as the intensity and ξ as the “direc-tion” of a “momentum” vector identifying a mode. The analogy in flat space would be parameterizingthe plane wave by the modulus of the momentum p = ( p · p ) and by its direction n: e ip n · x .This analogy goes one step further: we could have considered another set of plane waves characterizedby the opposite of the modulus of the momentum − p and by a direction n ′ : e − ip n ′ · x . A trivial remarkis that these modes already belong to the set { e ip n · x } since n can point in any direction.We will see in the following section, that also the mode ( x · ξ ) − d − − iq can be expressed as a superpositionon the absolute (i.e. as an integral over ξ ′ on a basis of the cone C + ) of the modes ( x · ξ ′ ) − d − + iq . Thismeans that to construct a basis of the Hilbert space H we may restrict our attention to the plane waves ψ iq ( x, ξ ), q ≥ ξ on the absolute.We end this section by remarking that there are purely imaginary values of q such that k ≥
0, namelythose purely imaginary q such that | q | ≤ d − . In particular the mode corresponding to k = 0 isconstant in space. These waves are not conventional in many respects; they are real functions, do notoscillate and (superposition of them) do not belong to the natural Hilbert space H . This means thata standard quantum mechanical interpretation is not immediately available for them. Such cases havebeen considered in the past [9, 13] but their status is not completely understood. We will discuss theirpossible role elsewhere [19]. There is no way to write the modes of the Laplace-Beltrami operator that be more symmetric thanthe expression (24): in that definition the modes appear as a complex power of a quantity invariantunder the action of the symmetry group, exactly as it happens for the exponentials exp( ip · x ) in theflat case. The exponentials have however the important property to be characters of the translation This can be shown by using a specific coordinate system, for instance (4), or either by using the inertial coordinatesof the ambient spacetime, by introducing the projection operator h and the tangential derivative D as follows: h µν = η µν − x µ x ν , D µ = h µν ∂ ν = ∂ µ − x µ x · ∂. (25)For any function that is smooth in a neighborhood of the manifold H d − one has that D µ D µ f = (cid:3) f − ( d − x · ∂f − x · ∂ ( x · ∂f ) , (26)where (cid:3) is the wave operator in the d -dimensional ambient spacetime. From this relation one can easily see that D µ D µ ( x · ξ ) α = α ( α − ξ ( x · ξ ) α − α ( α + d −
2) ( x · ξ ) α = − α ( α + d −
2) ( x · ξ ) α (27)Since − ∆ f = D µ D µ b f ˛˛˛ H d − , where b f is any smooth extension of the function f in a neighborhood of the manifold H d − Eq. (28) follows. ip · x ) exp( ip · y ) = exp( ip · ( x + y )). The lackof translation invariance of H d − is a major technical difficulty and, for our modes, there is nothingimmediately replacing this property of the exponentials. Therefore, from the viewpoint of practicalcalculations, it is useful to represent the modes (24) in terms of some integral transform which bereminiscent of translational invariance. Many representations are possible, in relation with differentchoices of coordinates on H d − and on the cone; we list only the two that are more relevant for ourpurposes. The following integral representations is an adaptation of the Euler integral of the second kind expressingthe Gamma function; an interesting geometrical interpretation can be based on the embedding of H d − in the ambient Minkowski spacetime M d :( x · ξ ) − d − + iq = 1Γ (cid:0) d − − iq (cid:1) Z ∞ dRR R d − − iq e − R x · ξ . (29)This representation is valid if Im q > Re d − and Re ( x · ξ ) > . More generally one can perform theintegration on the complex plane as follows:( x · ξ ) − d − + iq = i − ( d − − iq ) Γ (cid:0) d − − iq (cid:1) Z dRR R d − − iq e iR x · ξ (30)where the integration contour in the complex R plane is along any half-line issued from the origin inthe upper half-plane, i.e. 0 < Arg( R ) < π . Another useful integral representations can be obtained by inserting at the RHS of Eq. (29) the rep-resentation (14) of the scalar product x · ξ and Fourier transforming the Gaussian factor appearingthere: ( x · ξ ) − d − + iq = 1Γ (cid:0) d − − iq (cid:1) (cid:16) r π (cid:17) d − Z dκκ iq e − iκ · (x − η ) Z ∞ dRR e − rκ ( R + R ) R iq = 2Γ (cid:0) d − − iq (cid:1) (cid:16) r π (cid:17) d − Z dκ κ − iq e − iκ · (x − η ) K iq ( κr ) (31)where κ = √ κ · κ . In the second step we have used a well-known integral representation of the Bessel-Macdonald function K iq ( z ) that we recall here for the reader’s convenience: K iq ( z ) = 12 Z ∞ dRR R − iq e − z ( R + R ) . (32)The function K iq ( z ) decreases exponentially at large z ; near the origin one has that K iq ( z ) ∼ Γ( iq )2 (cid:0) z (cid:1) − iq + Γ( − iq )2 (cid:0) z (cid:1) iq . Therefore the integral (31) converges at κ = 0 and provides a representation of ( x · ξ ) − d − + iq for | Im q | < d − . There exists an abundant literature on Lobatchevski spaces that is based on the use of generalizedspherical harmonics in connection with spherical coordinate system. To render possible a comparisonof our results and methods with that approach let us work out the change of basis.9y adopting the spherical coordinates of Section 2.2 we can write (see Eq. 20) e iR x · ξ = e iR cosh ψ e − iR sinh ψ n φ · n θ . (33)The spherical leaves of our ( d − d −
2) dimensions and we mayexpand the second factor at the RHS in terms of generalized spherical harmonics Y l,M (n φ ) dependingon ( d −
2) angles, where M is a multi-index encoding ( d −
3) “magnetic” indices in addition to l ; inthe standard two-dimensional case (that corresponds to d = 4) Y l,M (n φ ) are the standard sphericalharmonics Y l,m ( φ , φ ). The starting point is as usual the well-known expansion of the exponential interms of Gegenbauer polynomials C νl and Bessel functions (see e.g. [24] Eq. (7.10;5)): e iγz = (cid:18) z (cid:19) ν Γ( ν ) ∞ X l =0 i l ( ν + l ) C νl ( γ ) J ν + l ( z ) . (34)The second step is to set ν to d − and take advantage of the known (see e.g. [25] Eq. (B.12)) expansionof C νl (n φ · n θ ) as a sum of the generalized spherical harmonics: e − iR sinh ψ n φ · n θ = (2 π ) d − ∞ X l =0 i − l ( R sinh ψ ) − d − J l + d − ( R sinh ψ ) X M Y l,M (n θ ) Y ∗ l,M (n φ ) . (35)By inserting this expression in the Euler representation (30) we get that( x · ξ ) − d − + iq = 2 π (cid:18) π sinh ψ (cid:19) d − ∞ X l =0 i − ( l + d − − iq ) Γ (cid:0) d − − iq (cid:1) X M Y l,M (n θ ) Y ∗ l,M (n φ ) Z ∞ dRR R − iq e iR cosh ψ J l + d − ( R sinh ψ ) . (36)The integral at the RHS is the Mellin transform of a product that can be evaluated by the Mellin-Barnesintegral. This is a way to directly check [24] Eq. (7.8;9): Z ∞ dRR R − iq e iR cosh ψ J l + d − ( R sinh ψ ) = i l + d − − iq Γ (cid:18) l + d − − iq (cid:19) P − l − d − − + iq (cosh ψ ) (37)that holds for Re ( l + d − − iq ) >
0, that is for all l ≥ Im q > − d − . Therefore( x ( ψ, φ ) · ξ ( θ )) − d − + iq == 2 π (cid:18) π sinh ψ (cid:19) d − ∞ X l =0 Γ (cid:0) l + d − − iq (cid:1) Γ (cid:0) d − − iq (cid:1) P − l − d − − + iq (cosh ψ ) X M Y ∗ l,M (n θ ) Y l,M (n φ ) == a ( q ) ∞ X l =0 X M Y ∗ l,M (n θ ) Z iq,l,M ( ψ, n φ ) (38)where a ( q ) = (2 π ) d − Γ ( − iq )Γ (cid:0) d − − iq (cid:1) . (39)and Z iq,l,M ( ψ, n φ ) = Γ (cid:0) l + d − − iq (cid:1) Γ ( − iq ) (sinh ψ ) − d − P − l − d − − + iq (cosh ψ ) Y l,M (n φ ) . (40)For q ≥
0, the latter are the orthonormal eigenmodes (see e.g. [5]; note a slight modification in ourdefinition of Z ) associated with the spherical coordinates. If we require ( x · ξ ) − d − − iq to bear simultaneously the same expansion, the condition of validity becomes | Im q | < d − . .4 Asymptotics. Using the coordinates (4) the ”boundary” at infinity of the manifold H d − is attained for either r → r → ∞ and the behaviour of the modes in these limits is of importance. Actually, only the behaviourat small r matters while the limit r → ∞ is rather an artifact of the coordinate system (see Fig. 4).Eq. (31) together with the behaviour of K iq ( κr ) at small r give the following asymptotics:( x ( r, x) · ξ ) − d − + iq ∼ r d − − iq ( ξ (x) · ξ ) − d − + iq + A ( q ) r d − + iq δ ( ξ (x) − ξ ) . (41)where A ( q ) = (2 π ) d − − iq Γ ( − iq )Γ (cid:0) d − − iq (cid:1) . (42)By sending the point x that appears in Eq. (24) to the ”boundary” at infinity of the manifold H d − one gets naturally the two-point kernel ( ξ · ξ ′ ) − d − + iq on the asymptotic cone. As before, this kerneladmits useful Euler and Fourier representations:( ξ · ξ ′ ) − d − + iq = 1Γ (cid:0) d − − iq (cid:1) Z ∞ dRR e − R ξ · ξ ′ R d − − iq (43)= 2 iq Γ ( iq )(2 π ) d − Γ (cid:0) d − − iq (cid:1) Z dκ κ − iq e − iκ · ( η − η ′ ) . (44)Let us discuss an immediate application of this definition by establishing the relation between modeswith negative and positive values of q . Suppose that ( x · ξ ) − d − − iq be superposition of the modes( x · ξ ′ ) − d − + iq , with q > x · ξ ) − d − − iq and R dµ ( ξ ′ ) ( ξ · ξ ′ ) − d − − iq ( x · ξ ′ ) − d − + iq are proportional.This can be explicitely shown by using the integral representations (31) and (44) in order to performthe latter integration:( x · ξ ) − d − − iq = 1 A ( q ) Z dµ ( ξ ′ ) ( ξ · ξ ′ ) − d − − iq ( x · ξ ′ ) − d − + iq (45) H d − . We are now ready to construct a basis for the Hilbert space L ( H d − , dµ ) that can be used to study”ordinary” quantum theories on the Lobatchevski space H d − . The word ordinary refers to theorieswhere the common wisdom and the standard tools of quantum mechanics, including the probabilisticinterpretation, apply. As we have already said, there are also non-standard theories, corresponding tothe allowed imaginary values of q . These theories will be examined elsewhere. The modes (24) correspond to eigenvalues of the continuous spectrum of the Laplacian. Of course modescorresponding to distinct values of q are orthogonal, because of the self-adjointness of the Laplacian.To find the correct normalization we study the distributional kernel constructed by taking the scalarproduct of two modes (24) for q and q ′ are real. It follows that (see Section 5.3) F q,q ′ ( ξ, ξ ′ ) = Z dµ ( x ) ( x · ξ ) − d − − iq ( x · ξ ′ ) − d − + iq ′ =11 1 n ( q ) (cid:20) δ ( q − q ′ ) δ ( η − η ′ ) + 1 A ( q ) ( ξ · ξ ′ ) − d − − iq δ ( q + q ′ ) (cid:21) (46)while the normalization reads1 n ( q ) = a ( q ) a ( − q ) = 2 πA ( q ) A ( − q ) = (2 π ) d − Γ ( iq ) Γ ( − iq )Γ (cid:0) d − + iq (cid:1) Γ (cid:0) d − − iq (cid:1) . (47)We may recall that a ( q ) is given in (39) and A ( q ) in (42). This result is indeed consistent with (45). It is appropriate to introduce normalized modes and the conjugate ones as follows. For q ≥
0, let usdefine ψ iq ( x, ξ ) = ( x · ξ ) − d − + iq a ( q ) , ψ ∗ iq ( x, ξ ) = ( x · ξ ) − d − − iq a ( − q ) (48)so that Z dµ ( x ) ψ ∗ iq ( x, ξ ) ψ iq ′ ( x, ξ ′ ) = δ ( q − q ′ ) δ ( ξ, ξ ′ ) . (49)The set { ψ iq ( x, ξ ) , q ≥ , ξ on the absolute } is then an orthonormal family of modes for the Hilbertspace L ( H d − , dµ ). The relation of these waves with the more commonly used waves in sphericalcoordinates is given by Eq. (38). The following suggestive expansion in term of generalized sphericalharmonics is worth to be mentioned: ψ iq ( x ( ψ, φ ) , ξ ( θ )) = ∞ X l =0 X M Y ∗ l,M (n θ ) Z iq,l,M ( ψ, n φ ) (50)and conversely Z iq,l,M ( ψ, n φ ) = Z Y l,M (n θ ) ψ iq ( x ( ψ, φ ) , ξ ( θ )) dµ ( θ ) (51)Our normalized plane waves are therefore superpositions of the spherical waves Z iq,l,M ( ψ, n φ ) withweights which are themselves normalized generalized spherical harmonics evaluated at the direction ofthe vector ξ ( θ ) parametrizing the plane wave itself.The advantage of the waves (48) is their independence on the choice of particular coordinate systemsand, above all, their maximal symmetry. They really encode the symmetry of the Lobatchevski space.Their representations in terms of exponentials given in Sect. (4) also renders feasible calculations thatare otherwise intractable (see below and [19]). Factorization of F q,q ′ ( ξ, ξ ′ ) . Let us insert in Eq. (46) the Fourier representation (31) in the previousexpression and change to the variables R = u , R ′ = u ′ : F q,q ′ ( ξ, ξ ′ ) = R duu du ′ u ′ drr dκ r i ( q − q ′ ) u − iq u ′ iq ′ e iκ · ( η − η ′ ) e − u + u ′ κ − ( u + u ′ )2 uu ′ r Γ (cid:0) d − + iq (cid:1) Γ (cid:0) d − − iq ′ (cid:1) . (52) From (46) it is however seen that the case d = 2 exhibits some pecularities since the term proportional to A ( q ) doesnot vanish for q = 0. κ can be factorized by the changes r = σ uu ′ κ , u = vκ and u ′ = v ′ κ : F q,q ′ ( ξ, ξ ′ ) = ∆ q,q ′ Γ (cid:0) d − + iq (cid:1) Γ (cid:0) d − − iq ′ (cid:1) Z dκ κ i ( q − q ′ ) e iκ · ( η − η ′ ) (53)∆ q,q ′ = Z dσσ σ i ( q − q ′ ) Z dvv v − i q + q ′ e − v − v σ Z dv ′ v ′ v ′ i q + q ′ e − v ′ − v ′ σ . (54) Evaluation of ∆ q,q ′ . q + q ′ = 0 . The second and third integrals at RHS can be evaluated as follows: Z dvv v − i q + q ′ e − v − v σ = 2 − iq + iq ′ (cid:0) σ (cid:1) iq + iq ′ Γ (cid:18) − iq + iq ′ (cid:19) . (55)The integral converges only if Im ( q + q ′ ) >
0. Otherwise it is defined in a generalized sense as ameromorphic function of the complex ( q + q ′ ) variable. It then follows that∆ q,q ′ = Γ ( − iq ) Γ ( iq ) 2 πδ ( q − q ′ ) . (56) q − q ′ = 0 . In this case we can extract a finite contribution by exchanging the integration order inEq. (54): Z dσσ σ i ( q − q ′ ) e − v + v ′ σ = 12 (cid:18) v + v ′ (cid:19) − iq − iq ′ Γ (cid:18) iq − iq ′ (cid:19) . (57)This expression is valid for Im ( q − q ′ ) <
0. Otherwise it is defined in a generalized sense as a meromorphicfunction of the complex ( q − q ′ ) variable. By introducing the variables λ = v ′ v and µ = v + v ′ Eq. (54)becomes ∆ q,q ′ = 12 Γ (cid:18) iq − iq ′ (cid:19) Z dλλ dµµ µ − iq − iq ′ λ iq + iq ′ e − µ = Γ ( iq ) Γ ( − iq ) 2 π δ ( q + q ′ ) . (58)Gathering all terms together finally yields∆ q,q ′ = 2 π Γ ( iq ) Γ ( − iq ) ( δ ( q − q ′ ) + δ ( q + q ′ )) . (59)and Eq. (46) follows from Eq. (53). Let f ( x ) be a smooth function defined on H d − . We define the following transform [20]: f ( x ) → ˜ f ( ξ, q ) = Z dµ ( x ) ( x · ξ ) − d − − iq f ( x ) . (60)Variables paired by the transform are H d − ∋ x ←→ ( ξ, q ) ∈ C + × R . (61)13ince the transformed function is an homogeneous function of the ξ variable, it is enough to take ξ onthe absolute and therefore there are ( d −
1) dimensions on both sides of this pairing. The aim is nowto reconstruct the function f ( x ) in terms of ˜ f ( ξ, q ) by inverting the previous transform. The inversionformula we will heuristically prove in this section is the following : f ( x ) = Z ∞ n ( q ) dq Z dµ ( ξ ) ( x · ξ ) − d − + iq ˜ f ( ξ, q ) (62)where n ( q ) is given in Eq. (47) Let us consider the (formal) integral operatorˆ δ ( x, x ′ ) = Z ∞ dq ′ Z dµ ( ξ ) ψ iq ′ ( x, ξ ) ψ ∗ iq ′ ( x ′ , ξ ) (63)in the Hilbert space L ( H d − , dµ ). Using the coordinates (4), this Hilbert space may be concretelyrealized as a tensor product: L ( H d − , dµ ) = L ( R + × R d − , r − d dr d x) = L ( R + , r − d dr ) ⊗ L ( R d − , d x) . (64)The space L ( H d − , dµ ) is therefore generated by finite linear combinations of factorized functions f ( r, x) = g ( r ) h (x) where the factors are such that g ( r ) ∈ L ( R + , r − d dr ) and h ∈ L ( R d − , d x). Considerone function of this type. Using the integral representation (31) we can express the operator (63) asfollows: Z dµ ( x ′ ) ˆ δ ( x, x ′ ) f ( x ′ ) = r d − (2 π ) d − Z dκ Z d x ′ e iκ · ( x − x ′ ) h (x ′ ) × Z ∞ dq π Γ ( iq ) Γ ( − iq ) K iq ( κr ) Z ∞ dr ′ r ′ K iq ( κr ′ ) g ( r ′ ) r ′ d − . (65)The function r − d − g ( r ) ∈ L ( R + , r − dr ) and this assures convergence of the inner integral. The integralover r ′ and q are then just an instance of the Kontorovich-Lebedev [26] inversion formula, that holdstrue for quite general classes of functions and distributions: g ( r ) r d − = Z ∞ dq π Γ ( iq ) Γ ( − iq ) K iq ( κr ) Z ∞ dr ′ r ′ K iq ( κr ′ ) g ( r ′ ) r ′ d − . (66)The remaining integral in Eq. (65) is then just Fourier inversion formula. Taking finite linear combina-tions we finally get Z dµ ( x ′ ) ˆ δ ( x, x ′ ) f ( x ′ ) = f ( x ) (67)on (a dense subset) of L ( H d − , dµ ). This shows the validity of the inversion formula (62). Another choice would be to define a normalized transform: f ( x ) → ˜ f ( ξ, q ) = Z dµ ( x ) ψ ∗ iq ( x, ξ ) f ( x ) . In this case the inversion formula would have unit weight f ( x ) = Z ∞ dq Z dµ ( ξ ) ψ iq ( x, ξ ) ˜ f ( ξ, q ) . Our above definition (60) follows similar choices of normalization such as the Mehler-Fock or the Kontorovich-Lebedevtransforms, see e.g. [24], Eqs. (3.15; 8-9). Projectors. Representations of the principal series.
At this point we may introduce the integral kernelsΠ q ( x, x ′ ) = Z dµ ( ξ ) ψ iq ( x, ξ ) ψ ∗ iq ( x ′ , ξ ) . (68)It is immediately seen that the kernels Π q satisfy the following projector relations: Z dµ ( x ′′ ) Π q ( x, x ′′ )Π q ′ ( x ′′ , x ′ ) = δ ( q − q ′ ) Π q ( x, x ′ ) . (69)The operator Π q ( x, x ′ ) is the projector on the subspace of a given q and as such Π q ( x, x ′ ) is a positive-definite kernel. Starting from the projector Π q ( x, x ′ ) we can construct a representation of the invariancegroup of H d − in the usual way: let us consider the space of smooth rapidly decreasing functions on S ( H d − ) endowed with the left regular action ( T g f )( x ) = f ( g − x ), g ∈ SO (1 , d − h f, f ′ i = Z H d − f ∗ ( x )Π q ( x, x ′ ) f ′ ( x ′ ) dxdx ′ (70) S ( H d − ) is a pre-Hilbert space. By quotienting and completing we obtain a Hilbert space carrying anirreducible unitary representation of the Lorentz group labeled by the real, non-negative parameter q .The set of such representations is called the principal series.Consider now any invariant (possibly positive-definite) two-point kernel W ( x, x ′ ) on the hyperboloid H d − . If we assume suitable growth properties of W at infinity we may expect that it can be decomposedas a superposition of the projectors Π q ( x, x ′ ) (representations of the principal series): W ( x, x ′ ) = Z ∞ ρ ( q )Π q ( x, x ′ ) dq. (71)In particular the kernel ˆ δ ( x, x ′ ), as defined in Section (6.2), determines the standard L Hilbert producton the hyperboloid H d − and the so-called regular representation:ˆ δ ( x, x ′ ) = Z ∞ Π q ( x, x ′ ) dq (72)which can be equivalently viewed as the decomposition of the regular representation into representationsof the principal series (Plancherel’s formula): Z dµ ( x ) f ∗ ( x ) g ( x ) = Z ∞ n ( q ) dq dµ ( ξ ) ˜ f ∗ ( ξ, q )˜ g ( ξ, q ) . (73) Evaluation of Π q ( x, x ′ ) . The explicit evaluation of the integral (68) is most easily done by integratingon the spherical basis (18) of the absolute ( ξ = 1). The result must depend only on the scalar product x · x ′ : without loss of generality we may choose x = (1 , , . . . ,
0) and x ′ = (cosh φ, − sinh φ, . . . ,
0) sothat x · x ′ = cosh φ. . Since x · ξ = 1 , and x ′ · ξ = cosh φ + cos θ sinh φ , the integral (68) becomes2 π d − Γ (cid:0) d − (cid:1) n ( q ) Z π dθ (cosh φ + cos θ sinh φ ) − d − − iq (sin θ ) d − == 2 π d − Γ (cid:0) d − (cid:1) n ( q )Γ (cid:18) d − (cid:19) d − (sinh φ ) − d − P − d − − + iq (cosh φ ) (74)15o that the final result reads Π q ( x, x ′ ) = ω d − n ( q ) P ( d ) − d − + iq ( x · x ′ ) . (75)The factor ω d − is the hypersurface of the sphere S d − (see Footnote 1) and n ( q ) is given by (47). Theresult is expressed in terms of the so-called generalized Legendre function: P ( d ) − d − + iq ( z ) = 2 d − Γ (cid:18) d − (cid:19) ( z − − d − P − d − − + iq ( z ) (76)where P µν ( z ) denotes the usual Legendre function of the first kind, defined and one-valued in the complex z -plane cut on the reals from −∞ to 1 [24]. The function ( z − α is defined and one valued on thesame cut complex plane (with the natural definition for real z >
1) so that the function P ( d ) − d − + iq ( z ) isregular at z = 1 and its cut goes from z = −∞ to z = − As a first application of the general construction displayed in the previous sections, we discuss hereQuantum Field Theory in the universe of Milne.Figure 5:
A view of the Milne universe. The spacetime curvature of this model is zero. Surfaces ofconstant time are copies of the Lobatchevski space H d − . The straight lines represented in figure are thegeodetic wordlines of particles having constant spatial coordinates. Milne’s universe is a simple model of an expanding universe obtained as a solution of the Einsteinequations in vacuo with zero spacetime curvature and nonzero spatial curvature. There is however noneed of General Relativity to talk about this model: Fig. 5 shows how the Milne universe can beconstructed as a foliation with Lobatchevskian leaves of the interior of the the future cone of an event(the ”Big Bang”) of a Minkowski spacetime. A quantitative description is very easy: let X µ denote16he coordinates of an event X of a ( d -dimensional) Minkowski spacetime. Consider the future cone ofthe origin of the chosen inertial system (as in Fig. 5) and introduce there the noninertial coordinatesystem: X µ ( t, x ) = t x µ , µ = 0 , . . . , d − , (77)where x · x = 1, i.e. x ∈ H d − . Milne’s line element is simply Lorentz invariant interval of the ambientspacetime expressed in the coordinates (77): ds = (cid:0) dX (cid:1) − (cid:0) dX (cid:1) − . . . − (cid:0) dX d − (cid:1) = dt − t dl d − . (78)Milne’s universe has therefore the structure of a warped product of a half line (the cosmic time) timesthe Riemannian manifold H d − ; the warping function is just the cosmic time t (see e.g [21]).As old as it is, this model has never become obsolete and disappeared from the scientific and cosmologicaldebate; its predictions are in surprisingly good agreement with the current cosmological observations[23]. A recent appearance of Milne’s model in M-theory is also worth to be mentioned [22].The simple question we ask ourselves here is that of finding the expansion of the exponential planewaves of the Minkowski spacetime on the base of modes (48). This is a preliminary step to studyquantum theories on the Milne universe in much the same way as finding an expansion of plane waves inspherical harmonics is a starting point in studying spherical symmetric potentials in ordinary quantummechanics. As an immediate bonus of our approach there is an easy construction the Wightman vacuumin Milne’s coordinates. We give here a new and we believe simple approach to solve this old problem[27]. Our approach may also be used to study other vacua, as for instance the thermal vacuum at agiven temperature. To fix ideas and notations, and also to put the results in perspective, it is useful to begin by shortlyreviewing the theory of a Klein-Gordon quantum field of mass m on a d -dimensional Minkowski spacetime M d with inertial coordinates ( X , X , . . . , X d − ): (cid:0) (cid:3) + m (cid:1) φ = 0 . (79)It is enough to solve the Klein-Gordon equation for the two-point vacuum expectation value W m ( X − Y ) = h Ω , φ ( X ) φ ( Y )Ω i . (80)The truncated n -point functions are assumed to vanish and the two-point function encodes all theinformation necessary to fully reconstruct the theory. Actually, for Klein-Gordon fields satisfying theWightman axioms the vanishing of the truncated n -point functions is not an assumptions but it is aresult [28] and a Klein-Gordon field is necessarily free. Eq. (79) is most easily solved in Fourier space,where it becomes algebraic: ( p − m ) ˜ W m ( p ) = 0 . (81)There are infinitely many inequivalent solutions and a criterium is to be found to select one amongthem. Assumption of positivity of the spectrum of the energy-momentum operator is the most popularpossibility and leads to f W m ( p ) ≃ θ ( p ) δ ( p − m ) , (82)where θ ( p ) denotes Heaviside’s step function. Inversion gives W m ( X, X ′ ) = h Ω , φ ( X ) φ ( X ′ ) Ω i = 1(2 π ) d − Z R d dp e − ip · ( X − X ′ ) θ ( p ) δ ( p − m ) . (83)The choice made in Eq. (82) selects the Wightman vacuum Ω which is uniquely characterized by thepositivity of the spectrum of the energy operator in any Lorentz frame. This property is equivalent to17ertain analyticity properties of the correlation functions that can be deduced by direct inspection ofEq. (83). One sees that the Wightman function W can be uniquely extended to a function holomorphicin the past tube T − as a function of the difference variable ( X − X ′ ) where T − = { X + iY, Y > , Y < } . (84)If we consider the plane waves on the mass shell we see that positive frequency waves exp( − i p p + m X + i~p · ~X ) admit a natural continuation in the past tube where they are decreasing while negative fre-quency waves exp( i p p + m X − i~p · ~X ) may be considered for complex events belonging to the futuretube T + = { X + iY, Y > , Y > } . (85)These properties are the link between the standard choices in the canonical quantization procedure andthe analyticity structure of the waves and the Wightman function. Let us consider therefore a Minkowskian plane wave on the mass shell p = m , p > ip · X ) = exp ( it p · x ) . (86)The wave is naturally extended in the future tube T + ; in particular we will consider the complexevents Z µ ( τ, x ) = τ x µ , τ = t + is, Im τ = s > , (87)that belong to the future tube. Similarly the wave exp ( − iτ p · x ) is naturally extended to the pasttube and in particular we will consider the events Z µ ( τ, x ) ∈ T − that belong in the past tube forIm τ < F ± q ( τ, ξ, p ) Z H d − dµ ( x ) ( x · ξ ) − d − − iq e ± iτ p · x . (88)Here F ± q are defined respectively for X ( τ, x ) ∈ T ± , i.e. Im τ > τ <
0. The Lorentz invarianceof the measure implies that F ± q may depend only on the invariant ( ξ · p ). Homogeneity of the integrandthen gives that F ± q ( τ, ξ, p ) = f ± q ( τ ) ( ξ · p ) − d − − iq . The steps to explicitly compute the function f are summarized at the end of this section; here is theresult: F + q ( τ ′ , ξ, p ) = iπ (cid:18) πimτ ′ (cid:19) d − (cid:18) p · ξm (cid:19) − d − − iq e − πq H (1) iq ( mτ ′ ) , Im τ ′ > F − q ( τ, ξ, p ) = πi (cid:18) πimτ (cid:19) d − (cid:18) p · ξm (cid:19) − d − − iq e πq H (2) iq ( mτ ) , Im τ < H (2) iq ( mτ ) ∝ e − imτ when τ → ∞ . As expected [27] the Hankel function H (2) iq plays the roleof the positive frequency solution of the Klein-Gordon equation when separated in the coordinates (77).The result in our construction comes out automatically from the known analyticity properties (84) ofthe Minkowskian waves. 18nversion is obtained by means of Eq. (62); this yields the expansion of the exponential plane wave (86)in terms of the wavefunctions (24); the ( d −
1) parameters p are described by the ( d −
2) degrees offreedom of ξ on the absolute plus one degree of freedom of the q variable:exp iτ ( p · x ) = iπ (cid:18) πimτ (cid:19) d − Z ∞ n ( q ) dq e − πq H (1) iq ( mτ ) × Z dµ ( ξ ) (cid:18) p · ξm (cid:19) − d − − iq ( x · ξ ) − d − + iq (91)and similarly for the other wave. The integration over the absolute at the RHS can be performed (seeEqs. (68) and (75)) and there results a one-dimensional integral expansion over the projectors Π q asfollows: e iτ ( p · x ) = iπ (cid:18) πimτ (cid:19) d − Z ∞ dq e − πq H (1) iq ( mτ ) Π q (cid:16) p · xm (cid:17) , Im τ > ,e − iτ ( p · x ) = − iπ (cid:18) πimτ (cid:19) d − Z ∞ dq e πq H (2) iq ( mτ ) Π q (cid:16) p · xm (cid:17) , Im τ < . (92)The details of the calculation are given at the end of the present section. As an immediate bonus theseformulae provide an expansion of the Wightman canonical Klein-Gordon quantum field theory expressedin Milne’s coordinates. Indeed, since the theory is completely encoded in the two-point function andthe invariant measure on the mass shell is proportional to the invariant measure dµ on H d − , we caninsert Eqs. (92) into Eq. (83) and use Eq. (69) to get W m ( X, X ′ ) = 1(2 π ) d − Z R d dp e − ip · ( X − X ′ ) θ ( p ) δ ( p − m )= 14 π ( τ τ ′ ) − d − Z dq H (2) iq ( mτ ) H (1) iq ( mτ ′ ) Π q ( x · x ′ ) , (93)where Im τ < , Im τ ′ >
0. As a verification, let us show that the theory is indeed canonical by computingthe following equal time commutation relations:[ φ ( t, x ) , π ( t, x ′ )] == πmt Z dq (cid:16) H (2) iq ( mt ) H (1) iq ′ ( mt ) − H (2) iq ′ ( mt ) H (1) iq ( mt ) (cid:17) Π q ( x · x ′ )= i δ ( x, x ′ ) , (94)where δ ( x, x ′ ) is understood in the sense of Section (6.2). Comments and details
Evaluation of F q . In this section we show that the expressions of F q as given in Eqs. (89) and (90)hold true. To this purpose, let us parametrize x as in Eq. (4) and similarly write the momentum vector p as follows: p = p = m λ (1 + κ + λ ) p i = m λ κ i p d − = m λ (1 − κ − λ ) . (95)This yields exp ip · X = exp iτ p · x = exp imτ rλ (cid:2) (x − κ ) + r + λ (cid:3) . (96)By using the integral representation (30) and performing the Gaussian integral one gets F + q = (2 π ) d − i − d − − i q Γ (cid:0) d − + i q (cid:1) Z dRR R d − + iq [ − i ( T + R )] d − Z drr r − d − e ir ( R + T )2 + iλ T r + i TR ( κ − η )22 r ( T + R ) (97)19here the integral over R is along an arbitrary straight half-line such that 0 < Arg( R ) < π ; to simplifynotations we have put T = mλ ( t + is ) = mτλ . (98)The evaluation of the remaining integrals is simplified by the introduction of a new complex variablereplacing the r -coordinate: given R and T in the upper complex plane we define v = (cid:18) R + 1 T (cid:19) r, − π < Arg( v ) < . (99)Let us use the freedom in choosing the integration path in the complex R -plane and take Arg( R ) =Arg( T ) (path γ ); it follows that Arg( v ) = − Arg( R ) (path ˆ γ ) and the previous expression becomes F + q = i − d − − i q Γ (cid:0) d − + iq (cid:1) (cid:18) iπT (cid:19) d − Z γ dRR R iq Z ˆ γ dvv v − d − e iRTv + iλ R + T )2 Rv e i ( κ − η )22 v . (100)In this expression we interchange the integration order, introduce the real variable S = Rv and theinversion u = λv (that implies 0 < Arg( u ) = Arg ( R ) < π ): F + q = ( iλ ) − d − − i q Γ (cid:0) d − + iq (cid:1) (cid:18) iπT (cid:19) d − Z γ duu u d − + iq e i ( κ − η )2+ iλ λ u Z ∞ dSS S iq e iST + iλ T S . (101)In the previous expression we recognize (see Eq. (30)) the scalar (cid:18) p · ξm (cid:19) − d − − iq = i − d − − iq Γ (cid:0) d − + i q (cid:1) Z γ duu u d − + i q e iu » ( κ − η )22 λ + λ – (102)and the representation (32) of the Bessel function K iq ( z ). Taking into account Eq. (98) and the phaseof τ we get F + q ( τ, ξ, p ) = 2 (cid:18) πimτ (cid:19) d − (cid:18) p · ξm (cid:19) − d − − iq K iq ( − imτ )= iπ (cid:18) πimτ (cid:19) d − (cid:18) p · ξm (cid:19) − d − − iq e − πq H (1) iq ( mτ ) , Im τ > . (103)This is the result given in Eq. (89). The expression (90) is obtained by complex conjugation, as can bereadily inferred from (88). This completes the proof of the expansions given in Eqs. (92). In this section we will apply our construction to the open de Sitter universe. The simplest possibledescription of this model is as follows: consider a ( d + 1)-dimensional Minkowski spacetime with innerproduct X · Y = X Y − X Y − . . . − X d Y d , (104)and the embedded Lorentzian d -dimensional manifold dS d with equation dS d = { X : X = X · X = − } ; (105)20his manifold models the whole de Sitter universe (the grey manifold in Fig. 6). Let us consider nowthe intersection of dS d with the future cone V + of a given event. Specifically, we consider the futureregion of the event O = (0 , . . . ,
1) (the ”origin”):Γ + = Γ +O = { O + V + } ∩ dS d = { X ∈ dS d : ( X − O) > , X > } . (106)Γ + can be thought as a warped manifold, foliated with ( d − H d − .Precisely, we have the following construction: X ( t, x ) = (cid:26) X i = sinh t x i , i = 0 , . . . , d − X d = cosh t (107)where t >
0, and x = 1, i.e. x ∈ H d − . In these coordinates the de Sitter metric is written asfollows: ds = n(cid:0) dX (cid:1) − (cid:0) dX (cid:1) − . . . − (cid:0) dX d (cid:1) o(cid:12)(cid:12)(cid:12) dS d = dt − sinh t dl d − ; (108)this is a warped product with warping function a ( t ) = sinh t ; in cosmology such a metric defines aparticular instance of a Friedmann-Robertson-Walker hyperbolic universe. We will call the region Γ + parametrized with the cosmic time given chosen in (107) the open de Sitter universe (Fig. 6).Figure 6: The open de Sitter model is the interior of the future cone Γ +X of any given event on thede Sitter manifold (in the figure X= O ). The surfaces of constant time are copies of the Lobatchevskispace H d − . The geodesic worldlines of particles having constant spatial coordinates are branches ofhyperbolae. As briefly mentioned in the Introduction, the open de Sitter universe was popular in the mid-ninetieswhen it was playing a central role in open models of inflation [5, 7, 8]. These models were abandonedwhen the microwave background fluctuation measurement showed that our universe is most likely flat.From both the physical and mathematical viewpoints, correctly quantizing a field in the open de Sittermanifold is an arduous task. In particular, a naive application of the procedures of canonical quantizationgives a ”wrong” result [9, 13] in the sense that when those procedures are applied to the open de Sittermanifold one does not en up with the standard de Sitter invariant vacuum [1, 10, 12, 30]. One reason isthat the spatial manifold H d − represented in Fig. 6 is a complete Cauchy surface for Γ + that is its ownCauchy development (i.e. its future), but it fails to be so for the whole de Sitter manifold where it cannotbe used to set up the usual canonical quantization. In [9] this difficulty was circumvented by finding anextension of the modes, originally defined only in the physical region Γ + , to the whole de Sitter manifoldand applying the canonical quantization there. The major drawback of this approach is that it is strictly21imited to the de Sitter geometry. The method used in [13] was not based on canonical quantization butalso necessitated the extension of the open de Sitter manifold to the whole manifold. These calculationsthus are flawed by the necessity, in order to explain local events, to use (possibly complex) extentions ofspace-time to classically unreachable regions, whose a priori existence or non-existence in more generalsituation cannot be simply established. This drawback is avoided in the present approach since by choicewe work solely in the physical space, here the open de Sitter manifold.In the following we will perform in the open de Sitter universe the same Fourier-type analysis alreadydescribed in the Milne’s case. As before, an immediate bonus is the complete resolution of the standardde Sitter QFT solely in terms of the modes of the physical region Γ + . One valuable aspect of themethod that we use is that it can also in principle be used to analyze observational data. Our discussionis limited here to the square-integrable case. Theories that involve modes that are not square-integrablecase as well as the implications of our analysis for general canonical quantum theory will be the objectof separate studies. In the study of quantum field theory on a given background, the complexification of the underlyingmanifold plays a central role in either the study of general properties (as for instance the PCT theorem)or in the construction of concrete models, which are essentially based on Euclidean methods, i.e. on theanalytical continuation in the time coordinate. The de Sitter case is no exception to this rule.The complex de Sitter spacetime can be described as a complex manifold embedded in the ( d + 1)-dimensional complex Minkowski spacetime with equation: dS ( c ) d = { Z = X + iY ∈ M ( c ) : Z · Z = − } (109)As in the flat case (see Eqs. 84 and 85) we introduce the tubular domains T + and T − : T + d = { Z ∈ dS ( c ) d : Y · Y > Y > } , (110) T − d = { Z ∈ dS ( c ) d : Y · Y > Y < } , (111)defined as the intersection of the complex Sitter manifold with the the forward and backward tubes inthe ambient complex Minkowski spacetime. These domains arise in connection with the thermal physicalinterpretation of de Sitter QFT [30, 31]. More precisely, assumption of analyticity of the correlationfunctions in (generalizations of) these domains give rise to the KMS property and therefore to thethermal interpretation: a de Sitter geodetic observer perceive a thermal bath of ”particles”.As in the Milne case, special attention will be devoted to the complex events Z ( τ, x ) = (cid:26) Z i = sinh τ x i , i = 0 , . . . , d − Z d = cosh τ (112)where only the cosmic time has been complexified. The complex coordinate τ = t + is is defined in thestrip Im τ = s ∈ ( − π, π ) (113)of the complex τ -plane. Events Z ( τ, x ) such that Im τ ∈ (0 , π ) belong to T + ; events Z ( τ, x ) such thatIm τ ∈ ( − π,
0) belong to T − .An alternative description of these events can be given by using the variable u = Z d = cosh τ . Theimage of both T + and T − in the u -variable is the cut plane∆ = C \ { ( −∞ , − ∪ [1 , ∞ ) } (114)22e are then led to consider the mappings u → Z ± ( u, x ) defined in ∆ × H d − as follows : Z ± ( u, x ) = ( Z l = ± i (cid:0) − u (cid:1) x l , l = 0 , . . . , d − Z d = u (115)It is readily seen that Z + ( u, x ) ∈ T + and Z − ( u, x ) ∈ T − . Indeed, consider for instance the mappingFigure 7: Two copies (left and right of the figure) of the cut plane ∆ of the complex u variable showingthe cuts at ( −∞ , − , + ∞ ) . The images of the de Sitter manifold correspond to u infinitesimallyclose to the cuts, at the place where the manifolds are drown. Left: points of ∆ are mapped through Z − ( u, x ) into T − . The imaginary part of these points is contained in V − (not represented). The copyat the lower right is the one which corresponds to the open de Sitter space that we solely consider in thepresent paper. Right: points of ∆ are mapped through Z + ( u, x ) into T + . The imaginary part of thesepoints is contained in V + (not represented). The copy at the upper right is the one which correspondsto the open de Sitter space that we solely consider in the present paper. u → Z + ( u, x ). When u is real and such that − < u <
1, the events Z + ( u, x ) evidently belong to thefuture tube T + because for such events Im Z + · Im Z + > Z = x Im (cid:0) i √ − u (cid:1) >
0. Now, thefirst of these conditions holds true for any u ∈ ∆. On the other hand, since the zeros of Im (cid:0) i √ − u (cid:1) all belong to the cuts of ∆ one also has thatIm Z + ( u, x ) = x Im (cid:16) i p − u (cid:17) > u ∈ ∆ , (116)and the result follows. Note that we have Z − ( u, x ) = [ Z + ( u, x )] ∗ (see App. (A), Footnote (7)). Let us consider an eigenfunction φ of the de Sitter Klein-Gordon operator: (cid:3) φ + m φ = 0 (117)here (cid:3) denotes the de Sitter-d’Alembert operator (i.e. the Laplace-Beltrami operator relative to thede Sitter metric). The usual approach to such an equation in curved spacetimes consists in trying tosolve it by separating the variables in a suitably chosen coordinate systems; here in the system (107). The change of complex variables τ → u maps the strip Im τ ∈ ( − π, π ) onto a two sheeted manifold defined by the cutsof ∆. With this mapping we could have considered a function Z ( u, x ) that coincides with Z + ( u, x ) on a sheet and with Z − ( u, x ) on the other sheet. Since the de Sitter global waves are defined either in T + or in T − it is simpler to use the(one-sheeted) cut-plane ∆ and two different functions mapping ∆ to T + and to T − . We use this viewpoint throughoutthe present section.
23f we do that and factorize the wave φ by separating the variables according with the reference system(107) φ ( X ) = f ( t ) ψ iq ( x, ξ ) (118)the time-dependent factor f ( t ) is required to satisfy the equation:1(sinh t ) d − ∂∂t (sinh t ) d − ∂f∂t + 1(sinh t ) "(cid:18) d − (cid:19) + q f + m f = 0 . (119)There is also the possibility to introduce global waves in a coordinate - independent way [12, 30] by usingthe embedding of the de Sitter hyperboloid in the Minkowski ambient spacetime. Their construction isidentical to that of the spatial wavefunctions (24) with an important difference: they are singular on( d − dS d . This difficulty can be overcome by moving to thecomplexification of the de Sitter spacetime (109). The physically relevant global waves can be definedas the functions Const ( Z · Ξ) − d − + iν (120)where, as before, Ξ = (Ξ , . . . , Ξ d ) belong to future lightcone in the ambieent spacetime, i.e. it is afuture directed null vector of the ambient space (Ξ · Ξ = 0 and Ξ > ν is a complexnumber. The physical values it may take are real, or purely imaginary with | ν | ≤ d − , and correspondto m = (cid:18) d − (cid:19) + ν ≥ . (121)The waves (120) are analytic for Z in the tubular domains T + or T − of dS ( c ) d , defined in (111). Theseanalyticity properties are the counterpart in the de Sitter universe of the spectral condition of theMinkowski case [30, 31].Figure 8: The most convenient choice to represent the absolute of the open de Sitter space (shown atthe right) is by a two-component manifold, labeled here by the two values of ǫ = ± . Each component isa copy of the Lobatchevski space H d − , similar to the mass shell of the Minkowski case. Given the ( d +1)-dimensional vector Ξ = (Ξ , . . . Ξ d ) as above the d -dimensional vector (Ξ , . . . , Ξ d − ) istimelike and forward directed. One has that (Ξ , . . . , Ξ d − ) = | Ξ d | ( a , . . . , a d − ) where a = 1. There isno loss of generality in setting | Ξ d | = 1; this is indeed another possible choice to represent the absoluteof the ambient spacetime, and indeed the most convenient for our purposes. This manifold has twodisconnected components: Ξ → ( a, ǫ ) , Ξ = ( a , . . . , a d − , ǫ ) , ǫ = ± Z ( τ, x ) · Ξ( a, ǫ ) = x · a sinh τ − ǫ cosh τ, (123)has an imaginary part which does not vanish for events strictly within T + or within T − . The waves(120) are therefore globally (but separately) well-defined in both these domains.24 epresentation of the plane waves. The embedding of the de Sitter hyperboloid in the Minkowski ambient spacetime is the foundation ofthe construction of the global waves. The same embedding leads naturally [21, 19] to an integral repre-sentation of the de Sitter waves (120) in terms of the Minkowski coordinates. With a convenient choiceof phase and normalization, the waves (120) may be expressed in the forward tube as follows:( − iZ · Ξ) − d − + iν = 1Γ (cid:0) d − − iν (cid:1) Z ∞ dRR R d − − iν e iR ( Z · Ξ) , Z ∈ T + . (124)The strictly positive imaginary part of Z · Ξ guarantees the proper definition of the wave as well (seeabove) as well as the convergence of the integral at infinity. The integral converges also at the originprovided | Im ν | < d − .A similar representation holds in T − :( iZ · Ξ) − d − + iν = 1Γ (cid:0) d − − iν (cid:1) Z ∞ dRR R d − − iν e − iR ( Z · Ξ) , Z ∈ T − . (125)Now it the strictly negative imaginary part of Z · Ξ that makes the integral converge.
In the following we want to provide an expansion of de Sitter waves in terms of the eigenmodes of thehyperbolic Laplacian. As before the relevant expansion can be obtained by by computing the followingintegral transform F ± q,ν ( τ, ξ, a, ǫ ) = Z H d − dµ ( x ) ( x · ξ ) − d − − iq ( ∓ iZ ( τ, x ) · Ξ( a, ǫ )) − d − + iν . (126)The integral is well-defined both for 0 < Im τ < π (corresponding to F + ) and − π < Im τ < F − ) so as to avoid the vanishing of X ( τ, x ) · Ξ( a, ǫ ). A glance to the large x ( r, x) behaviorof (126) that corresponds to r →
0, shows that the simultaneous convergence of F ± q,ν and F ± q, − ν is guar-anteed by the condition | Im ν | < , since we consider only real values of q . This condition sets a lowerbound of the masses of the field theories that are covered by the present treatment.As before SO (1 , d −
1) invariance tells us that result must be function of the scalar product ξ · a .Homogeneity then implies that F ± q,ν ( τ, ξ, a, ǫ ) = f ± q,ν ( τ, ǫ ) ( ξ · a ) − d − − iq . (127)The functions f ± q,ν ( t, ǫ ) are therefore the relevant solution of equation (119) that agree with the spectralcondition described above. We start by discussing the two-dimensional case, which deserves special consideration because of itssimplicity. The spatial manifold H is one-dimensional and can be parametrized by an hyperbolic angle v : x = (cosh v, sinh v ) , dµ ( x ) = dv (128)(i.e. r = e − v in Eq. (4)). Labeling the modes of the Laplacian is also quite simple: the spatial absolutehas only two possible direction; consequently, the spatial momentum vector ξ can take only two discretevalues: ξ l = (1 , −
1) and ξ r = (1 ,
1) so that ξ l · x = cosh v + sinh v = e v and ξ r · x = e − v .25s regards the plane waves, they are labeled as in Eq. (122) by the discrete variable ǫ = ± a ; the latter may in turn be also parameterized by an hyperbolic angle a = (cosh w, sinh w ) so that x · a = cosh( v − w )The computation of the scalar product is then straightforward F ± q,ν ( τ, ξ l , a, ǫ ) = f ± q,ν ( τ, ǫ )( ξ l · a ) − iq , F ± q,ν ( τ, ξ r , a, ǫ ) = f ± q,ν ( τ, ǫ )( ξ r · a ) − iq ,f ± q,ν ( τ, ǫ ) = Z + ∞−∞ dw e − iqw ( ∓ i cosh w sinh τ ± iǫ cosh τ ) − + iν . (129)The two-dimensional case also provides an easy direct check of the inversion formula. Indeed, with theabove choice of coordinates, it simply amounts to Fourier inversion. Expression of f in terms of Legendre functions. The functions f ± q,ν ( τ, ǫ ) are completely charac-terized by the integral representation (129). They can be however expressed in terms of the associatedLegendre functions. We use here and in the following the notations and the conventions of the Bate-man manuscript project [24] with one notable exception (the function Q , see below). In particular theLegendre functions P and Q are assumed to be analytic and one valued on the complex plane cut from z = −∞ to z = 1. The two cases to be considered look however at first rather different, since theintegrand never vanishes for ǫ = −
1, while it become singular along the integration path for ǫ = 1 (for τ real: τ = t > Case ǫ = − . In this case the integral representation (129) already coincides with a well-known integralrepresentation of a Legendre functions of the second kind [24] Eq. (3.7;12): f ± q,ν ( t, ǫ = −
1) = 2 e ± i π ( − iν ) Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) − iν (cid:1) e πq Q iq − − iν (cosh t ) (130) Case ǫ = 1 . Here the integral in (129) may be split into two parts according to the sign of theexpression cosh w sinh τ − cosh τ : one addendum may be evaluated by means of [24] Eqs. (3.7;8) and(3.3;14) and the second by means of [24] Eqs. (3.7;5) and (3.3;13), to yield f ± q,ν ( τ, ǫ = 1) = 2 ie ± i π ( − iν ) sinh πν Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) − iν (cid:1) × h e ∓ πν cosh ( π ( q + ν )) e πq Q iq − + iν (cosh t ) − cosh πq e πq Q iq − − iν (cosh t ) i (131) Expression of f in terms of Legendre functions ”on the cut”. There is a more elegant wayand synthetic way to express the modes in terms of Legendre functions based on the use of the variable u = Z d introduced in Sect. 9.2. This alternative procedure has also the advantage to fully exhibit theunderlying symmetries. The input of the relations (115) of Z into the expressions of the de Sitter waves(124) and (125) gives: f ± q,ν ( u, ǫ ) = Z ∞−∞ dv e − iqv h(cid:0) − u (cid:1) cosh v ± ǫiu i − + iν . (132)The two functions f ± q,ν ( u, ǫ ) are manifestly analytical in the cut plane ∆ introduced in Eq. (114): theterm in square brackets at the RHS vanishes for u = coth v for real v , and therefore the integral iswell-defined for u / ∈ ( −∞ , − ∪ (1 , + ∞ ) with no additional singularity in ∆. We see once more that the26omain ∆ is naturally related to the tuboids of analyticity of the de Sitter waves (120). It is thereforenatural to make use of the following ”Legendre function on the cut” (see App. A): P iq − − iν ( u ) = e ∓ πq P iq − − iν ( u ) , (133) Q iq − − iν ( u ) = e iπ ( − + iν ) iπq e − πq P iq − + iν ( u )Γ (cid:0) − iν + iq (cid:1) − e πq P − iq − + iν ( u )Γ (cid:0) − iν − iq (cid:1) . (134)The upper or lower signs of (133) refer to the imaginary part of u being positive or negative. Thesefunctions are analytic in the cut plane ∆. A brief summary of their properties and symmetries is givenin Appendix (A.2). We then see, with Q ∗ iq − + iν ( u ) = Q iq − − iν ( − u ), that Eqs. (130) and (131) areconveniently expressed as: f + q,ν ( u, ǫ ) = 2 π Γ (cid:0) − iν + iq (cid:1) Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) − iν (cid:1) Q iq − − iν ( ǫu ) ,f − q,ν ( u, ǫ ) = 2 π Γ (cid:0) − iν + iq (cid:1) Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) − iν (cid:1) Q ∗ iq − + iν ( ǫu ) . (135) Now that the two-dimensional case has been solved the general d -dimensional can be faced more easily.Let us go back to the complex time variable τ and consider say Im τ >
0. By using the parametrization(4), the scalar product X · Ξ and the integral representation (124) may be written as follows: x ( r, x) · a ( ρ, a) sinh τ − ǫ cosh τ = 1 ρ (cid:20) (x − a) + r + ρ r sinh τ − ǫρ cosh τ (cid:21) , (136)( − iZ · Ξ) − d − + iν = ( − ix · a sinh τ + iǫ cosh τ ) − d − + iν == ρ d − − iν Γ (cid:0) d − − iν (cid:1) Z ∞ dRR R d − − iν e iR » (x − a)2+ r ρ r sinh τ − ǫρ cosh τ – . (137)Given this formula, the steps to compute F + q,ν ( τ, ξ, a, ǫ ) are similar to those of the Milne case. At firstthe integral representations (30) and (137) are inserted into Eq. (126) and perform the Gaussian integralover x. By using the same change of variables as in (99) and (101) and identifying one factor with (theintegral representation of) ( a · ξ ) − d − − iq gives F + q,ν ( τ, ξ, a, ǫ ) = Γ (cid:0) − iν (cid:1) Γ (cid:0) d − − iν (cid:1) (cid:18) πi sinh τ (cid:19) d − ( a · ξ ) − d − − iq ×× Z ∞ dRR R − iq (cid:20) − i (cid:18) R R (cid:19) sinh τ + iǫ cosh τ (cid:21) − + iν . (138)As before, the integral at the RHS is symmetric in the exchange q → − q . This result holds for 0 < Im τ < π . The other case − π < Im τ < f ± q,ν ( τ, ǫ ) = Γ (cid:0) − iν (cid:1) Γ (cid:0) d − − iν (cid:1) (cid:18) ± πi sinh τ (cid:19) d − ×× Z ∞ dRR R − iq (cid:20) ∓ i (cid:18) R R (cid:19) sinh τ ± iǫ cosh τ (cid:21) − + iν (139)27here f + is defined in the strip 0 < Im τ < π while f − is defined in the strip − π < Im τ < ∓ iZ · Ξ) − d − + iν = Z ∞ n ( q ) dq Z dµ ( ξ ) f ± q,ν ( τ, ǫ ) ( a · ξ ) − d − − iq ( x · ξ ) − d − + iq (140)The upper or lower sign is to be taken accordingly as Im τ > τ <
0. As in the Milne case (Sect.8.2), it is more concise to rewrite this expansion as a one-dimensional integral by means of Eq. (68) theprojector Π q ( a, x ) onto the space of open waves with eigenvalue q . The latter is a particular solutionof the Laplace equation, which depends on the de Sitter wave parametrisation since it is labeled by a . ( ∓ iZ · Ξ) − d − + iν = Z ∞ dq f ± q,ν ( τ, ǫ )Π q ( a, x ) . (141)Note the analogy with the Milne case: here the role of p/m is played by the parameter a .The whole discussion of the two-dimensional case can be repeated and we get expressions for f ± q,ν ( u, ǫ )in terms of the Legendre functions ”on the cut”: f + q,ν ( u, ǫ ) = (2 π ) d Γ (cid:0) − iν + iq (cid:1) Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) d − − iν (cid:1) (cid:0) − u (cid:1) − d − Q iq − − iν ( ǫu ) f − q,ν ( u, ǫ ) = (2 π ) d Γ (cid:0) − iν + iq (cid:1) Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) d − − iν (cid:1) (cid:0) − u (cid:1) − d − Q ∗ iq − + iν ( ǫu ) (142)that are analytical functions defined on the cut plane ∆. The more elegant way to write the de Sitter two-point Wightman function of a massive Klein-Gordonfield is as a superposition of the global waves (120) [30]: W ( Z, Z ′ ) = γ d,ν Z dµ (Ξ) ( iZ · Ξ) − d − + iν ( − iZ ′ · Ξ) − d − − iν (143) γ d,ν = Γ (cid:0) d − + iν (cid:1) Γ (cid:0) d − − iν (cid:1) π ) d . (144)where Z ∈ T − and Z ′ ∈ T + . This writing is the one that is most similar to the Fourier plane waveexpansion of the two-point function of the flat case (83).By inserting in this representation formulae (141) one gets the spectral density ρ that provide theexpansion of the Wightman function in terms of the modes of the Lobatchevskian Laplace-Beltramioperator: W = Z ∞ dq ρ ( q, cosh τ, cosh τ ′ ) Π q ( x, x ′ ) . (145)with ρ ( q, u, u ′ ) = Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) + iν − iq (cid:1) Γ (cid:0) − iν + iq (cid:1) Γ (cid:0) + iν + iq (cid:1) − u ) − d − (1 − u ′ ) − d − × h Q iq − − iν ( u ) Q ∗ iq − − iν ( u ′ ) + Q iq − − iν ( − u ) Q ∗ iq − − iν ( − u ′ ) i . (146)28he spectral density ρ ( q, u, u ′ ) may also be expressed in terms of the functions P iq − + iν ( u ) (see Appendix(A.2)) as follows: ρ ( q, u, u ′ ) = Γ (cid:0) − iν − iq (cid:1) Γ (cid:0) + iν − iq (cid:1) Γ (cid:0) − iν + iq (cid:1) Γ (cid:0) + iν + iq (cid:1) π (1 − u ) − d − (1 − u ′ ) − d − × h P iq − + iν ( u ) P − iq − + iν ( u ′ ) + P iq − + iν ( − u ) P − iq − + iν ( − u ′ ) i . (147)These formulae provide the full solution of the difficult problem of describing the quantum Klein-Gordonfield on the open de Sitter universe, provided that | Im ν | < (see Sect. 9.4). The latter requirementcan be expressed as a condition on the mass parameter according to (121): m > m cr = d ( d − . (148)This is the condition that guarantees the convergence of the integral (126). When m < m cr modes thatare not square-integrable are necessary [9, 13, 19] for a full and correct description.
10 Conclusions
In this paper we have settled down the basic ingredients to work out quantum theories on homogeneousand isotropic spaces with negative curvature.While the state-of-the-art may be considered to be the expansion of the eigenmodes of the Laplacian interms of spherical harmonics (see e.g. [5]), we give here a new set of eigenmodes, based on a differentdecomposition. Most important, we provide also the way to deal with these new modes, that we believeis simpler than the standard approach. Simplicity is very often the source of new progress, that howeverwe have left for future work.The formalism we have developped here is the closest possible to the one employed in standard (text-book) quantum mechanics on the Euclidean space R , and is based on the suitable Fourier-type harmonicanalysis in terms of the eigenmodes of the Laplacian precisely in the same way as standard quantummechanics is based on the Fourier transform in terms of plane waves, i.e. the momentum space represen-tation of the wavefunctions. However, even though we have restricted our attention to square-integrablefunctions, the absence of translation invariance renders the task considerably more complicated thanin flat space. The eigenmodes of the Laplacian are labeled in the most convenient way by using [20]vectors of the cone asymptotic to the Lobatchevski space and a real number q . An important technicalpoint consist in finding suitable integral representations expressing such family of eigenmodes. Thisstep is in the spirit of our earlier work [21], where general embedded manifolds (branes) were stud-ied. These integral representations render quite simple calculations which at first sight might seemintractable.In the second part of this paper we have applied our general construction to the study of two geometrieswhich are quite popular nowadays: the universes of Milne and de Sitter. In both cases we have displayedexplicit formulae yielding the Fourier-type expansion of the relevant spacetime plane waves in terms ofthe eigenmodes of the Lobatchevski Laplacian. These expansions immediately provide also harmonicexpansions for the corresponding Wightman functions in terms of representations of the principal seriesof the SO (1 , d −
1) group. One important point of our treatment of the de Sitter case is that theresult is obtained by working solely in the physical region covered by the open chart, in contrast withprevious approaches [9, 13] that achieved a similar goal. The apparent simplicity of our approachshould not send to oblivion the fact the whole construction was thought impracticable and matter ofbig controversies. 29he quantization of theories implying the use of non-square integrable modes remains to be attacked. Aclean discussion of the square-integrable case as presented here is a necessary step forward. We expectthat our methods and results will allow to proceed one step further and to tackle the analysis of a largerclass of non square integrable functions.Due to their simplicity, the methods outlined in this paper set the premises of many future developmentsin various other, quite different, directions. 30
Legendre functions ”on the cut”
A.1 Legendre functions of the first kind
Let us introduce the function P iq − + iν ( u ) = e ∓ πq P iq − + iν ( u ) , (149)where the upper or lower signs refer to the imaginary part of u being positive or negative. The function P is an analytic continuation of the so-called ”Legendre function on the cut”, originally defined for real u such that | u | < P iq − + iν ( u ) is an entire functionof the complex parameters q and ν (the only singularities being at infinity). The functions P iq − + iν ( − u )and P ∗ iq − + iν ( u ) = P − iq − + iν ( u ) are two other solutions of the Legendre equation analytic on the samedomain ∆; one has the following relation:1Γ ( iq ) Γ (1 − iq ) P iq − + iν ( − u ) = 1Γ (cid:0) + iν (cid:1) Γ (cid:0) − iν (cid:1) P iq − + iν ( u ) − (cid:0) − iν − iq (cid:1) Γ (cid:0) + iν − iq (cid:1) P − iq − + iν ( u ) . (150)The following formula is useful to derive the spectral density of the de Sitter case and can be obtainedfrom the relation (150): P iq − + iν ( u ) P iq − + iν ( − u ′ )Γ (cid:0) + iq + iν (cid:1) Γ (cid:0) + iq − iν (cid:1) = P iq − + iν ( u ) P − iq − + iν ( − u ′ )Γ (cid:0) + iν (cid:1) Γ (cid:0) − iν (cid:1) + P iq − + iν ( u ) P − iq − + iν ( u ′ )Γ ( iq ) Γ (1 − iq ) , (151)or P − iq − + iν ( u ) P − iq − + iν ( − u ′ )Γ (cid:0) − iq + iν (cid:1) Γ (cid:0) − iq − iν (cid:1) = P iq − + iν ( u ) P − iq − + iν ( − u ′ )Γ (cid:0) + iν (cid:1) Γ (cid:0) − iν (cid:1) − P iq − + iν ( − u ) P − iq − + iν ( − u ′ )Γ ( iq ) Γ (1 − iq ) . (152) A.2 Legendre functions of the second kind
Let us define the function Q iq − − iν ( u ) by the following integral representation: Q iq − − iν ( u ) = Γ (cid:0) − iν (cid:1) π Γ (cid:0) − iν + iq (cid:1) Γ (cid:0) − iν − iq (cid:1) Z ∞−∞ dv e − iqv h(cid:0) − u (cid:1) cosh v + iu i − + iν . (153)The function Q iq − − iν ( u ) is readily seen to be a solution of the Legendre equation. It is manifestlyanalytic the cut-plane ∆ and invariant under the change q → − q . It is proportional to the standardLegendre function Q iq − + iν ( u ) [24] in the upper u half-plane: Q iq − − iν ( u ) = e − i π ( − iν ) π Γ (cid:0) − iν + iq (cid:1) e πq Q iq − − iν ( u ) , Im u >
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