aa r X i v : . [ h e p - t h ] A ug Localizability in de Sitter space
N. Yokomizo ∗ and J. C. A. Barata Instituto de F´ısica, Universidade de S˜ao Paulo,C.P. 66318, 05315-970 S˜ao Paulo-SP, Brazil (Dated: October 26, 2018)
Abstract
An analogue of the Newton-Wigner position operator is defined for a massive neutral scalarfield in de Sitter space. The one-particle subspace of the theory, consisting of positive-energysolutions of the Klein-Gordon equation selected by the Hadamard condition, is identified with anirreducible representation of de Sitter group. Postulates of localizability analogous to those writtenby Wightman for fields in Minkowski space are formulated on it, and a unique solution is shownto exist. Representations in both the principal and the complementary series are considered. Asimple expression for the time-evolution of the operator Newton-Wigner is presented.
PACS numbers: 04.62.+v, 03.65.Sq, 03.65.Db ∗ [email protected] . INTRODUCTION The question of the existence and usefulness of a notion of localization for quantumparticles moving at relativistic speed has a long history [1–4]. Although the idea of aposition measurement is one of the most intuitive ideas of a quantum observable, thereis no obvious mathematical counterpart to it in the relativistic domain. The conflict withintuition in such a fundamental subject is the main motivation to work on this problem. Butthere are also technical reasons for that. A quantum field theory is usually applied to thestudy of particle collision processes, so one needs to understand how to interpret the theoryin terms of particles. The main question is how to assign probabilities for the detection ofthe produced particles in detectors placed at specific regions of space [5–8], what amounts todefining a position probability distribution. Besides that, there is the very fact that classicalparticles do exist, i.e., that a classical limit of the underlying quantum theory exists whichdescribes particles. A position operator is the natural tool to deal with this limit [9, 10].Now the current widespread interest in quantum effects in curved spacetimes, boosted byexperimental and theoretical discoveries in cosmology, motivates the analysis of the problemof localizability in a more general context. In particular, the present accelerated expansionof the universe [11, 12] and the existence of an inflationary epoch in the very early universe[13–15] suggest that there should be eras in the beginning of the universe and in the distantfuture when the geometry of the universe is approximately a patch of de Sitter space, whatjustifies our interest in this special geometry. Local effects of de Sitter geometry on particledynamics have been investigated at the classical and quantum level [16–18].In flat Minkowski space, an early solution to the problem was provided by the work ofNewton and Wigner [1], later reformulated in more rigorous terms by Wightman [2]. Itwas proved that a natural set of postulates defines a unique position operator, at leastfor massive fields. But the operator found is frame-dependent, and alternative covariantnotions of localizability were put forward since then [3, 4]. The interpretation of theseoperators and the possibility of actually measuring them have been discussed in the contextof Quantum Field Theory in terms of specific models of interaction between a detector andthe quantum field (see [19, 20]). But if the particle moves in a curved spacetime, little isknown. There are additional complications in the analysis, mainly due to the existence ofmultiple vacua. In fact, the concept of particle is not strictly necessary for Quantum Field2heory in Curved Spacetimes—the general theory can be formulated without introducingthe notion of particles [21]. Only in special circumstances it still makes sense to speak ofparticles. In a flat Minkowski spacetime, for instance, that is certainly true. In the case ofultrastatic spacetimes, a notion of Newton-Wigner localization is available, as discussed in[22]. And it is also natural that in regions where the curvature is small one should be ableto speak of particle states—high-energy experiments are actually performed in a slightlycurved space, and particles are observed. However, there is no clear specification of thenecessary conditions for a particle interpretation to be available.We have studied the case of a neutral massive scalar field in 2d de Sitter space, and haveshowed that a particle interpretation of this theory is possible. This problem was previouslyinvestigated exploring special decompositions of de Sitter group [23] or an analogy with theMinkowski space case [24]. We have considered it in the context of the modern formulationof Quantum Field Theory in Curved Spacetimes, as described in [21, 22], for instance. Ourstrategy was the following. The main difficulty for the definition of particle states in ageneral curved spacetime lies on the existence of several inequivalent Fock representationsfor the canonical commutation relations. Since there is no preferred Fock representation,the concept of a particle becomes ambiguous. In de Sitter space, however, it is possible (see[25]) to select a unique vacuum state—the Bunch-Davies vacuum—by requiring: (i) physicalstates to satisfy the Hadamard condition, which corresponds to the requirement that theaveraged energy-stress tensor can be renormalized by a point-splitting prescription [21]; and(ii) the vacuum to be invariant under the action of de Sitter group [25, 26]. In this sense,there is a preferred Fock representation for massive free fields in de Sitter space. It is clearthat the maximal symmetry of the de Sitter space is crucial for that.There is an important remark that we should add here. In the Minkowski case, the notionof Newton-Wigner localization belongs to the realm of Relativistic Quantum Mechanics, notexactly to Quantum Field Theory, and is implemented in a one-particle Hilbert space. Incurved space-times, however, the choice of the adequate one-particle Hilbert space, even forfree fields, is dictated by the interest of the quantum field theoretical model one wishes toimplement. Hence, the possibility to define a reasonable localization concept is also relatedto the choice of representations for the canonical commutation relations and, therefore, isnot an exclusively relativistic quantum mechanical question.We discuss the canonical quantization of the scalar field in the Bunch–Davies represen-3ation in Section II. An important fact is that the one-particle subspace H of the Fockrepresentation can be interpreted as an irreducible representation of de Sitter group, as aresult of vacuum invariance. That allowed us to write localizability conditions on H analo-gous to those formulated on irreducible representations of the Poincar´e group in [1, 2]. Wedescribe these conditions in Section III, and prove that a unique solution exists at t = 0. Theposition operator which satisfies these conditions is the natural analogue in de Sitter spaceof the Newton-Wigner position operator. These results are then compared with previous in-vestigations of particle localization in de Sitter spacetime [24], and the time-evolution of theposition operator is described, based on an analogy with the case of Minkowski spacetime.Perspectives on future works are discussed in Section IV. II. THE QUANTIZED FIELD IN SPHERICAL COORDINATES
The choice of a particular vacuum state and the associated Fock representation of thequantized scalar field theory in de Sitter space is equivalent to the choice of a decompositionof the space S of solutions of the Klein-Gordon equation with smooth Cauchy conditionsas a direct sum S = S + ⊕ S − of subspaces of positive and negative energy. In this sectionwe construct the space S and describe the decomposition associated with the Bunch–Daviesvacuum. Canonical quantization based on such a decomposition is then briefly discussed.In the last subsection, the one-particle subspace of the Fock representation is interpreted asan irreducible representation of de Sitter group, and explicit expressions for the generatorsof the group and discrete symmetries are written. We consider de Sitter radii and particlemasses compatible with principal and with complementary series representations. A. Normal modes
We start our analysis describing in some detail the normal modes of the Klein-Gordonequation for the 2 d de Sitter space. This is required for the process of cannonical quantizationwe will present later.The simplest way of looking at the 2 d de Sitter space dS is to consider it a submanifoldembedded in a 3d Minkowski space M . Choosing a metric η ab = diag(1 , − , −
1) for M ,4ne has dS = { X ∈ M | X = X a X b η ab = − α } , where α > S × R . One may think of it as a spatial circle evolving in time. The symmetry group isthe de Sitter group O (2 , X = α sinh( t/α ) ,X = α cosh( t/α ) cos θ ,X = α cosh( t/α ) sin θ , the geometry of dS is described by the induced Lorentzian metric tensor, with components: g = 1 , g = 0 , g = − α cosh ( t/α ) . The volume density is √− g = α cosh( t/α ), and the D’Alembertian is (cid:3) = ∂ tt + 1 α tanh( t/α ) ∂ t − α cosh ( t/α ) ∂ θθ . The Klein-Gordon equation reads (cid:18) (cid:3) − m + ξR ~ (cid:19) φ = 0 . (1)The scalar curvature is related to the de Sitter radius by R = 2 /α . We put µ = m + ξR .After separation of variables, the Klein-Gordon equation becomes ψ ′′ = − k ψ ⇒ ψ k ( θ ) = 1 √ π e ikθ , k ∈ Z ,T ′′ + 1 α tanh( t/α ) T ′ + (cid:18) µ ~ + k α cosh ( t/α ) (cid:19) T = 0 . In order to solve the time-dependence of these “angular momentum modes” described bythe index k , put x = i sinh( t/α ), and get:(1 − x ) d Tdx − x dTdx + (cid:20) − α µ ~ − k − x (cid:21) T = 0 . (2)This is an associated Legendre equation. The solutions are associated Legendre functions P kν ( x ) and Q kν ( x ), with ν ( ν + 1) = − α µ / ~ . The coefficient ν is given by ν = − ± p − α µ / ~ . (3)5f µ is positive, then ν is either a real number in the interval [ − ,
0] or a complex numberwith real part equals to − / µ = 0, then ν = 0 , µ is negative, then ν may assume arbitrary real values.Throughout this work we will restrict to the case µ >
0. The squared mass is alwayspositive, so this restriction corresponds in fact to not allowing a large negative coupling withthe scalar curvature. In this case, a nice pair of linearly independent solutions of (2) is givenby T kν (cid:0) i sinh( t/α ) (cid:1) , T kν (cid:0) − i sinh( t/α ) (cid:1) , where T kν ( z ) := e ∓ ikπ/ P kν ( z ) for ± Im( z ) > T kν ( z ) is analytic in the whole complex plane, except for two branchingcuts on the real axis, one from −∞ to − ∞ . It doesn’t matterwhich root ν of (3) is taken: both give the same function (that follows from the symmetry T kν = T k − ν − ). The functions T kν ( z ) have the property that [ T kν ( ix )] ∗ = T kν ( − ix ), i.e., thelinearly independent solutions are complex conjugate (see Appendix A).Thus, there is a set of normal modes of the Klein-Gordon equation in de Sitter space (1)of the form: u k ( t, θ ) = r γ k T kν (cid:0) i sinh( t/α ) (cid:1) e ikθ √ π ,v k ( t, θ ) = r γ k T kν (cid:0) − i sinh( t/α ) (cid:1) e ikθ √ π , (4)with k ∈ Z , and where γ k := Γ( − ν − k )Γ( ν − k + 1) are conveniently chosen normalizationcoefficients, which will be discussed latter. Here, Γ is Euler’s gamma function. B. Space of solutions and positive-energy modes
The normal modes derived in the last section can be used for the construction of thespace S of complex solutions of the Klein-Gordon equation in de Sitter space (1) withsmooth Cauchy conditions. It is the vector space formed by wavefunctions of the form: φ ( t, θ ) = ∞ X k = −∞ (cid:2) c k u k ( t, θ ) + d k v k ( t, θ ) (cid:3) , (5)with coefficients c k , d k of rapid decay in | k | , such as P | c k || k | l < ∞ , ∀ l >
0, and similarly for d k . In order to see that, let us first prove that the series converges absolutely and uniformly6o a C -solution of the Klein-Gordon equation with smooth initial conditions. Consider thesum containing terms with c k first. From (4), one has at each fixed t a Fourier series: C ( t, θ ) := ∞ X k = −∞ p k e ikθ √ π , (6) p k := c k r γ k T kν (cid:0) i sinh( t/α ) (cid:1) . The asymptotic behavior of the coefficients p k can be obtained from the large k asymptoticrepresentation of the Legendre functions T kν given in Eq. VI.95b of [27]. One finds that (cid:12)(cid:12) √ γ k T kν (cid:0) i sinh( t/α ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Γ( − ν − k )Γ( ν − k + 1) (cid:12)(cid:12)(cid:12)(cid:12) / (cid:12)(cid:12) Γ( ν − k + 1) T kν ( i sinh( t/α ) (cid:1)(cid:12)(cid:12) ≃ | k | − / , for large k and ∀ t . Since the c k are of rapid decay, the series (6) converges absolutely and uniformly in spacetime.Similar arguments can be used for the terms with coefficients d k . Thus, the sum in Eq. (5)is uniformly convergent. It also follows that φ ( t, θ ) is continuous, since all terms in theuniformly convergent series are continuous. Consider now the derivatives ∂ mθ ∂ nt φ . A finitenumber n of spatial derivatives changes the coefficients p k by a factor ( ik ) n , and the timederivatives have the following form for large k : ddt T kν (cid:0) i sinh( t/α ) (cid:1) ≃ − ikα cosh( t/α ) T kν (cid:0) i sinh( t/α ) (cid:1) ,d dt T kν (cid:0) i sinh( t/α ) (cid:1) ≃ (cid:20) − ikα sinh( t/α )cosh ( t/α ) − k α cosh ( t/α ) (cid:21) T kν (cid:0) i sinh( t/α ) (cid:1) , (7)thus changing the Fourier coefficients only polynomially in k , and in a uniform manner in t . But then, rapid decay of the c k ’s ensure uniform convergence of all second derivatives of(6). Again, all arguments can be repeated for the sum involving the d k ’s. It follows that φ ( t, θ ) is C and, by construction, a solution of the Klein-Gordon equation (derivatives canbe applied inside the sum, and each term is a solution of the equation). Moreover, at eachfixed t = t , the restriction φ ( t , θ ) is a smooth function on the circle, since any numberof spatial derivatives can be applied to (6). From Eq. (7), the same is true for ˙ φ ( t , θ ).We will therefore consider only smooth Cauchy data φ (0 , θ ) , ˙ φ (0 , θ ). Finally, any smoothfunction on the circle has a Fourier series with rapidly decaying coefficients, so S containsall solutions with smooth initial coefficients.7he vector space of solutions S is equipped with an invariant Hermitian sesquilinear form: h f | g i = ia ( t ) Z S t d θ ( f ∗ ∂ t g − ∂ t f ∗ g ) , (8)where S t is any spatial slice of constant time t , and a ( t ) := α cosh( t/α ) is the correspondingscale factor –the radius of the circle S t . Notice that h f ∗ | g ∗ i = − h g | f i . Our choice of thenormalization coefficients γ k in Eq. (4) ensures that the normal modes are orthonormal, h u k | u l i = δ kl , h v k | v l i = − δ kl , h u k | v l i = 0 . (9)Let us prove that. The fact that h u k | v l i = 0 is a simple consequence of the definition of theinvariant form together with the identity [ T kν ( ix )] ∗ = T kν ( − ix ). For the modes u k (put now y = sinh( t/α )), h u k | u l i = − δ kl | γ k | cosh ( t/α ) (cid:2) T kν ( − iy ) T k ′ ν ( iy ) + c.c. (cid:3) . Invoke now the identity (from [27])(1 − z ) (cid:20) T kν ( z ) ddz T kν ( − z ) − ddz T kν ( z ) T kν ( − z ) (cid:21) = 2 γ k (10)in order to get h u k | u l i = δ kl | γ k | γ k . Let us show that γ k is positive for any k . Consider first the case k = 0. Then1Γ( − ν )Γ( ν + 1) = sin[( ν + 1) π ] π , where ν is either a real number in the interval ( − , − / iλ , λ ∈ R . In the first case, ν + 1 is in the interval (0 , ν + 1) π ] is positive.In the second case, sin[( ν + 1) π ] = cosh( πλ ), positive too. For general k , first note that, for k positive, 1 γ k = Q kl =1 ( − ν − l )( ν − l + 1)Γ( − ν )Γ( ν + 1) = 1 γ k Y l =1 ( α µ / ~ + l − l ) . It is clear that the product is positive ( l ≥ l when l is integer). A similar trick does thework for negative k . That completes the proof that h u k | u l i = δ kl . For the case of the normalmodes v k , one can use the identity √ γ k T kν ( z ) = √ γ − k T − kν ( z ) (11)8which is proved using the inversion formula for gamma functions and Eq. (A2)) to see that v k = u ∗− k . Then the result obtained for the modes u k implies that h v k | v l i = − δ kl .In order to proceed to the canonical quantization of the scalar field φ ( t, θ ), one needs tochoose a special decomposition of the space of solutions S as a direct sum S = S + ⊕ S − ,where S + ( S − ) is interpreted as the space of positive (negative) energy solutions. Thedecomposition must be such that: (i) positive energy solutions have positive norm and (ii)the complex conjugate of a positive energy solution is a negative energy solution. Fromprevious results, these conditions are satisfied if one picks a basis { u k ( t, θ ); k ∈ Z } for S + and a basis { v k ( t, θ ); k ∈ Z } for S − . This is the energy-splitting decomposition thatwill be used in this work. There are alternative valid decompositions which, after canonicalquantization, are associated with distinct choices of the vacuum state of the quantized theory.Our choice will lead to the so-called Bunch–Davies vacuum, in which we are interested dueto its invariance under de Sitter group actions and its Hadamard property. C. One-particle subspace and canonical quantization
The Newton-Wigner (NW) position operator will be defined in the so-called “one-particlesubspace” H . This Hilbert space is defined as the completion of S + in the scalar productdefined by the sesquilinear form (8) (which is positive when restricted to S + ). The vectors φ ∈ H are superpositions of positive-energy solutions, and can be represented explicitly as φ ( t, θ ) = X k φ k u k ( t, θ ) , X k | φ k | = 1 , φ k ∈ C . (12)The scalar product in H is simply h φ | ψ i = P φ ∗ k ψ k . We are going to think of the one-particle subspace as describing the quantum dynamics of a single relativistic particle inde Sitter space, following the usual physical interpretation: φ ( t, θ ) will be the spacetimerepresentation of the wavefunction associated with the particle. Some problems with thisinterpretation might be expected—it has been repeatedly remarked that the concept ofparticle for quantum fields in curved spacetimes is not well-defined. Nevertheless, it is justas clear that there are situations where a particle-like behavior is evident. As remarked in[30], particle physics experiments are actually performed in a curved spacetime, and we dosee particle tracks in experiments. To understand how to deal with a quantum field theoryin a curved spacetime under circumstances where a particle-like behavior is possible is one9f the purposes of this paper.After fixing the positive-energy modes of the classical field, canonical quantization inde Sitter space follows pretty much the same steps as in Minkowski space [28], as we nowbriefly describe. One considers the bosonic Fock space F built in the usual way from the one-particle state H as F := C ⊕ ∞ n =1 ( H ⊗ n ) s , where the index “ s ” indicates the symmetrization ofthe tensor products. Following the usual prescription, the quantized neutral massive scalarfield, acting in F , is expressed in the formˆ φ ( t, θ ) = ∞ X k = −∞ ( a k u k + a ∗ k u ∗ k ) , (13)where the u k are the chosen orthonormal positive energy modes, and the a k , a ∗ k are annihi-lation and creation operators satisfying the commutation relations[ a k , a ∗ l ] = δ kl , [ a k , a l ] = [ a ∗ k , a ∗ l ] = 0 . The vacuum | Ω i is defined as the vector state annihilated by all annihilation operators, a k | Ω i = 0, ∀ k , and many-particle states are created by repeated application of creationoperators to the vacuum.It is a well known fact that the quantization of free fields in curved space times is non-unique, and different choices of positive energy decompositions may lead to unitarily inequiv-alent representations of the algebra of cannonical commutation relations. These differentchoices reflect different possible choices for the vacuum state.Our particular choice of the previously defined positive energy modes u k in the expansion(13) corresponds to the choice of the so called “Bunch–Davies vacuum”. In order to establishthis claim, we present an explicit calculation of the two-point function in Appendix B, andcompare it to the two-point function obtained in the original work of Bunch and Davies[29] (where flat coordinates were used), showing that both results agree. The choice of theBunch–Davies vacuum is particularly relevant because of its previously mentioned relationto the Hadamard condition [25, 26]. D. Group action on the space of positive-energy solutions
The space S + of positive-energy solutions was described in a given system of sphericalcoordinates ( t, θ ), but there is a whole family of systems ( t ′ , θ ′ ) related by isometries in the de10itter group O (2 , S + is coordinate-independent,i.e., that the subspace of positive-energy modes is invariant under the action of the group(what is equivalent to the invariance of the vacuum state in the quantized theory), as wellas to find out how the group acts on these modes. That will lead to an interpretation of itscompletion H as an irreducible representation of de Sitter group.Any element of O (2 ,
1) is the product of an element of the restricted de Sitter group O (2 , ↑ + of Lorentz transformations of determinant 1 which do not reverse the directionof time, and possibly parity P and time reversal T . There are three linearly independentgenerators in the algebra of O (2 , N and N , and the generator of rotations, N . The question ishow these transformations act on the modes defined in Eq. (4). The case of rotations isquite simple. A transformation U ( φ ) = exp( φN ) which rotates the space by an angle φ changes angles in spherical coordinates according to θ θ − φ , while the coordinate t remains unaffected. The generator of rotations is N = − ∂/∂θ . Its action on the basisvectors is just N u k = − ik u k , i.e., the basis { u k } is that of the eigenvectors of the Hermitianoperator iN .Now consider the case of N . Since Lorentz transformations are naturally described inthe flat coordinates of the ambient Minkowski space M , let us describe the modes u k in thesame coordinates: u k = r γ k π T kν ( − iX /α ) ( X + iX ) k [ α + ( X ) ] k/ . An infinitesimal Lorentz transformation along the axis X is given by( X ) ′ = X − λX , ( X ) ′ = X − λX , ( X ) ′ = X , where λ is the infinitesimal parameter of the transformation (the transformation is Lorentzto first order in λ ). Thus, the variation of u k is N u k = X ∂u k ∂X + X ∂u k ∂X . A similar equation holds for boosts along the axis X . Evaluating the derivatives andusing a few relations between Legendre functions from [27], one finds that the action of the11enerators of de Sitter group is N u k = − iku k ,N u k = i | ( ν + k )( ν − k + 1) | / u k − + i | ( ν − k )( ν + k + 1) | / u k +1 , (14) N u k = 12 | ( ν + k )( ν − k + 1) | / u k − − | ( ν − k )( ν + k + 1) | / u k +1 . These equations show that H is closed under the action of the infinitesimal generators.Hence, H is a representation space for O (2 , ↑ + , the action of a Lorentz transformation L on a wavefunction φ ( x ) ∈ H being given by φ ( x ) φ ( L − x ). The Casimir operator whichcharacterizes the irreducible representations is C = N − N − N , and is easily verifiedto be C = − ν ( ν + 1) = α µ / ~ for the above expressions.With our restriction to µ >
0, the index ν may be: (a) a real number in the interval( − , ν = − / iλ , with λ ∈ R . In the case (a),one has 0 < C < /
4, what corresponds to a representation of de Sitter group in the so-called complementary series (the continuous representations C q in the exceptional interval0 < q < / C ≥ /
4, what correspondsto principal series representations (continuous representations C q with q ≥ / P and time-reversal T . Werepresent parity as the reversal of the axis X in the ambient Minkowski space. Then parityjust reverses the sign of the angular coordinate of a wavefunction in H , P φ ( t, θ ) = φ ( t, − θ ).In particular, for the basis vectors u k , one may use the identity (11) in order to get P u k = u − k . (15)The action of T has a peculiarity connected with the restriction to the space of positive-energy states. The geometrical realization of the transformation is the reversal of the timecoordinate in the ambient Minkowski space. But this cannot be represented as φ ( t, θ ) φ ( − t, θ ), since the result is a negative-energy state. In order that the transformation isclosed in H , we take the anti-unitary representation T φ ( t, θ ) = φ ∗ ( − t, θ ). But then theaction of the operator on modes u k is the same as that of parity, with the difference that12he action is anti-linear, T u k = u − k (anti-linear) . (16) III. NEWTON-WIGNER LOCALIZATIONA. Definition of the localization system
The notion of localization of relativistic particles in Minkowski space provided by theNewton-Wigner (NW) position operator was introduced in [1]. In that paper, a list ofproperties is postulated, which are assumed to hold for any reasonable relativistic positionoperator, and it is proved that there is a unique operator satisfying them. A more direct wayto understand this position operator is described in [32]. Let us review the basic argument.Consider a massive scalar field in Minkowski space. The one-particle subspace of the theoryconsists of vectors φ ( p ) ∈ L ( R , dp/ω ( p )), with ω ( p ) = p p + m , i.e., the scalar product is h φ | ψ i = Z d pω ( p ) φ ∗ ( p ) ψ ( p ) . Now absorb a factor p ω ( p ) in each wavefunction: i.e., consider the unitary transformation M ω : L ( R , dp/ω ( p )) → L ( R , dp ), whose action is φ ( p ) φ NW ( p ) = φ ( p ) / p ω ( p ). Then,introduce a unitary operator of time-evolution U t : L ( R , dp ) → L ( R , dp ), representedby the transformation φ NW ( p ) ( U t φ NW )( p ) = exp( − iω ( p ) t/ ~ ) φ NW ( p ). Finally, Fouriertransform the result in order to get a spatial representation, φ NW ( t, x ) = 1 √ π Z d p e ipx/ ~ e − iω ( p ) t/ ~ φ NW ( p ) . That gives the Newton-Wigner wavefunction. The probability density that the particle isdetected at the point x in time t is P ( t, x ) = | φ NW ( t, x ) | . The position operator itself, attime t , is the multiplication operator in the spatial representation at the same time,( q t φ ) NW ( t, x ) = xφ NW ( t, x ) . Some difficulties show up if one tries to repeat the same steps in the case of de Sitterspace. First, there is no canonical definition of a momentum space representation. Weovercome this problem by looking at the mode expansion as a convenient (for our purposes)de Sitter analogue of the Fourier transform. It is clear that a mode expansion is a coordinate13ependent concept, therefore the resulting position operator will depend on the choice ofcoordinates. But, as well known, the Newton-Wigner operator is not a covariant object evenin Minkowski space: there is a distinct operator associated with each reference frame. Theproblem found in Minkowski space is just carried over into de Sitter space, and we do notattempt to solve it here.The second point is the absence of a time-translation isometry in dS , what makes thetime-evolution of individual modes much more complicated than in Minkowski space. Twoaspects are relevant here: there is no definite frequency ω k associated with each mode,so that time-evolution in momentum space is not just multiplication by varying phasesexp( − iωt/ ~ ) as before; and the oscillation of the field goes on together with a damping ofthe field amplitude, forced by the expansion of the universe (for increasing | t | ). We will seethat these effects can be isolated: the damping factor will be analogous to the factor √ ω absorbed in the definition of the Newton-Wigner wavefunction in the Minlowski case, whilethe oscillating phases will be responsible for the time-evolution of the position operator.Let us now proceed to the definition of the de Sitter version of NW-localization. Laterwe will interpret the results drawing an analogy with the discussion above. We assume thata localization system in de Sitter space is: I: A family of unitary transformations W t : H → L ( S ), φ φ NW ( t, θ ), where L ( S ) isthe Hilbert space of square-integrable functions on the circle S ; II: If U ( α ) ∈ SO(1 ,
2) is a rotation by an angle α , then U ( α ) φ φ NW ( t, θ − α ); III: P φ φ NW ( t, − θ ), and T φ φ ∗ NW ( − t, θ ); IV:
In the large mass limit, one has φ NW ( t, θ ) ∝ φ ( t, θ ).Condition IV will be clarified below. As a regularity condition, we also assume that W − t : L ( S ) → H depends continuously on the mass m . Let us discuss the intuitive content ofthe Postulates above.The Newton-Wigner wavefunction φ NW ( t, θ ) is interpreted, for each time t , as describingquantum amplitudes for finding the particle at position θ . In other words, the probabilityof finding the particle in a Lebesgue measurable set I is P ( I ) = R I | φ NW ( t, θ ) | dθ . PostulateI corresponds to the basic requirement that such a probability distribution exists for eachtime t . 14he second postulate is that the Newton-Wigner representation is well-behaved underrotations. A rotation in de Sitter group, when seen from the Newton-Wigner spatial rep-resentation, must rotate the probability amplitudes on the circle by the same angle. Thiscondition can be reformulated as W t U ( ϕ ) W ∗ t = R ( ϕ ), where R is the operator of rotationfor square-integrable functions on the circle.Postulate III is the requirement that the discrete symmetries of parity and time-reversalact as geometrical transformations on the Newton-Wigner representations. The complexconjugation in the time-reversal condition is necessary because the image of an anti-unitaryoperator under a unitary equivalence must be anti-unitary too. A quantum symmetry isin general defined up to a phase, according to the celebrated Wigner’s theorem; we areassuming here that the phases are equal to 1, avoiding complications with the possibility ofa projective representation of the extended de Sitter group.Finally, the Postulate IV is necessary in order to fix some remaining ambiguities in W t ,as we shall see. It is motivated by the following fact. In the large mass limit, the scalarproduct (8) of one-particle states φ ( t, θ ) , ψ ( t, θ ) reduces to h φ | ψ i ≃ µa ( t ) ~ Z S t dθ φ ∗ ( t, θ ) ψ ( t, θ ) , (17)i.e., it becomes the scalar product of L functions on the circle. In this case, it is naturalto interpret | φ ( t, θ ) | directly as a probability distribution (up to the factor outside theintegral). The postulate IV ensures that the Newton-Wigner distribution agrees with suchan interpretation.The consequences of the postulates can now be evaluated. Let us start with postulateII. For each t , a suitable basis for L ( S ) is that composed of eigenvectors of the Hermitiangenerator of rotations. That is the same as describing the NW-wavefunction in its Fourierexpanded form, φ NW ( t, θ ) = X k q k ( t ) e ikθ √ π , X k | q k ( t ) | = 1 , with q k ∈ C . Consider the vector u k ∈ H . The action of a rotation U ( α ) on it is tomultiply the state by a phase, U ( α ) u k = exp( − ikα ) u k . Since U is linear, the same mustbe true for its image in L ( S W t ( U ( α ) u k ) = e − ikα W t ( u k ) , R ( α ) W t ( u k ) = e − ikα W t ( u k ) . (18)But then it must be W t ( u k ) = e − iϕ k ( t ) e ikθ √ π , (19)where ϕ k ( t ) is some arbitrary phase. For, suppose the space V of solutions W t ( u k ) ofEq. (18) has more than one dimension. Note that the action of the Hermitian generator J of rotations in V is multiplication by k . Then there would be at least two orthogonalvectors with the same eigenvalue k , what is impossible, since the eigenspaces of J are non-degenerate. Therefore, V is one-dimensional, the space of eigenvectors of J with eigenvalue k . Because the transformation W t is unitary, and u k has norm 1, there is just a phasefreedom, what corresponds to Eq. (19).The action of parity in H is given by Eq. (15). The first part of postulate III, whenapplied to the general form of the solution of postulate II described in Eq. (19), leads to W t ( u k ) = W t ( P u − k ) = e − iϕ − k ( t ) e ikθ √ π . (20)The action of time-reversal in H is given by Eq. (16). The second part of postulate III leadsto W t ( u k ) = W t ( T u − k ) = e iϕ − k ( − t ) e ikθ √ π , (21)Put s k ( t ) := e − iϕ k ( t ) . Comparing Eqs. (19), (20) and (21), one finds that s k ( t ) = s − k ( t ) , s k ( t ) = s ∗ k ( − t ) . (22)The form of the transformation W t is restricted, but not uniquely fixed by the axioms I–III.The varying phases must satisfy Eq. (22), but there remains a lot of freedom after theseconditions are imposed. In the next section we describe the additional restrictions whichfollow from postulate IV at t = 0, where a unique solution is obtained. Then we studythe case of generic t , and suggest a natural solution based on an analogy to the case ofMinkowski space. 16 . The case of t = 0 At t = 0, the identities in Eq. (22) simplify to s k (0) = s − k (0) = ±
1. Thus, the transfor-mation W determined by postulates I–III is given by: φ (0 , θ ) = X k φ k r γ k T kν (0) e ikθ √ π φ NW (0 , θ ) = X k φ k s k (0) e ikθ √ π . (23)As we see, there is a sign ambiguity in each term of the above series, due to the presence ofthe factor s k (0).This ambiguity was first pointed by Philips and Wigner in [24]. Below, we will discusshow these authors address this problem, but let us first show that our postulate IV fixesthese ambiguities in a more natural way, leading to a unique solution for W .From Eq. (3) and the definition µ = m + ξR , it follows that the large mass limit m → ∞ corresponds to the limit λ → ∞ in the index of the Legendre functions ν = − / λi . Inorder that the postulate IV is satisfied, it is necessary that φ NW (0 , θ ) = f ( m ) φ (0 , θ ) inEq. (23), with some mass-dependent normalization factor f ( m ). But from the asymptoticexpression for the Legendre function in the large ν limit (see Eq. VI.(93a) in [27]), and usingthe Stirling approximation for Gamma functions, one gets r γ k T kν (0) ≃ ( − k (2 λ ) − / . (24)The factor (2 λ ) − / is just a mass-dependent normalization coefficient, as can be seen fromEq. (17) with a (0) = α , since λ → αµ/ ~ for large masses. Therefore, postulate IV impliesthat, up to an irrelevant overall sign, one has s k (0) = ( − k . (25)The fact that the same choice is made for all masses m is a consequence of the asumptionthat W − is continuous with respect to the mass m . Since s k (0) = ±
1, it cannot changebut discontinuously. Summing up: the transformation W is completely determined by thepostulates I-IV, being given by Eq. (23) with s k (0) = ( − k . C. Heuristical discussion on the ambiguities of signs
Let us discuss some heuristical aspects of the position probability distribution we havefound and the origin of the sign ambiguities occuring in the coefficients s k (0) before the17ostulate IV is used. As we shall now explain, the existence of the sign ambiguities is aconsequence of the fact that the localization postulates I-III alone do not fix the localizationin the one-particle space H relative to the localization in the Newton-Wigner representationspace L ( S ).Consider a specific choice of the signs s k = ±
1. If these coefficients are changed accordingto s k s ′ k = ( − k , then the Newton-Wigner representation is rotated by an angle of π .Hence, part of the freedom in the choice of the s k is due to the possibility of applying arotation of π . In fact, analizing the action of the map W on a sufficiently large class ofcarefully chosen states, it is even possible to fix all sign ambiguities just by avoiding suchantipodal reflections.In order to see this, consider a simple example. Take a superposition of k = − , , a = s , a = a − = s / φ (0 , θ ) = r γ π T ν (0) s + r γ π T ν (0) s cos θ , (26) φ NW (0 , θ ) = 1 + cos θ √ π . (27)The Newton-Wigner wavefunction (27) has a maximum at θ = 0 and decreases monotonicallywith increasing | θ | , assuming the value zero at the antipodal point θ = π . That is, it describesa particle more likely to be found in the region | θ | < π/ T kν (0) = ( − k k √ π Γ (cid:0) ν − k + 1 (cid:1) Γ (cid:0) − ν − k +12 (cid:1) (28)and it can be proved that the product of Γ’s in the denominator is positive. Thus, thecoefficients T kν (0) have signs alternating in k , because of the factor ( − k . There are twodistinct possibilities for the action of W : either s = s or s = s . If s = s , then the twoterms in the r.h.s. of (26) have different signs at θ = 0, and the same sign at θ = π , and thewavefunction has higher amplitudes in the region π/ < | θ | < π , with its maximum at θ = π .On the other hand, if s = s , then φ (0 , θ ) has its maximum at θ = 0, and higher amplitudesin the region | θ | < π/
2. The two choices are related by a rotation of π . Therefore, in orderthat the wavefunction φ (0 , θ ) is concentrated at the same region as φ NW (0 , θ ), and not atthe antipodal related region, one must choose s = s .The same argument can be adapted to states constructed by superpositions of states with | k | = p, p + 1, allowing one to fix s p +1 in terms of s p . Notice that this argument works only18or very special states: in a generic state φ , distinct choices of s k (0) are related by morecomplicated transformations than simple rotations by π .In any case, as we saw above, different choices of signs reflect on the localization of statesin the one-particle space H relative to the Newton-Wigner representation spece L ( S ).According to the usual interpretation of the localization operators and of the wavefunctions,it is natural to choose the signs in a way that the same interpretation of localization isfound in both spaces: if the Newton-Wigner wavefunction is concentrated about some θ ,the corresponding one-particle state should be concentrated at the same region, and not atthe antipodal point.The choice s k (0) = ( − k can be obtained, alternatively, from a condition of “maxi-mal localization” of position eigenstates. Consider a sequence of localized functions in theNewton-Wigner representation, δ KNW ( θ ) = 1 √ π X | k | 0) = 12 π X | k | An earlier discussion of localizability in de Sitter space was presented by Philips andWigner in [24], and we would like to compare our results to theirs. A brief review of [24]19s presented in Appendix C, to which we refer for more details. The main result obtainedin that paper was a description of (improper) states φ ( θ ) localized at position θ in t = 0.Such states were described in terms of their Fourier coefficients in an explicit principal seriesrepresentation of de Sitter group on a space L ( S ) of square-integrable functions on thecircle. We want to compare these with the localized states η ( θ ) = δ NW ( θ − θ ) of our NW-representation in t = 0, whose Fourier coefficients in the NW-representation are given byexp( − ikθ ) / √ π . From Eqs. (23) and (25), these localized states correspond to (improper)one-particle states in H of the form: η ( θ ) ( t, θ ) = X k ( − k √ π e − ikθ r γ k T kν (cid:0) i sinh( t/α ) (cid:1) e ikθ √ π , (31)that is, they have components η ( θ ) k = ( − k √ π e − ikθ (32)in H . We shall restrict in this section to representations of the principal series, on whichthe work [24] is based.Let us start by discussing the relation between the representation of Sitter algebra in thespace H of one-particle states described in Section II D and the more traditional Bargmann’srepresentations used in [24]. The principal series Bargmann representation on H ′ := L ( S )is briefly reviewed in Appendix C. Let {| k i} be the basis of H ′ composed of normalizedeigenstates of the generator of rotations, | k i = exp( ikθ ) / √ π . The action of de Sitteralgebra in this basis is given by: N | k i = − ik | k i ,N | k i = ν + k | k − i + ν − k | k + 1 i , (33) N | k i = i ν + k | k − i − i ν − k | k + 1 i , with ν = − / λi . These expressions are direct translations of Eqs. (C1) and (C2),discussed in more detail in [24]. On the other hand, the representation of de Sitter algebraon H is described explicitly in Eq. (14). The principal series representations are those with ν = − / λi , λ ∈ R , in which case Eq. (14) reduces to the simpler form: N u k = − iku k ,N u k = i | ν + k | u k − + i | ν − k | u k +1 , (34) N u k = − | ν + k | u k − + 12 | ν − k | u k +1 . U † B : H ′ → H is given by | k i 7→ χ k u k , where χ k is a complex number defined by the recurrencerelations: χ = 1 , χ k +1 = − i ν + k + 1 | ν + k + 1 | χ k . The last relation is equivalent to χ k − = − i ν − k + 1 | ν − k + 1 | χ k . Hence, the coefficients can be written as ( k > χ k = χ − k = ( − i ) k (cid:0) + iλ (cid:1) (cid:0) + iλ (cid:1) . . . (cid:0) k − + iλ (cid:1)(cid:12)(cid:12)(cid:0) + iλ (cid:1) (cid:0) + iλ (cid:1) . . . (cid:0) k − + iλ (cid:1)(cid:12)(cid:12) . (35)Now let H (0) NW := L ( S ) denote the space of Newton-Wigner wavefunctions at time t = 0, and build the composition U (0) B := U B ◦ W † . This transformation maps a NW-wavefunction to the corresponding state in Bargmann’s representation. Choosing the basis (cid:8) | k i = exp( ikθ ) / √ π, k ∈ Z (cid:9) for H (0) NW , one has W † | k i = ( − k u k . Therefore, a NW-wavefunction φ NW (0 , θ ) = P φ k | k i corresponds to a vector φ B ( θ ) = P ( − k χ ∗ k φ k | k i inBargmann’s representation.A Philips-Wigner state φ ( π/ localized at θ = π/ √ πl k in H ′ given by the explicit formula displayed in Eq. (C5). Such state can be mapped to H with the help of the unitary transformation U † B . One finds that U † B φ ( π/ has coefficients φ ( π/ k = √ πl k χ k in H . From Eq. (35) and Eq. (C5), it follows that l k χ k = i k . Now comparewith our localized states. From Eq. (32), a NW-state η ( π/ localized at θ = π/ t = 0has coefficients η ( π/ k = i k / √ π . Therefore, up to an irrelevant normalization factor, thelocalized states at θ = π/ t = 0, and with therestriction to the principal series, to recover the results of [24].Therefore, working in a quite different setting, we have arrived at the same localizedstates as obtained in [24] in the context of group theory. We would like to discuss now themain differences and similarities between these approaches, and emphasize some technicalsimplifications and a conceptual clarification we believe our work brings to the discussion oflocalizability in de Sitter spacetime.Compare with what happens in Minkowski space. The work of Newton and Wigner [1]was written in terms of distributions describing improper states localized at specific points.21he results were latter reformulated by Wightman [2] in terms of projectors E ( S ) in aHilbert space associated with observables describing the property of the particle being in aregion S of space. So the idea of localization at a point was replaced by localization in afinite region. The main advantage in doing this is that technical complications associatedwith the theory of distributions are avoided. In short, the approach of Wightman allowedthe results of [1] to be derived in a rigourous manner in the simplest context of a Hilbertspace of square-integrable functions. In de Sitter space, the work of Philips and Wignerfollows the original idea of looking for localized states, while we have studied the analogueof Wightman’s localizability postulates, describing the probability P ( S ) of detection of theparticle in a finite, measurable region S in terms of the norm of its state φ projected to asuitable subspace in the Hilbert space H , i.e., P ( S ) = R S dθ | φ NW ( θ ) | = k χ S φ k , where χ S isthe projection operator which acts as the characteristic function of S in the Newton-Wignerrepresentation.In Minkowski space, the approaches of [1] and [2] are essentially equivalent: the conditionsused by Wightman were direct translations of the original conditions on distributions. Inde Sitter spacetime, however, there are important differences between our approach andthat of [24]. Firstly, distinct sets of axioms are used: one of the axioms of [24] (Axiom cin Appendix C) is replaced in our approach by postulate IV. These conditions play similarroles in each approach, being required for the elimination of sign ambiguities encoded in thefactors s k (0) = ± 1. However, the physical motivation of Axiom (c)—minimal disturbanceof a localized state at θ under the action of boosts which leave the point θ invariant—is unclear. According to our previous discussion, the ambiguities can be fixed in a morenatural way by a condition of ‘optimal localization’—that the probabilities should be asconcentrated as possible about the wavefunction—and can be easily implemented requiringa reasonable large mass limit.Moreover, there is a residual ambiguity in [24]. This is removed by requiring that thelocalized states have positive energy in the Minkowski space limit, what is done using acomplicated process of contraction (roughly speaking, by taking the limit α → ∞ ) of Liealgebra representations of the de Sitter group. This step is not necessary in our approach.Such a simplification is due to the fact that negative energy states are not allowed here,the starting point being a space of positive-energy solutions of the Klein-Gordon equation.Importantly, we employ the Hadamard condition to select states with positive energy, making22t unnecessary to check the sign of the energy of the localized states through a Minkowskispace limit.A second remark concerns the connection between representations of de Sitter group andwave equations in de Sitter space. As widely known, the irreducible representations of deSitter group were classified by Bargmann in [31]. Yet, when one considers applications toquantum field theory, it is natural to ask for an interpretation of the representations in spacesof solutions of wave equations in de Sitter space. More than that, one wants to restrict topositive-energy solutions. Here we have used the Hadamard condition in order to select asuitable space of positive-energy solutions of the Klein-Gordon equation, and displayed suchan interpretation for the principal and complementary series representations.In doing so, we identified the one-particle subspace of the quantized massive scalar fieldtheory with a particular irreducible representation of the de Sitter group. In an intuitivesense, that identification provides a spacetime representation for vectors in the more abstract(from a physicist’s perspective) Bargmann’s representations: it allows one to see these vec-tors as wavefunctions in de Sitter space. In particular, it becomes possible to determine howthe localized states are spread in spacetime, i.e., Eq. (31) (remember that relativistic local-ized states are not strictly localized, but “as localized as possible” states). This questioncould not be dealt with without a prescription for the choice of the positive-energy states,and was not investigated in [24].In Minkowski space, if one wants to see how a localized state defined in momentum spacelooks like in spacetime, one just goes to configuration space, using the well-known transfor-mation φ ( p ) φ ( x , t ), the relativistic Fourier transform. The result is a Hankel function,with an exponential decay for large spatial distances [1]. Such a familiar transformation hasno natural analogue in curved spacetimes. A Bargmann’s representation can be seen as asort of ‘momentum representation’, but the transformation to configuration space—that ofwavefunctions in de Sitter space—is not unique: it depends on the choice of a vacuum, orequivalently, of the positive-energy states. It is necessary to combine purely group theoret-ical results with the modern specification of positive-energy states given by the Hadamardcondition in order to find the spacetime representation of states of interest.23 . Time-evolution of the Newton-Wigner wavefunction The postulates I–IV determine uniquely the form of the Newton-Wigner wavefunction attime t = 0. They also impose restrictions on the time-evolution of the wavefunction, butdo not fix it uniquely. In this section we discuss a solution of these conditions, suggestedby an analogy with the definition of the position operator in Minkowski space discussedin the beginning of Section III A. It is natural that in generalizing structures defined inMinkowski space to the context of curved spacetimes some non-uniqueness might be metwith. Nevertheless, one would certainly like to restrict it as much as possible, and in deSitter space there is the advantage of dealing with a maximally symmetric spacetime. Wediscuss later in this section the possibility of using the group symmetry operations in deSitter space in order to fix the Newton-Wigner dynamics, but we will answer this questionin the negative, at least for a simple implementation of these symmetries.So, let us describe a solution of the time-evolution W t of the Newton-Wigner wavefunc-tion. Keep in mind the discussion in Section III A. From Eq. (12) and the explicit form ofthe normal modes given in Eq. (4), a generic vector in H can be written as φ ( t, θ ) = X k φ k r γ k T kν (cid:0) i sinh( t/α ) (cid:1) e ikθ √ π , (36)with P k | φ k | = 1. The scalar product is given by Eq. (8), which reduces to h φ | ψ i = P φ ∗ k ψ k in this representation. Now, introduce N k ( t ) := 1 γ k (cid:12)(cid:12) T kν (cid:0) i sinh( t/α ) (cid:1)(cid:12)(cid:12) , and ϕ k ( t ) := − arg (cid:16) T kν (cid:0) i sinh( t/α ) (cid:1)(cid:17) (37)and define a time-dependent unitary transformation W t given for each t by φ ( t, θ ) φ NW ( t, θ ) = X k φ k e − iϕ k ( t ) e ikθ √ π . (38)Above, a factor [2 N k ( t )] − / is absorbed in each coefficient φ k , and a time-evolutionexp[ − iϕ k ( t )] is associated with each mode. The analogy with the definition of the positionoperator in Minkowski space should be clear. The absorption of the factor [2 N k ( t )] − / is a consequence of postulates I–III; what is added is the choice of the phases ϕ k ( t ) pre-scribed by Eq. (37). Note that for t = 0, it follows from Eq. (28) and Eq. (37) that24xp( − iϕ k (0)) = ( − k . Substituting that in Eq. (38), we get the operator W obtained inthe previous section. Hence, we have the right transformation at t = 0. Moreover, it is easyto check that the postulates I–IV are satisfied. It must be verified that e − iϕ k ( t ) = e − iϕ − k ( t ) (parity), e − iϕ k ( t ) = e iϕ k ( − t ) (time-reversal), and that the large mass limit is correct. Thatthe parity condition is satisfied is a consequence of Eq. (11), which shows that T − kν ( z ) and T kν ( z ) are proportional, with a positive proportionality factor. The time-reversal conditionis a consequence of the identity h T kν (cid:0) i sinh( t/α ) (cid:1)i ∗ = T kν (cid:0) − i sinh( t/α ) (cid:1) . And the large masslimit can be established using Eq. VI.(90) of [27]: P µν ( z ) ≃ ν µ − / √ π ( z − / (cid:2) − e ± i ( µ − / π e i ( ν +1 / ω + e − i ( ν +1 / ω (cid:3) , for ν → ±∞ , where ν = ν + iν and z = cos ω , together with the Stirling approximation for complexnumbers with large imaginary part [33], which gives (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) − k − λi (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≃ √ πλ − k e − πλ/ , so that the analogue of Eq. (24) is r γ k T kν (cid:0) i sinh( t/α ) (cid:1) ≃ ( − k (2 λ cosh (cid:0) t/α ) (cid:1) − / e − iλt/α for large m . The factor (cid:0) λ cosh( t/α ) (cid:1) / is a mass-dependent normalization coefficient,from Eq. (17).There is a simple physical interpretation for the prescribed choice of the phases ϕ k ( t ).The Newton-Wigner wavefunction at time t is described in Eq. (38) as a square-integrablefunction on a circle of radius 1. Its squared value gives the probability of finding the particlein an infinitesimal interval of angles. But the actual spatial radius of the corresponding timeslice is the scale factor a ( t ) = α cosh( t/α ), so if one wants to get the probability density inthe spatial slice itself, a factor of p a ( t ) must be included, leading to˜ φ NW ( t, θ ) = 1 p α cosh( t/α ) X k φ k e − iϕ k ( t ) e ikθ √ π . In this case, the transformation which defines ˜ φ NW ( t, θ ) involves the absorption of a factor[2 ω dSk ( t )] − / , with ω dSk ( t ) := 1 γ k (cid:12)(cid:12) T kν (cid:0) i sinh( t/α ) (cid:1)(cid:12)(cid:12) α cosh( t/α ) . (39)25here is an interesting relation between the derivative of the phases ϕ k ( t ) and the factors ω dSk ( t ). Pick Eq. (10) and consider it on the imaginary axis, with z = iy . Divide it by (cid:12)(cid:12) T kν ( − iy ) (cid:12)(cid:12) = T kν ( iy ) T kν ( − iy ): − γ k (1 + y ) | T kν ( − iy ) | = (cid:20) T k ′ ν ( − iy ) T kν ( − iy ) + T k ′ ν ( iy ) T kν ( iy ) (cid:21) = 2Re (cid:20) T k ′ ν ( iy ) T kν ( iy ) (cid:21) . (40)The last identity follows from the fact that the derivative of a Legendre function can bewritten as a (real) linear combination of Legendre functions, which are complex conjugatedby the inversion iy → − iy . Now, Eq. (37) implies ϕ ′ k ( t ) = − α cosh( t/α ) Re " T k ′ ν (cid:0) i sinh( t/α ) (cid:1) T kν (cid:0) i sinh( t/α ) (cid:1) , what in turn, using Eq. (39) and Eq. (40), leads to ϕ ′ k ( t ) = ω dSk ( t ) . Therefore, the dynamics described by Eq. (38) corresponds to that generated by normalmodes k with a time-dependent energy ω dSk ( t ), with the time t = 0 representation fixed bypostulates I–IV. In other words, we are looking at Eq. (39) as a time-dependent dispersionrelation giving the energy of a mode k as a function of time.The solution we have found for the localizability postulates was written in a form validboth for the principal and complementary series, i.e. Eq. (38). But it must be noticed thatfor representations in the complementary series the normal modes T kν (cid:0) i sinh( t/α ) (cid:1) are purelyreal and not oscillatory, implying that the phases exp( − iϕ k ( t )) are constant. Therefore, thedynamics of the Newton-Wigner function in this case is trivial; there is no time-evolution.Notice that complementary series representations are associated with particle masses andde Sitter radii satisfying 4 α µ / ~ < 1. This inequality essentially states that the Comptonwavelength of the scalar particle is bigger than the radius of de Sitter spacetime. Underthese circumstances the NW-wavefunction describing the position of the particle is not ableto move, and the notion of localization is trivial. In the principal series representations,on the other hand, the Newton-Wigner function has a nontrivial dynamics, and the time-evolution of the distribution of probabilities might be used in order to study how wavepacketsmove. 26t is a well-known fact that the Newton-Wigner operator displays acausal features in thepropagation of wavepackets in Minkowski space, i.e. superluminal propagation is possible[34], and the same phenomenon should be expected in the case of de Sitter spacetime. Thecentral result of [34] was a determination of upper bounds for the probability of detectionof superluminal propagation. These depend on the experimental techniques available, butusing some generous estimates of experimental parameters it was found in [34] that theprobability of detecting superluminal propagation in a single experiment is smaller than10 − . It was also argued that a more carefull exam should reduce this bound considerably.It is natural to expect that curvature-dependent corrections to this result should be presentin de Sitter spacetime, but given the order of magnitude of the effect, there shall be aconsiderable window in the space of parameters ( m , α , a typical time-scale T , etc) whereacausal effects are negligible.For particles in the principal series, the position operator should be useful for the studyof the dynamics of relativistic particles in de Sitter spacetime in a semiclassical regime. Onecould prepare an initially localized wavepacket, and study how the probability density prop-agates and spreads. It is clear that the expansion of the universe will enforce an additionalspreading in the dispersion of wavepackets compared to that in Minkowski space, so that awavepacket resembling a localized particle will remain so only for a finite amount of time.But that is a general feature in the study of classical limits of quantum systems, true even fornon-relativistic systems: typically a classical limit is approached only inside a finite intervalof time, as discussed in the classical work [9]. For how long the classical limit is reasonable,and how exactly the wavepackets spread is described by the time-evolution of the positionprobability distribution.Finally, we should briefly mention that, following a suggestion made in [24], but notfurther developed there, we analysed the possibility of deriving the dynamics of the NW-wavefunction from the action of the de Sitter group on the NW-wavefunctions at time t = 0,thus trying to remediate with boots and rotations the absence of time-translation isometries.However, our attempts led to incompatibilities with our more natural postulates I–III andwe came to the conclusion that the implementation of this seemingly sound idea is actuallyquite subtle, perhaps impossible. 27 V. PERSPECTIVES We have showed that a notion of localization exists for massive neutral scalar fields in deSitter space compatible with the prescription for the choice of positive energy modes encodedin the Hadamard condition. In de Sitter space, this condition is equivalent to the choice of theBunch–Davies vacuum as the “physical vacuum” among the family of α -vacua. Therefore,we have proved that localizability is compatible with this choice of vacuum. A naturalquestion arises whether other choices of vacuum are compatible or not with localizability.If they are not, that would be another argument in favor of the Bunch–Davies vacuum. Weexpect to investigate this problem in a future work.Another direction of research is related to the problem of understanding the classicallimit of quantum field theories in curved spacetimes. Following the general procedure forstudying classical limits introduced by Hepp in [9], we have proved in a previous work [10]that the quantum theory of the free neutral massive scalar field in Minkowski space has twodistinct kinds of classical limits: one of them describing a classical field theory, the otherone a classical particle dynamics. The Newton-Wigner position operator is used in orderto prove the existence of the latter. We expect that the same problem can be investigatedin de Sitter space along similar lines, with the position probability distributions discussedherein playing the role of the Newton-Wigner operator.We have considered particles with positive mass in the principal and complementaryseries of representations of de Sitter group. In the case of the complementary series, theposition operator was found to be trivial, without dynamics. Hence, a nontrivial classicallimit should exist only for representations in the principal series. Moreover, it has beenrecently discovered that it is also possible to formulate sensible free quantum fields in deSitter space using representations with a negative mass, the so-called tachyonic fields of[35]. The question of the localizability of these fields was not treated here, and could beinvestigated in a future work. ACKNOWLEDGMENTS This work was supported by FAPESP under Grant 2007/55450-0. NY thanks the ErwinSchr¨odinger Institute, Vienna, for support and hospitality during the program “Quantum ield Theory on Curved Spacetimes and Curved Target Spaces” . He also thanks U. Moschellafor discussions during the initial stages of his studies of quantum fields in de Sitter spacetime.Both authors specially wish to thank the referees for important remarks on a previous versionof this manuscript that led to considerable improvements in our results. Appendix A: Proof of [ T kν ( iy )] ∗ = T kν ( − iy ) The functions T kν ( z ) are defined for | z − | < T kν ( z ) := (1 − z ) k/ Γ( ν + k + 1)2 k Γ( k + 1)Γ( ν − k + 1) f ( z ) , (A1)with f ( z ) := F (cid:18) − ν + k, ν + k + 1 , k + 1; 1 − z (cid:19) . Let us see what happens to the function under complex conjugation. For k ≥ 0, thehypergeometric function can be represented as a convergent power series in the radius | z | < j + 1)-th term in the expansion of f in powers of (1 − z ) / Q jl =0 ( − ν + k + l )( ν + k + 1 + l ) Q jl =0 ( k + 1 + l ) . The denominator is real, so ignore it. Recall that ν is a root of the quadratic equation ν ( ν + 1) = − α µ , so whenever ν ( ν + 1) makes an appearance, it is a real number. It followsthat every factor in the product is real. Thus, the power function has real coefficients, and[ f ( z )] ∗ = f ( z ∗ ).Now, the gamma functions. The factor Γ( k + 1) is real. The part that matters isΓ( ν + k + 1)Γ( ν − k + 1) = ( ν + k )( ν + k − · · · ( ν − k + 2)( ν − k + 1) . This can be rewritten as k − Y l =0 ( ν + k − l )( ν + 1 − k + l ) = k − Y l =0 [ ν ( ν + 1) − ( k − l ) + ( k − l )] , which is also real. Besides that, each factor in the product is a negative number: ν ( ν + 1) = − α µ , and ( k − l ) ≥ ( k − l ), since k − l is an integer. Therefore, the product is negativefor odd k , and positive for even k . This result will be needed somewhere else.Finally, take z = iy , y ∈ R , and consider the factor (1 − z ) k/ . Here one must be careful.The functions T kµ are defined with square roots cut along distinct lines: the factor √ − z 29s cut along x > √ z has a cut along x < − 1. Withthese choices, p − ( ix ) = | iy | . Then it follows that { [1 − ( ix ) ] k/ } ∗ = [1 − ( − ix ) ] k/ ,so that [ T kν ( iy )] ∗ = T kν (( iy ) ∗ ) = T kν ( − iy ), at least in the radius | z | < k ≥ T kν , introduce an auxiliary analytic function[ T kν ( − x, y )] ∗ . This function coincides with T kν ( − z ) along the imaginary axis inside the radius | z | < 1. Moreover, both functions are defined on the same domain: T kν ( z ) is single-valuedon a domain invariant both under inversion z → − z , and inversion of the real part ( x, y ) → ( − x, y ). Hence, [ T kν ( − x, y )] ∗ = T kν ( − z ). Restricting to the imaginary axis, [ T kν ( iy )] ∗ = T kν ( − iy ). The result is extended to negative k using the relation T − kν ( z ) = ( − k Γ( ν − k + 1)Γ( ν + k + 1) T kν ( z ) . (A2)It was already proved that the factor with the Γ’s is real. Appendix B: Two-point function The two-point function, G := h Ω | φ ( t, θ ) , φ ( t ′ , | Ω i , is given by: G = X k u k ( t, θ ) u ∗ k ( t ′ , X k Γ( − ν − k )Γ( ν − k + 1)4 π e ikθ T kν ( iy ) T kν ( − iy ′ )= 14 | sin νπ | X k ( − ) k Γ( ν − k + 1)Γ( ν + k + 1) e ikθ T kν ( iy ) T kν ( − iy ′ ) , with y = sinh( t/α ). Call the sum in the last line S . Using (A2), it can be written as S = X k = −∞ (cid:2) e ikθ T − kν ( iy ) T kν ( − iy ′ ) (cid:3) − T ν ( iy ) T ν ( − iy ′ )+ ∞ X k =0 (cid:2) e ikθ T kν ( iy ) T − kν ( − iy ′ ) (cid:3) = 2 ∞ X k =0 cos( kθ ) T kν ( iy ) T − kν ( − iy ′ ) − T ν ( iy ) T ν ( − iy ′ )= 2 ∞ X k =0 ǫ k Γ( ν − k + 1)Γ( ν + k + 1) cos( k ( π − θ )) T kν ( iy ) T kν ( − iy ′ ) , ǫ k = 1 − δ k / 2, i.e., ǫ k is 1 except for k = 0, when it is 1 / 2. There is a nice summationtheorem for Legendre functions [27], P ν (cid:18) z z + q − z q − z cos θ (cid:19) = 2 ∞ X k =0 ǫ k Γ( ν − k + 1)Γ( ν + k + 1) cos( kθ ) T kν ( z ) T kν ( z ) , which leads to S = P ν (cid:16) yy ′ − p − ( iy ) p − ( − iy ′ ) cos θ (cid:17) = P ν ( Z ) . The argument in the function P ν ( Z ) can be written in invariant form: Z = sinh( t/α ) sinh( t ′ /α ) − cosh( t/α ) cosh( t ′ /α ) cos θ = α − X · X ′ , where X is the vector in the Minkowski space M corresponding to the point ( t, θ ) in deSitter space, while X ′ corresponds to the point ( t ′ , G = G ( Z ) = 14 | sin( νπ ) | P ν ( Z ) . The Legendre function is singular at Z = − 1, where a cut begins which extends along thereal axis to −∞ . This value has a simple geometric interpretation. Recall that the causalityrelations on de Sitter hyperboloid are inherited from the Minkowski ambient space: twopoints x, x ′ are space (light,time) related if their corresponding vectors are space (light,time)-like. In particular, light-like related vectors satisfy ( X − X ′ ) = 0 ⇒ X · X ′ = − α , so that Z = − G ( Z ) = 14 | sin( νπ ) | F (cid:18) ν + 1 , − ν, 1; 1 − Z (cid:19) . Compare with the original Bunch and Davies work [29]. In their notation, a coefficient µ isintroduced: µ = r − ξ − m α , in terms of which our ν becomes ν = − / µ . It is easy to check this relation. Ourdefinition (3) of ν can be rewritten using R = − /α as ν = − ± r − ξ − m α . νπ ) = sin(( − / µ ) π ) = ( − 1) cos( µπ ), where µ ∈ ( − / , / 2) or is purelyimaginary, so that cos( µπ ) is positive either way. Thus, G ( Z ) = 14 sec( µπ ) F (cid:18) ν + 1 , − ν, 1; 1 − Z (cid:19) . That is just the expression in Eq. (2.13) for the two-point function in [29]. Appendix C: Philips and Wigner localized states Our work has a close relation with that of Philips and Wigner [24], and for the sake ofthe reader, we present now a brief review of their little known article. Their purpose wasto investigate how the existence of localized states is related to the condition of positivityof the energy. But it was not known at that time how to define positive energy states incurved spaces, so in order to check the sign of the energy of a given state it was necessary tostudy the limit where the geometry of de Sitter approached that of Minkowski space, whatwas done invoking a contraction of the group representation. Although the problem remainsunsolved in general, it is now known that, at least in the case of spacetimes with a compactCauchy surface, the Hadamard condition is sufficient to fix the ambiguity in the choice ofthe positive energy solutions [21].Let us describe the unitary representation of the de Sitter group used in [24]. We restrictto the case of O (2 , 1) which is relevant here. Let H ′ be the set of square-integrable functions ψ ( θ ) on the unitary circle S on the Euclidean plane R . Extend these functions to the wholeplane: ψ ( θ ) f ( ρ ) ψ ( θ ), where ρ is the radius ρ = p ( X ) + ( X ) , and f ( ρ ) is a fixedfunction, smooth and square-integrable on the plane. Rotations are realized as rotations onthe circle, i.e., N = − ∂/∂θ , (C1)and infinitesimal boosts are represented by N = − sin θ ∂∂θ + ν cos θ ,N = cos θ ∂∂θ + ν sin θ , (C2)where ν = − / iλ . The generators can be integrated to give finite boosts and rotations,so that there are unitary operators U ( S ) corresponding to each element S of the restrictedde Sitter group. The parity operator P , understood as the representation of the geometric32peration p of reversing the axis X in the ambient Minkowski space, must satisfy the grouprelations up to some projective factor, P U ( S ) = ω ( S ) U ( pSp ) P , where S is any Lorentz transformation in the restricted de Sitter group. But it can be provedthat ω ( S ) = 1, and that P ψ ( X , X ) = ± ψ ( − X , X ) , where the choice of the sign must be the same for all ψ . This choice is physically irrelevant,so just pick the sign +1. For the time-reversal operator T , the group relations lead toessentially two possibilities, corresponding to a unitary T u or an anti-unitary T a , given by T u ψ ( X , X ) = ± ψ ( − X , − X ) , and T a ψ ( θ ) = Z K ( θ − θ ′ ) ψ ∗ ( θ ′ ) dθ ′ , (C3)where the kernel K is given in Fourier expanded form by K ( θ − θ ′ ) = X a k e ik ( θ − θ ′ ) , a k +1 a k = − + k − iλ + k + iλ , (C4)with a = 1 / (2 π ). The coefficients automatically satisfy a k = a − k . In order that T a isuniquely defined, it is assumed that T = 1 (it could be − TP = PT (therecould be a phase difference).The definition of the localized states is based on a set of three postulates, which repre-sent the de Sitter version of the postulates of Newton and Wigner adopted in the case ofMinkowski space [1]. The postulates are: a: A localized state is invariant under reflections that leave the point of localization invari-ant. b: A rotation applied to a localized state gives a new localized state –the point of localizationis just rotated accordingly. c: A boost which keeps the point of localization invariant changes the state as little aspossible. 33he first result is that the postulates cannot be satisfied with a unitary time-reversaloperator. Hence, the existence of localized states implies that T is anti-unitary –it must bethe T a defined in Eqs. (C3), (C4). In this case, the postulates are satisfied by two distinctsets of localized states.Consider a state ψ ( θ ) localized at θ = π/ t = 0. It must be invariant under parityand time-reversal. Writing a Fourier expansion ψ ( θ ) = X l k e ikθ , invariance under parity implies l − k = ( − k l k , while invariance under time-reversal leads to2 πa k l ∗− k = l k . Combining these results, and using (C4), it follows that l k +1 l k = ζ k +1 / + k − iλ (cid:2) ( + k ) + λ (cid:3) / , where the ζ ’s are real numbers satisfying ζ k +1 / ζ − k − / = 1 . Then condition (b), together with (c), which reduces here to minimal deformation underboosts along X , fixes ζ = 1 or ζ = − 1. The first possibility is ruled out by looking whathappens in the contraction of the de Sitter group representation to a representation of theinhomogeneous Lorentz group. The choice ζ = 1 corresponds to a state of negative-energyin Minkowski space in this limit. So it must be ζ = − 1. The Fourier coefficients of ψ arethen completely determined, being given by ( k > l k = ( − k (cid:0) − iλ (cid:1) (cid:0) − iλ (cid:1) . . . (cid:0) k − − iλ (cid:1)(cid:12)(cid:12)(cid:0) − iλ (cid:1) (cid:0) − iλ (cid:1) . . . (cid:0) k − − iλ (cid:1)(cid:12)(cid:12) ,l = 1 (C5) l − k = ( − k (cid:0) − + iλ (cid:1) (cid:0) − + iλ (cid:1) . . . 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