GGeneralized Wentzell boundary conditions andquantum field theory
Jochen ZahnInstitut f¨ur Theoretische Physik, Universit¨at LeipzigBr¨uderstr. 16, 04103 Leipzig, [email protected] 10, 2018
Abstract
We discuss a free scalar field subject to generalized Wentzell bound-ary conditions. On the classical level, we prove well-posedness of theCauchy problem and in particular causality. Upon quantization, weobtain a field that may naturally be restricted to the boundary. Wediscuss the holographic relation between this boundary field and thebulk field.
Holography has been a main theme in theoretical high energy physics andquantum gravity in the last two decades [1–4]. Inspired by the gauge/gravityduality, studies of holographic aspects were often considering d + 1 dimen-sional Anti deSitter space (AdS) and its (conformal) boundary, d dimen-sional Minkowski space. However, holography is a generic aspect of quantumfield theory on space-times with time-like boundaries, raising the questionwhich of the properties of holography on AdS are generic, and which onesare special to AdS.By holography we here mean an isomorphism of the algebras of bulkand boundary observables. For AdS such an isomorphism was abstractlyconstructed in [5], using the fact that wedges in AdS have double cones asboundaries: one identifies the algebras of observables localized in wedges ofAdS with the algebras of observables localized in the corresponding doublescones on the boundary. One can also proceed via Wightman functions,and define the Wightman function of the boundary theory to coincide withappropriately scaled limits of bulk Wightman functions [6]. This amounts todefining the boundary field as an appropriately scaled limit of the bulk field,cf. also [7] for the relation between the boundary value of the bulk field and1 a r X i v : . [ m a t h - ph ] D ec he “dual field” in the original formulation of the AdS/CFT correspondence.Let us list some of the properties of holography on AdS: • The boundary theory is a conformal field theory. This is a genericfeature on AdS [5, 6], independent of the concrete choice of the bulkfields. • The correspondence maps bulk observables localized in a compact re-gion to boundary observables localized in compact regions. • For a bulk theory with local observables, the boundary theory will notfulfill the time-slice axiom [5]. • The boundary conformal field in general has a positive anomalousdimension. The basic example is the massive scalar field [4], wherethe anomalous dimension (( d ) + µ ) + 1 of the dual field is strictlypositive.In the following, we study the holographic relation between a massivescalar field on d + 1 dimensional Minkowski space with d dimensional time-like boundaries. Not surprisingly, we find that the boundary field theory isnot conformal, and that the bulk observables localized in compact space-time regions are mapped to boundary observables that are delocalized. Thefirst two properties in the above list thus seems to be specific to AdS. Alsoin our setting, the time-slice axiom does not hold for the boundary theory.Regarding the last point in the above list, it seems obvious that theboundary field, being the boundary limit of the bulk field, inherits its short-distance behavior. Hence, for a scalar field one would expect a short-distancesingularity ( x − y ) − ( d − for the two-point function. However, it turns outthat there are boundary conditions ensuring that the boundary two-pointfunction has short-distance singularity ( x − y ) − ( d − , as one expects fora scalar field in d space-time dimensions. These are so-called generalizedWentzell boundary conditions. These are of interest in their own right, andthe first part of this work will be devoted to their study.Concretely, we consider a free scalar field on the ( d + 1)-dimensionalspace-time M = R × Σ, with the spatial slices Σ having a boundary ∂ Σ.The two main examples will be the half-space Σ = R d + = R d − × [0 , ∞ )and the strip Σ = R d − × [ − S, S ]. We do not impose boundary conditionsby hand, but supplement the bulk action with an action for the boundary,which is of the same form. Concretely, S = S bulk + S boundary = − (cid:90) M g µν ∂ µ φ∂ ν φ + µ φ − c (cid:90) ∂M h αβ ∂ α φ∂ β φ + µ φ (1)where g is the Minkowski metric on the bulk and h is the induced metric onthe boundary. c is a positive constant with the dimension of a length. The2implest case where actions of this type appear may be the open Nambu-Goto string with masses at the ends [8], i.e., S = − γ (cid:90) Ξ (cid:112) | g | − m (cid:90) ∂ Ξ (cid:112) | h | , where Ξ is the world-sheet, g the induced metric in the bulk and h the in-duced metric on the boundary. In fact, for rotating string solutions to thisaction, the quadratic part of the action for fluctuations normal to the plane ofrotation are exactly of the form (1) with d = 1, µ = 0 and Σ = [ − S, S ], cf. [9].Higher-dimensional examples can be straightforwardly obtained by general-izing the Nambu-Goto action to higher dimensional objects, i.e., branes.Interestingly, actions of the above type were also considered in the con-text of the AdS/CFT correspondence. In treatments of the gauge/gravityduality, one supplements the bulk Einstein-Hilbert action with cosmologicalconstant with a counterterm boundary Einstein-Hilbert action with cosmo-logical constant [10]. The role of the length scale c is there played by theradius of curvature of the AdS space-time. Note, however, that the coun-terterm action is fixed by the requirement of obtaining a finite stress-energytensor by variations of the metric on the boundary, whereas in (1) c can bechosen arbitrarily. Also note that the analogue of c is negative for the grav-itational counterterms. In holographic renormalization [11], also boundarycounterterms for scalar fields are introduced, of the same form as in (1).Still, c is negative, but it is no longer simply given by the radius of curva-ture. As we will see, negative c leads to severe difficulties already in thetreatment of the classical system.Another instance where boundary terms of the above form arise natu-rally is in models with a non-renormalizable bulk action. For a boundary ofco-dimension k , the power-counting degree of divergence of terms localizedat the boundary is reduced by k . This was discussed in [12] in the contextof a scalar field with Dirichlet boundary conditions. Hence, in renormaliz-able theories, where the field strength counterterm has at most a logarith-mic divergence, no kinetic counterterm is necessary at the boundary. This,however, changes in non-renormalizable theories, where the occurrence of akinetic term at the boundary is thus generic.Variation of the action (1) yields, upon integration by parts, the equa-tions of motion − (cid:50) g φ + µ φ = 0 in M, (2) − (cid:50) h φ + µ φ = c − ∂ ⊥ φ in ∂M. (3)Here ∂ ⊥ denotes the inward pointing normal derivative. Using (2), one maywrite (3) alternatively as ∂ ⊥ φ = c − ∂ ⊥ φ in ∂M. (4)3hese equations may be read as an equation of motion for the bulk (2),supplemented by boundary conditions (3) or equivalently (4). Boundaryconditions of this type are known as generalized Wentzell or Feller-Wentzelltype boundary conditions [13, 14]. It turns out that it is physically more appropriate to consider (2) and (3)as two wave equations, which are coupled by the solution of the bulk equa-tion (2) providing a source on the r.h.s. of the boundary equation (3) andthe solution of the boundary equation (3) providing a Dirichlet type bound-ary condition for the bulk equation (2). This is reminiscent of the fact thatin the AdS/CFT correspondence the boundary values of the bulk fields actas sources for the boundary field. Mathematically, this interpretation isimplemented by considering Cauchy data in ( L (Σ) ⊕ L ( ∂ Σ)) , or relatedSobolev spaces. The fact that the Hilbert space L (Σ) ⊕ L ( ∂ Σ) is appro-priate for Wentzell boundary conditions was already noted in [13–15, 17,19].Physically, this means that the boundary can carry energy. We will provewell-posedness of the Cauchy problem and causal propagation. This may beinteresting in itself, as it contradicts the folklore wisdom that Robin bound-ary conditions are the most general sensible linear boundary conditions forthe wave equation.As a by-product of the discussion of the wave equation (2), (3), we ob-tain a (generalized) orthonormal basis of solutions. This can be used for acanonical quantization of the system. The unusual space of Cauchy datathen has interesting physical implications. In particular, it turns out that itis possible to restrict the field to the boundary in a natural way, resultingon a dimensionally reduced field, which, interestingly, has the short-distancebehavior of a free scalar field theory on ∂M . Furthermore, bulk observablescan be holographically mapped to boundary observables. This is possible asthe boundary field is a generalized free field [20]. Heuristically, the transver-sal degree of freedom that is lost by the restriction to the boundary is tradedfor the possibility to excite these higher modes.The classical aspects of the field equations (2), (3) are discussed in thenext section. In Section 3, the quantization of the system is performed andthe holographic aspects are discussed. In Section 4, we comment on therelation to other boundary conditions and holography on AdS. I am grateful to Konstantin Pankrashkin for pointing out to me the mathematicalliterature on the subject. Note, however, that in the context of the wave equation, thegeneralized Wentzell boundary conditions considered in the mathematical literature areusually slightly different, with (cid:50) h on the l.h.s. of (3) replaced by ∂ [15], i.e., the termi-nology used here does only coincide with the one used in some of the literature in thecase d = 1. An example where generalized Wentzell boundary conditions are formulatedso general that the case (3) is also covered for d > µ = 0 and additional damping or source terms, was studied in [17, 18], where theseboundary conditions were called “kinetic” or “dynamical”. otation and conventions We use signature ( − , + , . . . , +). The coordinates on Σ = R d − × R + or R d − × [ − S, S ] are usually denoted ( x, z ). For x ∈ R ,d − , the spatial part isdenoted by x . In the case Σ = R d − × [ − S, S ], ∂ ± Σ denotes the componentat ± S , and analogously for ∂ ± M . Fourier transformation is defined asˆ f ( k ) = (2 π ) − d (cid:90) f ( x ) e − ikx d d x. The sign of the Laplacian is defined as ∆ = ∂ i ∂ i . For a set Σ ⊂ M , we denoteby D + (Σ) its future domain of dependence, i.e., the set of all points p suchthat all past inextendable causal piecewise C curves through p intersectΣ. These curves may also be part of the boundary. As usual, H s ( R d ) and S ( R d ) denote Sobolev and Schwartz spaces, the latter being the space ofrapidly decreasing smooth functions. Let us study the wave equation (2), (3). It is straightforward to check thatthe symplectic form σ (( φ, ˙ φ ) , ( ψ, ˙ ψ )) = (cid:90) Σ φ ˙ ψ − ˙ φψ + c (cid:90) ∂ Σ φ ˙ ψ − ˙ φψ (5)is conserved for solutions φ , ψ . It is thus natural to introduce the scalarproduct (cid:104) φ, ψ (cid:105) = (cid:90) Σ ¯ φψ + c (cid:90) ∂ Σ ¯ φψ on functions on Σ. Completion in the corresponding norm yields the Hilbertspace H = L (Σ , (cid:37) + cδ ∂ Σ ), where (cid:37) is the Lebesgue measure, and δ ∂ Σ theDirac measure on the boundary. Under completion of the space of continuousfunctions, bounded functions that are localized at the boundary ∂ Σ are notequivalent to zero. Hence, H is canonically isomorphic to L (Σ) ⊕ L ( ∂ Σ).An element of H is typically written as Φ = ( φ, φ | ) (but note that φ | neednot coincide with the boundary value φ | ∂ Σ of φ , even if it is well-defined),with the scalar product (cid:104) Φ , Ψ (cid:105) = (cid:104) φ, ψ (cid:105) L (Σ) + c (cid:104) φ | , ψ |(cid:105) L ( ∂ Σ) . (6)On H , the equation of motion (2), (3) can be written as − ∂ t Φ = ∆Φ , with ∆ = (cid:18) − ∆ Σ + µ − c − ∂ ⊥ · | − ∆ ∂ Σ + µ (cid:19) , D = (cid:8) ( φ, φ | ) ∈ H | φ ∈ H (Σ) , φ | ∈ H ( ∂ Σ) , φ | ∂ Σ = φ | (cid:9) . (7)Note that the Sobolev space H s (Σ) is given by [21, Section 4.5] H s (Σ) = H s ( R d ) (cid:110) u ∈ H s ( R d ) | supp u ⊂ R d \ Σ (cid:111) . Also note that for s > , the restriction map H s (Σ) → H s − ( ∂ Σ) is well-defined and continuous [21, Prop. 4.4.5], so that the boundary conditionmakes sense.In the following, we consider the case of the half-space Σ = R d + . Thecase Σ = R d − × [ − S, S ] is discussed at the end of the section.
Proposition 1.
With
Σ = R d + and on the domain D , ∆ is self-adjoint withspectrum contained in [ µ , ∞ ) . A normalized complete system of generalizedeigenfunctions of ∆ is given by Φ k,q = (cid:16) (2 π ) d − π (cid:0) c q + 1 (cid:1)(cid:17) − (cid:16) e ikx (cos qz − cq sin qz ) , e ikx (cid:17) , (8) with k ∈ R d − , q ∈ R + and the eigenvalue ω k,q = k + q + µ . (9) Proof.
We compute (for simplicity, we here set µ = 0) (cid:104) Φ , ∆Ψ (cid:105) = − (cid:90) Σ ¯ φ ∆ Σ ψ − (cid:90) ∂ Σ ¯ φ | ∂ ⊥ ψ | ∂ Σ − c ¯ φ | ∆ ∂ Σ ψ | = (cid:90) Σ ∂ i ¯ φ∂ i ψ + (cid:90) ∂ Σ ¯ φ | ∂ Σ ∂ ⊥ ψ | ∂ Σ − ¯ φ | ∂ ⊥ ψ | ∂ Σ − c ∆ ∂ Σ ¯ φ | ψ | = − (cid:90) Σ ∆ Σ ¯ φψ − (cid:90) ∂ Σ ∂ ⊥ ¯ φ | ∂ Σ ψ | ∂ Σ − ¯ φ | ∂ Σ ∂ ⊥ ψ | ∂ Σ + ¯ φ | ∂ ⊥ ψ | ∂ Σ + c ∆ ∂ Σ ¯ φ | ψ | , which for ψ | = ψ | ∂ Σ equals (cid:104) ∆Φ , Ψ (cid:105) iff φ | = φ | ∂ Σ . In particular, it followsthat the domain of ∆ ∗ is contained in { ( φ, φ | ) | φ | ∂ Σ = φ |} . But for ∆ ∗ tobe well-defined on ( φ, φ | ), we also need φ ∈ H (Σ), φ | ∈ H ( ∂ Σ). For theclaim on the spectrum, we compute, for Ψ ∈ D , (cid:104) Ψ , ∆Ψ (cid:105) = (cid:90) Σ ∂ i ¯ ψ∂ i ψ + c (cid:90) ∂ Σ ∂ a ¯ ψ | ∂ a ψ | ≥ , where the index a runs over the coordinates on ∂ Σ. A separation ansatz forthe generalized eigenfunctions of − ∆ Σ is φ ( x, z ) = e ikx ( A cos qz + B sin qz ) , (10) Normalization is understood in the distributional sense, i.e., (cid:104) Φ p , Φ p (cid:48) (cid:105) = δ ( p − p (cid:48) ). k ∈ R d − is real. Obviously, (cid:18) − ∆ Σ + µ − c − ∂ ⊥ · | ∂ Σ − ∆ ∂ Σ + µ (cid:19) (cid:18) φφ | ∂ Σ (cid:19) = ω (cid:18) φφ | ∂ Σ (cid:19) with ω = k + q + µ implies B = − cqA. From this, the generalized basis (8) follows by normalization.In order to discuss the regularity of the solutions, it is advantageous torestrict to µ > K r = D r +1 ⊕ D r , where for r ∈ N , D r = dom ∆ r is a Hilbert space with inner product (cid:104)· , ·(cid:105) D r = (cid:104) ∆ r · , ∆ r ·(cid:105) . (11)For r ∈ Z , r <
0, one defines D r as the completion of H w.r.t. the innerproduct (11). In particular, D ∗ r = D − r . Obviously,∆ D r = D r − . The equation of motion can on K r now be written as i∂ t (cid:18) Φ Φ (cid:19) = (cid:18) (cid:19) (cid:18) Φ Φ (cid:19) = A (cid:18) Φ Φ (cid:19) . The operator A is self-adjoint on K r with domaindom( A ) = dom ∆ r +22 ⊕ dom ∆ r +12 . In particular, the equation of motion is solved by (cid:18) Φ ( t )Φ ( t ) (cid:19) = e − iAt (cid:18) Φ (0)Φ (0) (cid:19) . The time-evolution leaves the domain invariant.It is instructive to determine the domain of ∆ . Obviously,dom(∆ ) = { Φ ∈ dom(∆) | ∆Φ ∈ dom(∆) } . In order to fulfill ∆Φ ∈ dom(∆), we certainly have to require that φ ∈ H (Σ), φ | ∈ H ( ∂ Σ). However, we also have to ensure that for ∆Φ the7oundary value of the bulk field coincides with the boundary field. Thisyieldsdom(∆ )= (cid:8) ( φ, φ | ) | φ ∈ H (Σ) , φ | ∈ H ( ∂ Σ) , φ | ∂ Σ = φ | , ∂ ⊥ φ | ∂ Σ = c − ∂ ⊥ φ | ∂ Σ (cid:9) , i.e., the bulk field has to fulfill the boundary condition (4). Analogously,one findsdom(∆ k ) = (cid:110) ( φ, φ | ) | φ ∈ H k (Σ) , φ | ∈ H k ( ∂ Σ) , φ | ∂ Σ = φ | ,∂ j − ⊥ φ | ∂ Σ = c − ∂ j − ⊥ φ | ∂ Σ ∀ j ≤ k (cid:111) . Now consider Cauchy data Ψ = (Φ , Φ ) ∈ K ∞ , with K ∞ = ∩ r K r . For time derivatives of the corresponding solutions Ψ( t ) = e − iAt Ψ, we com-pute (cid:107) ∂ mt Ψ( t ) (cid:107) K s = (cid:107) A m Ψ( t ) (cid:107) K s = (cid:107) Ψ( t ) (cid:107) K s + m = (cid:107) Ψ(0) (cid:107) K s + m , where the last equality follows from the unitarity of the time evolution. Wecan thus bound arbitrary derivatives on future Cauchy surfaces by the initialdata. It is instructive to compute (cid:107) Φ (cid:107) D = (cid:104) Φ , ∆Φ (cid:105) = (cid:104) φ, ( − ∆ Σ + µ ) φ (cid:105) L (Σ) + c (cid:104) φ | , ( − ∆ ∂ Σ + µ ) φ |(cid:105) L ( ∂ Σ) − (cid:104) φ | , ∂ ⊥ φ (cid:105) L ( ∂ Σ) = (cid:107)∇ Σ φ (cid:107) L (Σ) + µ (cid:107) φ (cid:107) L (Σ) + c (cid:107)∇ ∂ Σ φ |(cid:107) L ( ∂ Σ) + cµ (cid:107) φ |(cid:107) L ( ∂ Σ) , where we assumed that φ | ∂ Σ = φ | . Using (cid:107) Φ (cid:107) D k = (cid:107) ∆Φ (cid:107) D k − , we obtain,for k odd and Φ ∈ dom(∆ k +12 ), (cid:107) Φ (cid:107) D k = (cid:107)∇ Σ ( − ∆ Σ + µ ) k − φ (cid:107) L (Σ) + µ (cid:107) ( − ∆ Σ + µ ) k − φ (cid:107) L (Σ) + c (cid:107)∇ ∂ Σ ( − ∆ Σ + µ ) k − φ | ∂ Σ (cid:107) L ( ∂ Σ) + cµ (cid:107) ( − ∆ Σ + µ ) k − φ | ∂ Σ (cid:107) L ( ∂ Σ) . (12)For even k and Φ ∈ dom(∆ k +1 ), we analogously obtain (cid:107) Φ (cid:107) D k = (cid:107) ( − ∆ Σ + µ ) k φ (cid:107) L (Σ) + c (cid:107) ( − ∆ Σ + µ ) k φ | ∂ Σ (cid:107) L ( ∂ Σ) . (13)Denoting by H ∞ ( · ) = ∩ r H r ( · ) the intersection of Sobolev spaces, we havethus shown: 8 roposition 2. For smooth Cauchy data ( φ , φ ) ∈ H ∞ ( R d + ) × H ∞ ( R d + ) such that ∂ k +2 ⊥ φ i | ∂ Σ = c − ∂ k +1 ⊥ φ i | ∂ Σ , ∀ k ∈ N , (14) for i = 0 , , there is a unique smooth solution φ ( t ) to the wave equation (2) , (3) with µ > . The properties of the Cauchy data are conserved under timeevolution. Furthermore, denoting Φ( t ) = ( φ ( t ) , φ ( t ) | ∂ Σ ) , we have (cid:107) ∂ mt Φ( t ) (cid:107) D k +1 + (cid:107) ∂ m +1 t Φ( t ) (cid:107) D k = (cid:107) Φ (cid:107) D k + m +1 + (cid:107) Φ (cid:107) D k + m . (15)As usual, local energy estimates are very useful to prove causal propaga-tion. As can be expected from the action (1), the boundary should be takeninto account with its own weight: Proposition 3.
Let Σ , Σ be two equal time surfaces, with Σ in the futureof Σ . Let S ⊂ Σ and S = D + ( S ) ∩ Σ , cf. Figure 1. Then for a solution φ to (2) , (3) , it holds (cid:90) S ( ∂ φ ) + g ij ∂ i φ∂ j φ + µ φ + c (cid:90) S ∩ ∂M ( ∂ φ ) + h ij ∂ i φ∂ j φ + µ φ ≤ (cid:90) S ( ∂ φ ) + g ij ∂ i φ∂ j φ + µ φ + c (cid:90) S ∩ ∂M ( ∂ φ ) + h ij ∂ i φ∂ j φ + µ φ . (16) Proof.
We consider the bulk and boundary stress-energy tensors T µν = ∂ µ φ∂ ν φ − g µν (cid:16) ∂ λ φ∂ λ φ + µ φ (cid:17) ,T | ab = c (cid:2) ∂ a φ∂ b φ − h ab (cid:0) ∂ c φ∂ c φ + µ φ (cid:1)(cid:3) , where roman indices refer to the coordinates on ∂M . Obviously, T µν isconserved on-shell. For the boundary stress-energy tensor one finds ∂ a T | ab = − T ⊥ b . (17)Furthermore, both T µν and T | ab fulfill the dominant energy condition, i.e.,for future pointing time-like ξ µ , η a , we have T µν ξ µ ξ ν ≥ , T µν ξ ν time-like or null ,T | ab η a η b ≥ , T ab η b time-like or null . Now we choose ξ = e and integrate ∂ µ T µν ξ ν over D = D + ( S ) ∩ J − (Σ ),obtaining (cid:90) S T = (cid:90) S T + (cid:90) S (cid:96) µ T µ − (cid:90) ∂D T ⊥ , + ( S ) S S ∂M Σ Σ S Figure 1: Illustration of the geometric setup in Proposition 3.where (cid:96) is the future directed normal to S , cf. Figure 1. We also choose η = e and integrate ∂ a T | ab η b over ∂D , obtaining (cid:90) S ∩ ∂M T | = (cid:90) S ∩ ∂M T | + (cid:90) S ∩ ∂M p a T | a + (cid:90) ∂D T ⊥ , where we used (17), and p is the future directed normal to S ∩ ∂M . Summingthe previous two equations and using the dominant energy condition, onefinds (16).As usual, such energy estimates establish the causal propagation ofsmooth solutions. Because of the causal propagation, one may refine (15)by considering the norms on the l.h.s. w.r.t. the Sobolev spaces on S and S ∩ ∂M . Given arbitrary smooth initial data that fulfills (14), one can con-struct solutions up to time T as follows: Consider a dense enough coveringof the Cauchy surface with open regions U i of diameter 3 T . For each i , cutoff the initial data outside of U i and solve the wave equation. Then glue thesolutions together. Hence, we obtain: Proposition 4.
The wave equation (2) , (3) for Σ = R d + and µ > is well-posed for smooth initial data ( φ , φ ) ∈ C ∞ ( R d + ) × C ∞ ( R d + ) fulfilling (14) ,i.e., there exists a unique smooth solution φ which depends continuously and ausally on the initial data in the sense that, for Φ( t ) = ( φ ( t ) , φ ( t ) | ∂ Σ ) , (cid:107) ∂ mt Φ( t ) (cid:107) D k +1 ( S ) + (cid:107) ∂ m +1 t Φ( t ) (cid:107) D k ( S ) ≤ (cid:107) Φ (cid:107) D k + m +1 ( S ) + (cid:107) Φ (cid:107) D k + m ( S ) . (18) Here we used the same convention as in Proposition 3, and (cid:107) Φ (cid:107) D k ( S ) standsfor the restriction to S and S ∩ ∂ Σ of the integrals in (12) , (13) . As usual, cf. [22] for example, the estimates on the l.h.s. of (18) may beconverted to supremum estimates using the Sobolev embedding theorems.The causal propagation can also be established for distributions, using theduality of D r and D − r , analogously to the Dirichlet case discussed in [21,Sect. 6.1].The global energy estimate (15) for µ = 0 and m = k = 0 was alreadyderived in [18]. However, to the best of my knowledge, local energy estimatesand thus causal propagation have not yet been proven in the literature. Remark . With small adjustments, one can also treat the massless case µ = 0. The technical difficulty is that ∆ is not strictly positive so that D r = dom ∆ r is not complete w.r.t. the scalar product (11). One may forexample make use of the causal propagation to restrict the discussion aboutpropagation in a bounded region to the case of a supplementary Dirichletboundary at some z = L . This yields a strictly positive ∆, so that the abovetechniques can be applied. Also the case with different masses for the bulkand the boundary may be treated. For µ bulk ≥ µ boundary , the adjustmentsare minor. However, for µ bulk < µ boundary , there will be a bound state, but∆ is still strictly positive for µ > Remark . For the case c <
0, there seem to be severe difficulties. First ofall, H is then no longer a Hilbert, but a Krein space. This may be possibleto deal with, one would have to show that ∆ is definitizable and regular atinfinity. But even then the energy estimates will no longer work, so it will bedifficult, if not impossible, to establish continuous dependence on the initialdata and causality.As can be seen by the form of the energy estimates, the boundary maycarry some energy. It is instructive to consider a concrete example. Consider µ = 0 and a singularity φ = δ ( t + z )infalling to the boundary for t <
0. The full solution to the equations (2),(3) is then (see below) φ = δ ( t + z ) − δ ( t − z ) + 2 c − e − t − zc θ ( t − z ) , (19) φ | = 2 c − e − tc θ ( t ) , with θ the Heaviside function. We see that the boundary absorbs someenergy and radiates it off on the time-scale c . It is also obvious from thisexample why negative c seems physically unacceptable.11he solution (19) has to be understood in the weak sense, i.e., for anytest function ϕ ∈ D ( R d +1 ) with (cid:50) h ϕ | z =0 = − c − ∂ z ϕ | z =0 , we have (cid:90) R d +1+ φ (cid:50) ϕ = 0 . To check this, it is convenient to introduce light cone coordinates u = t + z, v = t − z. For convenience, we may assume d = 1. We then compute (cid:90) R φ (cid:50) ϕ = − (cid:90) v
0) + 4 c − (cid:90) ∞ e − vc ∂ v ϕ | u = v d v = − ∂ t ϕ (0 ,
0) + 2 c − (cid:90) ∞ e − tc ( ∂ t − ∂ z ) ϕ ( t, t = − ∂ t ϕ (0 ,
0) + 2 c − (cid:90) ∞ e − tc (cid:0) ∂ t − c∂ t (cid:1) ϕ ( t, t, where in the last step we used the boundary condition for ϕ . One easilychecks by integration by parts that this vanishes. Remark . The fact that the restriction of the normalized modes withtransversal momentum q falls off like q − , cf. (8), will be important in thefollowing. This property is generic and does not depend on the specific formof the boundary, the equality of the two masses in the bulk and the bound-ary or the presence of curvature and curvature couplings. Choosing normalcoordinates, we may write the boundary condition as ∂ ⊥ φ = c − ∂ ⊥ φ + Rφ, with R some operator that does not act in the transversal direction. Forlarge transversal momentum the mode is well approximated by φ q,k ( x ) (cid:39) Cf k ( x ) sin( qz + φ q )with f k normalized modes on the boundary and a normalization constant C which is essentially independent of q . We conclude that for large transversalmomentum, where the term Rφ may be neglected, and the boundary at z = 0, we have tan( φ q ) (cid:39) ( cq ) − . For large q , this means C sin( φ q ) (cid:39) C ( cq ) − , so that the normalization constant of the modes on the boundary indeedfalls off like q . 12e close this section by briefly discussing the case of the strip Σ = R d − × [ − S, S ]. The statement analogous to Proposition 1 is now:
Proposition 8.
With
Σ = R d − × [ − S, S ] and on the domain D , ∆ is self-adjoint with spectrum contained in [ µ , ∞ ) . A normalized complete systemof generalized eigenfunctions of ∆ is given by φ k,m = c m (2 π ) − d − S − e ikx (cid:40) cos q m z m even sin q m z m odd (20)Φ k,m = ( φ k,m , φ k,m | ∂ Σ ) , with k ∈ R d − , m ∈ N and the eigenvalue ω k,m = k + q m + µ . (21) Here { q m } is an increasing sequence of non-negative real numbers with q =0 and q p ∈ (( p − ) πS , p πS ) , q p − ∈ (( p − πS , ( p − ) πS ) (22) for all p ≥ . For large enough m , the q m are bounded by π S ( m −
1) + (1 − δ ) 2 c − π ( m − ≤ q m ≤ π S ( m −
1) + 2 c − π ( m −
1) (23) for any δ > . For large m , the normalization constants behave as c m =1 + O ( m − ) . The restriction to the boundary is given by φ k,m | ∂ ± Σ ( x ) = ( ± ) m (2 π ) − d − d m e ikx (24) where the d m are real, non-zero and fulfill (1 − δ ) 2 c − √ Sπ ( m − ≤ | d m | ≤ (1 + δ ) 2 c − √ Sπ ( m − for any δ > and large enough m .Proof. The statement on self-adjointness and the spectrum of ∆ follows asin the proof of Proposition 1. Due to the symmetry of the problem, theseparation ansatz (20) is general enough. For the even/odd modes, theboundary condition then implies c − tan qS = − q, (25) q tan qS = c − , (26)which only have real solutions q due to (cid:61) tan( x + iy ) ≷ y ≷
0. Thestatement (22) for odd m follows from the monotonicity of tan on the interval( π ( p − ) , π ( p − )). Hence, for odd m , we must have q m = π S (( m −
1) + ε m )13ith some ε m >
0. Clearly, for π S ( m − (cid:29) c − , we must have ε m (cid:28) ε ≤ tan( πN + ε ) ≤ (1 + δ ) ε for N ∈ N , δ > ε small enough, we have π S ( m − ε m ≤ q m tan Sq m ≤ (1 + δ ) π S ( m − ε m for m large enough. With (26), this can be used to bound ε m to show (23).For even m one argues analogously. For the statement on the normalization,we compute, for odd m , (cid:104) Φ k,m , Φ k (cid:48) ,m (cid:105) = | c m | S − δ ( k − k (cid:48) ) (cid:20)(cid:18) S − sin(2 q m S )2 q m (cid:19) + 2 c sin ( q m S ) (cid:21) By (23), the expression in square brackets on the r.h.s. is S + O ( m − ). Thebounds on d m then follow again from (23) and the behavior of sin x near x = N π . For even m , one argues analogously. That the d m are non-zerofollows from (22).With this result, one may continue as for the half-space to establishwell-posedness of the Cauchy problem and causal propagation. We begin to study the quantization for the case Σ = R d − × [ − S, S ], wherethe holographic mapping has nicer properties than for the half-space R d + .Given the orthonormal basis { Φ k,m } k ∈ R d − ,m ∈ N , cf. Proposition 8, quan-tization proceeds canonically, i.e., we define the one-particle Hilbert space H = L ( R d − ) ⊗ l ( N ) , the corresponding symmetric Fock space F and the usual annihilation andcreation operators a m ( k ), a m ( k ) ∗ fulfilling[ a m ( k ) , a m (cid:48) ( k (cid:48) ) ∗ ] = δ mm (cid:48) δ ( k − k (cid:48) ) . For µ >
0, we define, for F = ( f, f | ) ∈ dom(∆ − ) and G ∈ dom(∆ ), thetime zero fields as φ ( F ) = (cid:88) m (cid:90) d d − k (cid:112) ω k,m (cid:0) (cid:104) ¯ F , Φ k,m (cid:105) a m ( k ) + (cid:104) Φ k,m , F (cid:105) a m ( k ) ∗ (cid:1) ,π ( G ) = − i (cid:88) m (cid:90) d d − k √ ω k,m √ (cid:0) (cid:104) ¯ G, Φ k,m (cid:105) a m ( k ) − (cid:104) Φ k,m , G (cid:105) a m ( k ) ∗ (cid:1) . φ ( F ) , φ ( F (cid:48) )] = 0 , [ π ( G ) , π ( G (cid:48) )] = 0 , [ φ ( F ) , π ( G )] = i (cid:104) ¯ F , G (cid:105) , cf. the symplectic form (5). Interestingly, the time-zero fields may be re-stricted to the boundary: Inserting F = (0 , f | ), one obtains, using (24) φ (0 , f | ) = (cid:88) m (cid:90) d d − k (cid:112) ω k,m d m (cid:16) ˆ f | ( − k ) a m ( k ) + ˆ f | ( k ) a m ( k ) ∗ (cid:17) . Due to the decay of the coefficients d m , cf. Proposition 8, this operator iswell defined on a dense domain for f | ∈ L ( ∂ Σ). However, this is not thecase of the momentum π .Note that for µ = 0 and d = 1, the zero mode has to be treated sepa-rately, using position and momentum operators q , p , corresponding to thelinear growth of the classical mode. For µ = 0 and d = 2, we have the in-frared problems for the zero mode that are usually present in 1+1 space-timedimension.When defining space-time fields, we have to decide on the appropriatetest function space. In view of the definition of the time-zero fields, it seemsnatural to allow for test functions F = ( f, f | ) ∈ S ( M ) ⊕S ( ∂M ) and defining φ ( F ) = (cid:88) m (cid:90) d x d d − k (cid:112) ω k,m (cid:16) (cid:104) ¯ F ( x ) , Φ k,m (cid:105) e − iω k,m x a m ( k )+ (cid:104) Φ k,m , F ( x ) (cid:105) e iω k,m x a m ( k ) ∗ (cid:17) . (27)These have the usual properties of Wightman fields (generalized to thepresent setting): Proposition 9.
Let µ > . The field φ ( f, f | ) , with ( f, f | ) ∈ S ( M ) × S ( ∂M ) real, is essentially self-adjoint on a dense invariant linear domain D ⊂ F .For Ω , Ω ∈ D the maps S ( M ) × S ( ∂M ) (cid:51) ( f, f | ) (cid:55)→ (cid:104) Ω , φ ( f, f | )Ω (cid:105) ∈ C are linear and continuous. The field φ is causal, i.e., [ φ ( f, f | ) , φ ( g, g | )] = 0 if the supports of ( f, f | ) and ( g, g | ) are space-like separated. There is a uni-tary representation U of the proper orthochronous Poincar´e group R ,d − (cid:111) SO + (1 , d − , under which the domain D is invariant and such that U ( a, Λ) φ ( f, f | ) U ( a, Λ) ∗ = φ ( f ( a, Λ) , f | ( a, Λ) )15 ith f ( a, Λ) ( x, z ) = f (Λ − ( x − a ) , z ) , f | ( a, Λ) ( x ) = f | (Λ − ( x − a )) . The vacuum vector Ω ∈ D is invariant under U , cyclic w.r.t. polynomials ofthe fields φ ( f, f | ∂ Σ ) or φ (0 , f | ) , and the spectrum of U on Ω ⊥ is containedin V µ = (cid:110) p ∈ R ,d − | − p ≥ µ , p ≥ (cid:111) . Parts of the proof consist in mapping to the generalized free field ψ on R ,d − with momentum space weight (cid:88) m δ ( − k − q m − µ )d d k. Obviously, the Fock spaces F and F ψ on which φ and ψ act can be triviallyidentified by the isomorphism (cid:16) ( ı ˜Ω) j (cid:17) m ...m j ( k , . . . , k j ) = (cid:16) ˜Ω j (cid:17) m ...m j ( k , . . . , k j ) . With this identification, the field ψ can be expressed as ψ ( x ) = (2 π ) − d − (cid:88) m (cid:90) d d − k (cid:112) ω k,m (cid:16) e − iω k,m x + ikx a m ( k ) + e iω k,m x − ikx a m ( k ) ∗ (cid:17) . (28)For the mapping of the fields, we need the following lemma: Lemma 10.
Let µ > and ˆ f ± ∈ S ( R d − ) ⊗ s ( N ) , with s ( N ) the space ofsequences that fall off faster than any power. There is a linear continuousmapping ( S ( R d − ) ⊗ s ( N )) (cid:51) ( ˆ f + , ˆ f − ) (cid:55)→ ˆ f (cid:48) ∈ S ( R d ) such that ˆ f ± m ( k ) = ˆ f (cid:48) ( ± ω k,m , ± k ) ∀ k ∈ R d − , m ∈ N , with ω k,m given by (21) .Proof. Choose χ : R → R , smooth and supported in [ − / , / χ (0) =1. Then choose a > µ < /a and q m − q m − < /a for all m ≥ f (cid:48) ( ω, k ) = (cid:88) m (cid:88) s ∈± θ ( sω ) χ ( a ( ω − ω k,m )) ˆ f sm ( sk ) . Clearly, this fulfills the requirement and is a Schwarz function. Furthermore,each seminorm (cid:107) ˆ f (cid:48) (cid:107) αβ = sup k ∈ R d | k α || ∂ β ˆ f (cid:48) ( k ) | α , β , can be bounded by the seminorms (cid:107) ˆ f ± (cid:107) αβc = (cid:88) m ∈ N sup k ∈ R d − m c | k α || ∂ β ˆ f ± m ( k ) | of S ( R d − ) ⊗ s ( N ), due to the growth of ω k,m with m , cf. Proposition 8. Proof of Proposition 9.
Causality is a consequence of the canonical equal-time commutation relations and the causal propagation, proved in the pre-vious section. The action of the Poincar´e group is straightforwardly definedby its action on the one-particle space H ,( U ( a, Λ) f ) m (˜ k ) = e ia ˜ k f m (Λ˜ k ) , where ˜ k = ( ω k,m , k ) and by an abuse of notation we identified the function f m ( k ) on R d − with the the function f m (˜ k ) on the mass hyperboloid formass (cid:112) q m + µ . This also entails the claim on the spectrum.To prove continuity, we define continuous linear maps S ( M ) × S ( ∂M ) (cid:51) ( f, f | ) (cid:55)→ ˆ f ± ∈ S ( R d − ) ⊗ s ( N )by ˆ f − m ( k ) = (2 π ) − (cid:90) (cid:104) ( ¯ f , ¯ f | )( x ) , Φ k,m (cid:105) e − iω k,m x d x , ˆ f + m ( k ) = (2 π ) − (cid:90) (cid:104) Φ k,m , ( f, f | )( x ) (cid:105) e iω k,m x d x . Continuity can be shown by using the growth of ω k,m with m , Proposition 8.Using the previous lemma, we can map this pair further to ˆ f (cid:48) ∈ S ( R d ). Byconstruction, we then have φ ( f, f | ) = ı − ◦ ψ ( f (cid:48) ) ◦ ı, with ψ the generalized free field (28), and ı the canonical isomorphism ofthe Fock spaces. For ψ , self-adjointness on the invariant domain D = ˜Ω ∈ n (cid:77) j =0 ( S ( R d − ) ⊗ s ( N )) ⊗ s j | n ∈ N , with ⊗ s j the j symmetric tensor power was shown in [23, Section II.6].Also the cyclicity of the vacuum w.r.t. polynomials of the field ψ wasproven in [23, Section II.6]. In order to prove the cyclicity w.r.t. polynomialsof φ (0 , f | ), it thus suffices to construct a map S ( R d ) (cid:51) f (cid:48) (cid:55)→ f | ∈ S ( ∂M )such that φ (0 , f | ) = ı − ◦ ψ ( f (cid:48) ) ◦ ı.
17e define ˆ f (cid:48)± m ( k ) = ˆ f (cid:48) ( ± ω k,m , ± k ) d m , with d m the constants defined in Proposition 8, and then use Lemma 10 toobtain ˆ f such thatˆ f ( ± ω k,m , ± k ) = ˆ f (cid:48)± m ( k ) ∀ k ∈ R d − , m ∈ N . Placing f at the ∂ + M component of the boundary yields the desired result.For cyclicity w.r.t. polynomials in φ ( f, f | ∂M ), given f (cid:48) , we define f ( x, z ) = (2 π ) − (cid:88) m (cid:90) d ω d d − k θ ( ω ) φ k,m ( x, z ) χ ( ω − ω k,m ) × (cid:16) ˆ f (cid:48) ( − ω k,m , − k ) e iωx + ˆ f (cid:48) ( ω k,m , k ) e − iωx (cid:17) where χ (0) = 1 and the support of the test function χ is taken sufficientlysmall, so that there is no overlap of the various mass shells. This has thedesired property and using the explicit form of φ k,m derived in Proposition 8,one can show that f ∈ S ( M ).Proposition 9 states that the restriction of the field to the boundary israther naturally possible. Hence, we define, for f ∈ S ( ∂M ), φ | ( f ) = φ (0 , c − f ) , (29)where the factor c − is introduced to cancel the factor c from the scalarproduct. We may also restrict to the two boundary components separately.This yields φ | ± ( x ) = (2 π ) − d − (cid:88) m ( ± ) m d m (cid:90) d d − k (cid:112) ω k,m ×× (cid:16) e − i ( ω k,m x − kx ) a m ( k ) + e i ( ω k,m x − kx ) a m ( k ) ∗ (cid:17) , (30)i.e., a generalized free field. The spectrum contains all the mass shells forthe masses q m . Its two-point function is given by∆ + | ± ( x ) = (cid:88) m | d m | ∆ µ m + ( x ) , (31)where we made use of translation invariance, µ m = (cid:112) q m + µ and ∆ m + isthe two-point function for the free field of mass m on the d dimensionalMinkowski space. For d = 1, this is∆ µ + ( x ) = 12 µ e − iµx .
18y Proposition 8, the coefficients d m are square summable, which impliesthat the two-point function has the same degree of short-distance singularityas the vacuum two-point function in d space-time dimensions. This can beformalized by the concept of the scaling degree of a distribution, cf. [24].We then have: Proposition 11.
Let µ > or d > . Then ∆ + | ± is a tempered distribution.Its singular support is contained in { x ∈ R d | x ≤ } and the projection ofits analytic wave front set to the cotangent space is given by { k ∈ R d | k ≤ , k > } . For d ≥ , the scaling degree of ∆ + | ± at coinciding points is d − .Proof. We consider its Fourier transformˆ∆ + | ± ( k ) = (cid:88) m | d m | ˆ∆ µ m + ( k ) . For the individual ˆ∆ µ + ( k ) = (2 π ) − θ ( k ) δ ( − k − µ )we can derive a bound |(cid:104) ˆ∆ µ + , φ (cid:105)| ≤ C (cid:107) φ (cid:107) d , with the seminorm (cid:107) φ (cid:107) d = sup k ∈ R d (1 + | k | ) d | φ ( k ) | of the space of Schwarz distributions. This proceeds as follows: |(cid:104) ˆ∆ µ + , φ (cid:105)| ≤ (2 π ) − (cid:90) d d − k ω µ ( k ) | φ ( ω µ ( k ) , k ) |≤ (2 π ) − (cid:107) φ (cid:107) d (cid:90) d d − k ω µ ( k ) 1(1 + (cid:112) k + µ ) d . For the latter integral, one can derive a bound which is independent of µ .Temperedness then follows from the square summability of the d m . Forthe singular support, consider any compact set K space-like to the origin.There, ∆ + | ± is given by∆ + | ± ( x ) = (cid:88) m | d m | π ) − d/ µ d/ − m | x | / − d/ K d/ − ( (cid:112) µ m x )Due to [25, 9.6.29] ∂ z K ν ( z ) = − ( K ν − ( z ) + K ν +1 ( z ))and the exponential decay of K ν ( z ) for z → ∞ , all derivatives of ∆ + | ± can be uniformly bounded on K , so that the series converges to a smooth19unction outside of the light cone. That only positive frequency momenta arecontained in the analytic wave front set follows from the support propertiesof ˆ∆ + | ± . That positive frequency momenta with k < K ν ( z ) for z → d m .For time-like separations, one expects supplementary singularities com-ing from reflections at the other boundary.Obviously, the representation of φ | + on the Fock space F coincides withthe GNS representation of the Borchers-Uhlmann algebra for fields on R ,d − w.r.t. the two-point function (31). As a consequence of the above bound onthe analytic wave front set and [26, Cor. 5.5], the boundary field φ | + fulfillsthe Reeh-Schlieder property, i.e., the set of states obtained by acting withthe bounded operators localized in a fixed, arbitrarily small region of theboundary on the vacuum is dense in F , cf. [26] for details.In analogy with (29), we may also define the bulk field as φ bulk ( f ) = φ ( f, f ∈ S ( M ). Admitting more singular smearing functions f , we readilyarrive at φ | ± ( f ) = φ bulk ( f δ ( z ∓ S )) . (32)Furthermore, it is straightforward to check that the boundary value of ∂ ⊥ φ bulk is indeed the source for the boundary field φ | : φ | ± (( − (cid:50) + µ ) f ) = ∓ c − φ bulk ( f δ (cid:48) ( z ∓ S )) . (33)It is straightforward to define Wick powers of the bulk and boundaryfields, either by a coinciding point limit with subtraction of the correctcombination of lower order Wick powers and two-point functions (dictatedby Wick’s theorem), or by normal ordering the creation and annihilationoperators. In any case, the boundary Wick power is given by the boundaryvalue of the bulk Wick power, i.e., φ | k ± ( f ) = φ k bulk ( f δ ( z ∓ S )) . Remark . Instead of defining Wick powers globally by normal ordering,it may be more appropriate to choose a local prescription, as advocated inthe context of quantum field theory on curved space-times [27]. However,problems may then occur at the boundary, cf. [28], for example, i.e., it maybe necessary to restrict to bulk test functions supported in the interior of thebulk. Also the treatment of the boundary part needs to be clarified then.20or each mode m , one may also define local fields φ m | ± , localized at theboundary: φ m | ± ( x ) = ( ± ) m (2 π ) d − (cid:90) d d − k (cid:112) ω k,m (cid:16) e − i ( ω k,m x − kx ) a m ( k ) + e i ( ω k,m x − kx ) a m ( k ) ∗ (cid:17) . The relation between the fields φ m | ± and φ | ± is clearly non-local, i.e., for φ m | ± ( f m ) = φ | ± ( f ) (34)to hold, the test function f must be de-localized. This can be made quiteprecise: Proposition 13.
Let d = 1 . For (34) to hold, f can not be supported onan interval smaller than π S .Proof. Assume f is localized in an interval of length L . By the Paley-Wienertheorem [29], its Fourier transform is an entire function which is boundedby | ˆ f ( ξ ) | ≤ Ce L |(cid:61) ( ξ ) | for some constant C , i.e., it is of order 1 and type τ ≤ L [30]. On the otherhand, in order for (34) to hold, we must have ˆ f ( ± q m (cid:48) ) = 0 for all m (cid:48) (cid:54) = m .From Proposition 8 and [30, Thm. 9.1.4] it follows that τ > π S unless f vanishes.From causality, i.e., local commutativity, it is clear that the boundaryfield does not fulfill the time-slice axiom: All boundary observables containedin a small time-slice commute with space-like separated bulk observables. Itfollows that the time-slice axiom does not hold for time slices smaller than2 S , cf. [31] for the discussion for a generalized free field with continuousK¨allen-Lehmann weight. Whether the time-slice axiom holds for time sliceslarger than 2 S is however not clear. The previous proposition suggest thateven in the case d = 1 the minimal time slice should be at least π S .We now want to construct the holographic map, i.e., for a given bulkfield φ bulk ( f ) we want to find a test function f (cid:48) on ∂ + M such that φ bulk ( f ) = φ | + ( f (cid:48) ) . (35)We denote ˆ f − m ( k ) = (2 π ) − (cid:90) d x (cid:104) ( ¯ f ( x ) , , Φ k,m (cid:105) e − iω k,m x , ˆ f + m ( k ) = (2 π ) − (cid:90) d x (cid:104) Φ k,m , ( f ( x ) , (cid:105) e iω k,m x . For simplicity, we restrict to the right boundary. One may of course also consider ∂ − M or ∂M .
21y Proposition 8, we have to find f (cid:48) ∈ S ( ∂ + M ) such thatˆ f ± m ( k ) = d m ˆ f (cid:48) ( ± ω k,m , ± k ) ∀ k ∈ R d − , m ∈ N . As d m (cid:54) = 0 and { d − m } is polynomially bounded, cf. Proposition 8, there isan f (cid:48) ∈ S ( ∂ + M ) with this property, cf. Lemma 10. We have thus proven: Proposition 14.
Let µ > . Then to each f ∈ S ( M ) there exists f (cid:48) ∈S ( ∂ + M ) such that (35) with φ | ± defined by (30) holds. An interesting open question is whether f (cid:48) can be chosen to be compactlysupported if f is. By the Paley-Wiener theorem, a necessary condition forthis is that ˆ f (cid:48) can be chosen to be an entire function of exponential type. For d = 1, this is possible due to Proposition 8 and [32, Thm. 2]. But it is notclear whether one can also achieve the necessary fall-off in the real direction.In any case, one can not expect this to work in higher dimensions, due to thenon-analyticity of ω k,m in k . This is in contrast to the case of holography onAdS, where a localized bulk observable can always be mapped to a localizedboundary observable [5].A similar mapping can be constructed for Wick powers. However, onecan not expect that one obtains local Wick powers at the boundary. Instead,a function f (cid:48) ∈ S ( ∂ + M k ) such that φ k bulk ( f ) = (cid:90) d d x . . . d d x k : φ | + ( x ) . . . φ | + ( x k ) : f (cid:48) ( x , . . . , x k )can easily be constructed along the lines discussed above.For illustration, we can consider how a holographic image of a localobservable actually looks like. We choose d = 1, S = 1, c = 1, µ = 0 and f ( t, x ) = (cid:40) e − t +1 / e − / − t e − x +1 / e − / − x x, t ∈ ( − / , / . (36)A holographic dual f (cid:48) to this is shown in Figure 2. We see a sequence ofoscillations, around t = ± , ± , ±
5. These correspond to the times when thepropagation of the peak of the test function hits the boundary after 0 , , R d + . We thenhave the orthonormal basis { Φ k,q } k ∈ R d − ,q ∈ R + , cf. Proposition 1, so that theone-particle Hilbert space is given by H = L ( R d + ) , and the annihilation and creation operators a ( k, q ), a ( k, q ) ∗ fulfilling[ a ( k, q ) , a ( k (cid:48) , q (cid:48) ) ∗ ] = δ ( k − k (cid:48) ) δ ( q − q (cid:48) ) . I am grateful to Michael Bordag for helping with the translation of [32]. - - t - - - Figure 2: A holographic image f (cid:48) ( t ) of (36). The dashed curve shows f ( t, φ ( F ) = (cid:90) d x d d − k (cid:112) ω k,m d q (cid:16) (cid:104) ¯ F ( x ) , Φ k,q (cid:105) e − iω q,m x a ( k, q )+ (cid:104) Φ k,q , F ( x ) (cid:105) e iω k,q x a ( k, q ) ∗ (cid:17) , for F = ( f, f | ). Note that for µ = 0 and d = 1, we have the infraredproblems that are usually present in 1+1 space-time dimension.As for the case of the strip, the fields can be restricted to the boundary,yielding a generalized free field with the two-point function∆ + = (cid:90) ∞ d q π ( c q + 1) ∆ √ µ + q + . This is again a tempered distribution with scaling degree d − d ≥
2. Aholographic map S ( M ) (cid:51) f (cid:55)→ f (cid:48) such that φ bulk ( f ) = φ | ( f (cid:48) )can be defined as follows. Defineˆ f − ( k, q ) = (2 π ) − (cid:90) d x (cid:104) ( ¯ f ( x ) , , Φ k,q (cid:105) e − iω k,q x , ˆ f + ( k, q ) = (2 π ) − (cid:90) d x (cid:104) Φ k,q , ( f ( x ) , (cid:105) e iω k,q x , and set ˆ f (cid:48) ( ω, k ) = (cid:114) π ( c q + 1)2 (cid:40) ˆ f + ( k, q ) , ω − k > µ , ω > , ˆ f − ( − k, q ) , ω − k > µ , ω < . q = (cid:112) ω − k − µ . The problem is that this can in general not besmoothly continued to the region ω − k ≥ µ . Hence, the resulting f (cid:48) will ingeneral be a smooth L function. In the half-space setting, the holographicmap is thus more delocalizing than in the strip case.Also thermodynamically, the boundary field in the strip-space case isless appealing, as it does not fulfill Buchholz-Wichmann nuclearity [33].Nevertheless, for µ >
0, one can straightforwardly define KMS states.
First of all, let us compare to more common boundary conditions. Forconcreteness, let us consider Neumann boundary conditions. Restriction oftime-zero fields to the boundary is then certainly not possible. But whensmeared in time, the bulk field may be restricted to the boundary, so that(32) holds. However, the resulting theory has bad short distance behavior:It has the same degree of singularity as the original d + 1 dimensional fieldtheory, but only lives in d space-time dimensions. This is due to the fact thatfor Neumann boundary conditions the coefficients { d m } would be constant.Apart from the statement about the degree of singularity, all propertieslisted in Proposition 11 also hold for the two-function of boundary fieldsin Neumann boundary conditions. Obviously, (33) does not hold for Neu-mann boundary conditions, but the source on the r.h.s. would be given by ∂ ⊥ φ bulk | ∂M .One could also consider Dirichlet boundary conditions, but then therestriction to the boundary would be trivial. However, one may restrict ∂ ⊥ φ to the boundary, yielding a field theory on the boundary whose singularbehavior is even worse than in the case of Neumann boundary conditions.The Dirichlet case is quite similar to what happens in the AdS/CFTcorrespondence as applied to massive scalar fields [4]. There, one has twobulk propagators G ± , whose leading behavior near the boundary z = 0 is z ∆ ± with ∆ + > ∆ − . Hence, one may understand G + as the analog of theDirichlet propagator and G − as the analog of the Neumann propagator onflat space (where we would have ∆ + = 1 and ∆ − = 0). The dual field canbe understood as the limit O φ ( x ) = lim z → z − ∆ + φ ( x, z ) [7], whose analogin flat space is ∂ ⊥ φ restricted to the boundary. Its source φ is interpretedas the boundary value of the bulk field in the sense that φ ∼ z ∆ − φ . It’sflat space analog is thus the restriction to the boundary. Hence, in Wentzellboundary conditions, the analog the dual field is the source for the analog ofthe boundary value, in the sense of (33). However, both fields are quantizedin that setting.In holographic renormalization [11], one introduces, in suitable coordi-nates, a boundary at z = ε and considers the limit ε →
0. Now the classical24ction, evaluated on a solution with the boundary value φ in the abovesense, diverges as ε →
0. The cure is to introduce ε dependent countertermslocalized on the boundary. For the massive scalar field, these boundarycounterterms are of the type considered here, but with µ bulk (cid:54) = µ boundary and c <
0. As discussed in Section 2, negative values of c lead to severedifficulties already at the classical level. But as holographic renormaliza-tion is a formal mathematical trick, it is not clear whether this poses realproblems. In any case, working formally with this negative value of c , onefinds that for finite ε , the singularities of the boundary field are indeed ofthe expected form, i.e., the two-point function is singular of degree d −
2, cf.also the discussion in Remark 7.
We established well-posedness of the Cauchy problem for (2), (3), includingcausality. We also quantized the system and discussed some properties ofthe resulting quantum field. In particular, we showed that the field may berestricted to the boundary, yielding a generalized free field with the degree ofsingularity that one would expect for a scalar field in d dimensions. Finally,we constructed an explicit holographic correspondence between bulk andboundary fields.Our results lead to a couple of new questions. Regarding generalizedWentzell boundary conditions, a proper microlocal calculus should be setup, which would allow to prove propagation of singularities and thus givesome control over the wave front sets of the relevant propagators.One could also work out generalized Wentzell boundary conditions forother fields. In particular, it would be interesting to see whether they canbe defined for Dirac fields. This may involve chiral fields on the bound-aries. For gauge fields, there naturally appears a supplementary scalar onthe boundaries, the boundary value of the normal component of the bulkvector potential. Hence, such models are potentially interesting also phe-nomenologically.Regarding the holographic aspect, it would be interesting to investigatewhether the holographic relations still hold for interacting theories. Acknowledgments
I would like to thank Claudio Dappiaggi, Stefan Hollands, Karl-HenningRehren, Ko Sanders, Rainer Verch, Ingo Witt, and Michal Wrochna for help-ful discussions or remarks. This is a post-peer-review, pre-copyedit versionof an article published in Annales Henri Poincar´e. The final authenticatedversion is available online at: https://doi.org/10.1007/s00023-017-0629-325 eferenceseferences