Local zeta regularization and the scalar Casimir effect III. The case with a background harmonic potential
aa r X i v : . [ m a t h - ph ] J u l Local zeta regularizationand the scalar Casimir effect III.The case with a background harmonic potential
Davide Fermi a , Livio Pizzocchero b ( ) a Dipartimento di Matematica, Universit`a di MilanoVia C. Saldini 50, I-20133 Milano, Italye–mail: [email protected] b Dipartimento di Matematica, Universit`a di MilanoVia C. Saldini 50, I-20133 Milano, Italyand Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italye–mail: [email protected]
Abstract
Applying the general framework for local zeta regularization proposed inPart I of this series of papers, we renormalize the vacuum expectation value ofthe stress-energy tensor (and of the total energy) for a scalar field in presenceof an external harmonic potential.
Keywords:
Local Casimir effect, renormalization, zeta regularization.
AMS Subject classifications:
PACS : 03.70.+k, 11.10.Gh, 41.20.Cv . Corresponding author ontents H ( u ) µν . . . . . . . . . . . . . . . . . . . . 133.5 Analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . 143.6 Asymptotics for r = k | x | → + . . . . . . . . . . . . . . . . . . . . . 163.7 Asymptotics for r = k | x | → + ∞ . . . . . . . . . . . . . . . . . . . . 173.8 The total energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 d ∈ { , , } d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Case d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Case d = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A Appendix. Asymptotic expansions for certain integrals 36
A.1 Derivation of Eq.s (3.34-3.37). . . . . . . . . . . . . . . . . . . . . . . 36A.2 The incomplete gamma functions and the integral g ( s, x ); asymptoticsand bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.3 Derivation of Eq.s (3.45) (3.46). . . . . . . . . . . . . . . . . . . . . . 40 B Appendix. An alternative representation for the bulk energy 42
B.1 The integrals I n ( s ) and a recursive relation. . . . . . . . . . . . . . . 42B.2 The bulk energy in terms of the integrals I n ( s ) . . . . . . . . . . . . . 43B.3 The bulk energy in terms of the Riemann zeta function for d ∈ { , , } . 441 Introduction
In Part I of this series of papers [10, 11, 12] (partly inspired by [9]) we have consideredthe general framework of local (and global) zeta regularization of a neutral scalarfield in a d -dimensional spatial domain Ω, in the environment of ( d + 1)-dimensionalMinkowski spacetime; the possible presence of an external potential V was takeninto account as well.In the present Part III we consider a massless field on Ω = R d , and choose for V aharmonic potential. The potential is assumed to be isotropic, i.e., to be proportionalto the squared radius | x | ( x ∈ R d ); nevertheless, our approach could be extendedwith little effort to anisotropic harmonic potentials and to the case of a massivescalar field.The main result of this paper is the renormalization of the vacuum expectation value(VEV) for the stress-energy tensor, obtained applying the general framework of PartI to the present configuration; we also discuss the total energy in this configuration.After producing general integral representations for both the renormalized stress-energy VEV and the total energy, we consider in detail the cases of spatial dimension d ∈ { , , } .The above mentioned integral representations are derived analytically and are fullyexplicit; however, the integrals therein must be computed numerically. We haveperformed these latter computations for d ∈ { , , } , using Mathematica .Using the previously cited integral representation, it is also possible to derive theasymptotics for the stress-energy tensor components when the radius | x | goes tozero or to infinity; we derive, as well, remainder estimates for these asymptoticexpansions as well.The idea to replace sharp boundaries with suitable background potentials is well-known in the literature on the Casimir effect. Typically (see, e.g., [3, 5, 15, 17, 19]),delta-like potentials are introduced in order to mimic boundary conditions in a“physically more realistic” framework; the ultimate purpose is to obtain less singu-lar behaviours of the renormalized quantities, avoiding, e.g., boundary divergencessuch as the ones pointed out in Parts I and II [10, 11]. The case of a scalar field inter-acting with an external harmonic potential has been formerly considered by Actorand Bender [1, 2], who have determined the renormalized VEV of the total en-ergy via global zeta regularization; to the best of our knowledge, local aspects suchas the stress-energy tensor have never been considered previously for the presentconfiguration.The paper is organized as follows. In Section 2 we present a summary of basic resultsfrom Part I; as in Part II, the purpose of the summary is to save the reader fromthe tedious task of recovering from Part I the general identities applied here. InSection 3 (and in the related Appendix A) we discuss the general approach to treatthe case of a scalar field interacting with a classical, isotropic harmonic potential in2rbitrary spatial dimension d ; this includes general expressions for the renormalizedstress-energy VEV, its asymptotics for | x | small or large and for the renormalizedVEV of the total energy (more precisely, for the bulk energy as defined in Part I).The main ingredients in the derivation of the above results are:i) the general results of Part I relating the zeta regularization to the heat kernel offundamental operator A = − ∆ + V ( x );ii) the fact that, when V ( x ) is harmonic, the related heat kernel is the well-knownMehler kernel [18] (also see [4, 6, 8, 16]).In Section 4, the analysis presented in the preceeding sections is specified to thecases of spatial dimension d ∈ { , , } .We point out that results on the renormalized bulk energy agree with the ones ofActor and Bender [2] for d ∈ { , , } . Nevertheless, it should be mentioned thatthese authors compute the analytic continuations required by the zeta approachusing ad hoc results on continuation the special functions involved in this specificcase; on the contrary, here we are applying mechanically the general setting discussedin Part I, in the spirit of the present series of papers. Throughout the paper we use natural units, so that c = 1 , ~ = 1 . (2.1)Our approach works in ( d + 1)-dimensional Minkowski spacetime, which is identifiedwith R d +1 using a set of inertial coordinates x = ( x µ ) µ =0 , ,...,d ≡ ( x , x ) ≡ ( t, x ) ; (2.2)the Minkowski metric is ( η µν ) = diag( − , , ... ,
1) . We fix a spatial domain Ω ⊂ R d and a background static potential V : Ω → R . We consider a quantized neutral, scalarfield b φ : R × Ω → L sa ( F ) ( F is the Fock space and L sa ( F ) are the selfadjoint operatorson it); suitable boundary conditions are prescribed on ∂ Ω. The field equation reads0 = ( − ∂ tt + ∆ − V ( x )) b φ ( x , t ) (2.3)(∆ := P di =1 ∂ ii is the d -dimensional Laplacian). We put A := − ∆ + V , (2.4)keeping into account the boundary conditions on ∂ Ω, and consider the Hilbert space L (Ω) with inner product h f | g i := R Ω d x f ( x ) g ( x ). We assume A to be selfadjointin L (Ω) and strictly positive (i.e., with spectrum σ ( A ) ⊂ [ ε , + ∞ ) for some ε > F k ) k ∈K of (proper or improper) eigen-functions of A with eigenvalues ( ω k ) k ∈K ( ω k > ε for all k ∈ K ). Thus F k : Ω → C ; A F k = ω k F k ; h F k | F h i = δ ( k, h ) for all k, h ∈ K . (2.5)The labels k ∈ K can include both discrete and continuous parameters; R K dk indi-cates summation over all labels and δ ( h, k ) is the Dirac delta function on K .We expand the field b φ in terms of destruction and creation operators correspondingto the above eigenfunctions, and assume the canonical commutation relations; | i ∈ F is the vacuum state and VEV stands for “vacuum expectation value”.The quantized stress-energy tensor reads ( ξ ∈ R is a parameter) b T µν := (1 − ξ ) ∂ µ b φ ◦ ∂ ν b φ − (cid:18) − ξ (cid:19) η µν ( ∂ λ b φ ∂ λ b φ + V b φ ) − ξ b φ ◦ ∂ µν b φ ; (2.6)in the above we put b A ◦ b B := (1 / b A b B + b B b A ) for all b A, b B ∈ L sa ( F ), and all thebilinear terms in the field are evaluated on the diagonal (e.g., ∂ µ b φ ◦ ∂ ν b φ indicatesthe map x ∂ µ b φ ( x ) ◦ ∂ ν b φ ( x )). The VEV h | b T µν | i is typically divergent. The zeta-regularized field operator is b φ u := ( κ − A ) − u/ b φ , (2.7)where A is the operator (2.4), u ∈ C and κ > b φ u | u =0 = b φ , at least formally. The zeta regularized stress-energy tensor is b T uµν := (1 − ξ ) ∂ µ b φ u ◦ ∂ ν b φ u − (cid:18) − ξ (cid:19) η µν (cid:16) ∂ λ b φ u ∂ λ b φ u + V ( b φ u ) (cid:17) − ξ b φ u ◦ ∂ µν b φ u . (2.8)The VEV h | b T uµν | i is well defined for ℜ u large enough (see the forthcoming subsec-tion 2.5); moreover, in the region of definition it is an analytic function of u . Thesame can be said of many related observables (including global objects, such as thetotal energy VEV).For any one of these observables, let us denote with F ( u ) its zeta-regularized versionand assume this to be analytic for u in a suitable domain U ⊂ C . The zeta approachto renormalization can be formulated in two versions.i) Restricted version . Assume the map U → C , u
7→ F ( u ) to admit an analyticcontinuation (indicated with the same notation) to an open subset U ⊂ C with U ∋ F ren := F (0) . (2.9)4i) Extended version . Assume that there exists an open subset
U ⊂ C with U ⊂ U ,such that 0 ∈ U and the map u ∈ U
7→ F ( u ) has an analytic continuation to U \{ } (still denoted with F ). Starting from the Laurent expansion F ( u ) = P + ∞ k = −∞ F k u k ,we introduce the regular part ( RP F )( u ) := P + ∞ k =0 F k u k and define F ren := ( RP F )(0) . (2.10)Of course, if F is regular at u = 0 the defnitions (2.9) (2.10) coincide.In the case of the stress-energy VEV, the prescriptions (i) and (ii) read, respectively, h | b T µν ( x ) | i ren := h | b T uµν ( x ) | i (cid:12)(cid:12)(cid:12) u =0 , (2.11) h | b T µν ( x ) | i ren := RP (cid:12)(cid:12)(cid:12) u =0 h | b T uµν ( x ) | i . (2.12) Theseare indicated by the superscripts ( ♦ ) and ( (cid:4) ), respectively; they are defined by h | b T µν | i ren = h | b T ( ♦ ) µν | i ren + ( ξ − ξ d ) h | b T ( (cid:4) ) µν | i ren , (2.13)where we are considering for the parameter ξ the critical value ξ d := d − d . (2.14) If B is a linear operator in L (Ω), its integral kernel is the(generalized) function ( x , y ) ∈ Ω × Ω
7→ B ( x , y ) := h δ x |B δ y i ( δ x is the Dirac deltaat x ). The trace of B , assuming it exists, fulfills Tr B = R Ω d x B ( x , x ) .In the following subsections A is a strictly positive selfadjoint operator in L (Ω), witha complete orthonormal set of eigenfunctions as in Eq. (2.5). In typical applications, A is the operator (2.4). For (suitable) s ∈ C , the s -th Dirichlet kernel of A is D s ( x , y ) := A − s ( x , y ) = Z K dkω sk F k ( x ) F k ( y ) . (2.15)If A = − ∆ + V (with V a smooth potential) is strictly positive, the map D s ( , ) :Ω × Ω → C , ( x , y ) D s ( x , y ) is continuous along with all its partial derivatives upto order j ∈ N , for all s ∈ C with ℜ s > d/ j/ h | b T u ( x ) | i = κ u (cid:20)(cid:18)
14 + ξ (cid:19) D u − ( x , y )+ (cid:18) − ξ (cid:19) ( ∂ x ℓ ∂ y ℓ + V ( x )) D u +12 ( x , y ) (cid:21) y = x , (2.16)5 | b T u j ( x ) | i = h | b T uj ( x ) | i = 0 , (2.17) h | b T uij ( x ) | i = h | b T uji ( x ) | i == κ u (cid:20)(cid:16) − ξ (cid:17) δ ij (cid:16) D u − ( x , y ) − ( ∂ x ℓ ∂ y ℓ + V ( x )) D u +12 ( x , y ) (cid:17) ++ (cid:18)(cid:16) − ξ (cid:17) ∂ x i y j − ξ ∂ x i x j (cid:19) D u +12 ( x , y ) (cid:21) y = x (2.18)( h | b T uµν ( x ) | i is short for h | b T uµν ( t, x ) | i ; indeed, the VEV does not depend on t );of course, the map Ω → C , x
7→ h | b T uµν ( x ) | i possesses the same regularity asthe functions x ∈ Ω D u ± ( x , x ) , ∂ zw D u +12 ( x , x ) ( z, w any two spatial variables);so, due to the previously mentioned results, x
7→ h | b T uµν ( x ) | i is continuous for ℜ u > d + 1.The above framework relates the renormalized stress-energy VEV h | b T µν ( x ) | i ren := RP | u =0 h | b T uµν ( x ) | i to the analytic continuation at u = 0 of the maps u κ u D u ± ( x , y )and of their derivatives.In the sequel we will also consider the total energy VEV and express it in terms ofthe trace Tr A − s , fulfilling Tr A − s = Z Ω d x D s ( x , x ) . (2.19) For t ∈ [0 , + ∞ ), this is given by K ( t ; x , y ) := e − t A ( x , y ) = Z K dk e − t ω k F k ( x ) F k ( y ) . (2.20)If A = − ∆ + V ( V smooth) is strictly positive, the map K ( t ; , ) : Ω × Ω → C ,( x , y ) K ( t ; x , y ) is continuous along with all its partial derivatives of any order,for all t > heat trace (assuming it to exists) is K ( t ) := Tr e − t A = Z Ω d x K ( t ; x , x ) . (2.21) For suit-able values of s ∈ C (see Part I), there holds D s ( x , y ) = 1Γ( s ) Z + ∞ d t t s − K ( t ; x , y ) . (2.22)Similar results hold for Tr A − s ; for example, using the heat trace K ( t ) of Eq. (2.21),we obtain Tr A − s = 1Γ( s ) Z + ∞ d t t s − K ( t ) . (2.23)6 .8 Analytic continuation of Mellin transforms via integration by parts. Let F : (0 , + ∞ ) → C be a function of the form F ( t ) = 1 t ρ H ( t ) (2.24)for some ρ ∈ C and some smooth function H : [0 , + ∞ ) → C , vanishing exponentiallyfor t → + ∞ . Consider the Mellin transform of F , i.e. M ( σ ) := Z + ∞ d t t σ − F ( t ) ; (2.25)this is an analytic function of σ for ℜ σ > ℜ ρ . Integrating the above equation byparts n times (for any n ∈ { , , , ... } ) we obtain the relation M ( σ ) = ( − n ( σ − ρ ) ... ( σ − ρ + n − Z + ∞ d t t σ − ρ + n − d n H d t n ( t ) , (2.26)giving the analytic continuation of M ( σ ) to a meromorphic function of σ in theregion {ℜ σ > ℜ ρ − n } , possibly with simple poles at σ ∈ { ρ , ρ − , ... , ρ − n + 1 } .The above results can be employed to obtain the analytic continuations of the reg-ularized stress-energy VEV h | b T uµν ( x ) | i (treating x ∈ Ω as a fixed parameter) andof the trace Tr A − s ; for example, assuming the heat trace K ( t ) to have the form K ( t ) = 1 t p H ( t ) , (2.27)for some p ∈ R and some smooth function H : [0 , + ∞ ) → R , vanishing rapidly for t → + ∞ , starting from Eq. (2.23) we obtain (for n ∈ { , , , ... } )Tr A − s = ( − n Γ( s )( s − p ) ... ( s − p + n − Z + ∞ d t t s − p + n − d n Hd t n ( t ) . (2.28) Let A := − ∆ + V and consider the case whereΩ = Ω × Ω ∋ x = ( x , x ) , y = ( y , y ) , (2.29) V ( x ) = V ( x ) + V ( x ) (2.30)(Ω a ⊂ R d a is an open subset, for a ∈ { , } ; d + d = d ); assume the boundaryconditions on ∂ Ω to arise from suitable boundary conditions prescribed separatelyon ∂ Ω and ∂ Ω so that, for a = 1 ,
2, the operators A a := − ∆ a + V ( x a ) (2.31)7with ∆ a the Laplacian on Ω a ) are selfadjoint and strictly positive in L (Ω a ). Then,the Hilbert space L (Ω) and the operator A can be represented as L (Ω) = L (Ω ) ⊗ L (Ω ) , A = A ⊗ + ⊗ A . (2.32)This implies, amongst else, that the heat kernels K ( t ; x , y ) := e t A ( x , y ), K a ( t ; x a , y a ):= e t A a ( x a , y a ) ( a = 1 ,
2) are related by K ( t ; x , y ) = K ( t ; x , y ) K ( t ; x , x ) . (2.33)Similarly, writing K ( t ), K a ( t ) ( a ∈ { , } ) for the heat traces of A and A a ( a ∈{ , } ), respectively, we have K ( t ) = K ( t ) K ( t ) . (2.34)Let us briefly recall that a special subcase of the present framework is the case of a slab , i.e.,Ω = Ω × R d ∋ x = ( x , x ) , y = ( y , y ) , V ( x ) = V ( x ) (2.35)with Ω ⊂ R d an open subset ( d + d = d ); see Part I for more details on this topic. The zeta-regularized total energy is E u := Z Ω d x h | b T u ( x ) | i = E u + B u ; (2.36)the second equality is proved after defining the regularized bulk and boundary en-ergies , which are E u := κ u Z Ω d x D u − ( x , x ) = κ u A − u , (2.37) B u := κ u (cid:18) − ξ (cid:19)Z ∂ Ω da ( x ) ∂∂n y D u +12 ( x , y ) (cid:12)(cid:12)(cid:12)(cid:12) y = x . (2.38)If Ω is unbounded, the integral over ∂ Ω in Eq. (2.38) has to be intended aslim ℓ → + ∞ R ∂ Ω ℓ da , where (Ω ℓ ) ℓ =0 , , ,... is a sequence of bounded subdomains withΩ ℓ ⊂ Ω ℓ +1 for ℓ ∈ { , , , .. } , and ∪ + ∞ ℓ =0 Ω ℓ = Ω.Assuming E u and B u (see Eq.s (2.37) (2.38)) to be finite and analytic for suitable u ∈ C , we define the renormalized energies by the generalized (or restricted) zetaapproach; for example, we put E ren := RP (cid:12)(cid:12)(cid:12) u =0 E u (cid:16) or E ren := E u (cid:12)(cid:12)(cid:12) u =0 (cid:17) . (2.39)8 .11 Curvilinear coordinates. In some applications it is natural to employ onΩ some set of curvilinear coordinates ( q i ) i =1 ,...,d ≡ q , inducing a set of spacetimecoordinates q ≡ ( q µ ) ≡ ( t, q ); the spatial and space-time line elements are, respec-tively, dℓ = a ij ( q ) dq i dq j ; ds = − dt + dℓ = g µν ( q ) dq µ dq ν ,g := − , g i = g i := 0 , g ij ( q ) := a ij ( q ) for i, j ∈ { , ..., d } . (2.40)The analogue of Eq. (2.8) in the coordinate system ( q µ ) is b T uµν := (1 − ξ ) ∂ µ b φ u ◦ ∂ ν b φ u − (cid:18) − ξ (cid:19) η µν (cid:16) ∂ λ b φ u ∂ λ b φ u + V ( b φ u ) (cid:17) − ξ b φ u ◦ ∇ µν b φ u (2.41)( ∇ µ is the covariant derivative induced by the metric (2.40)). For any scalar function f there hold ( γ kij are the Christoffel symbols for the spatial metric ( a ij ( q )), D i thecorresponding covariant derivative) ∇ µ f = ∂ µ f , ∇ ij f = D ij f = ∂ ij f − γ kij ∂ k f , ∇ i f = ∂ ( ∂ i f ) = ∂ i ( ∂ f ) = ∇ i f , ∇ f = ∂ f . (2.42)In the present paper we often work in a curvilinear coordinate system; more pre-cisely we use a set of (rescaled) spherical coordinates, fitting the symmetries of theconfiguration under analysis. Most of the results of the previous subsections arereadily rephrased in this framework. We consider the case of a massless scalar fieldon R d in presence of a classical isotropic harmonic potential. More precisely, weassume Ω := R d , V ( x ) := k | x | , (3.1)where k > | x | := p ( x ) + ... + ( x d ) ; the constant k is, dimensionally, a mass(or an inverse length) like the parameter κ employed for the field regularization (2.7).All the considerations reported in the following could be generalized, with somecalculational effort, to the anisotropic case V ( x ) = P di =1 ( k i ) ( x i ) where k i > i ∈ { , ..., d } , also including slab cases where some of the k i vanish; a furthervariation would concern a massive field in one of the above mentioned configurations,so that, e.g., in place of (3.1) we would have V ( x ) := m + k | x | . None of thesegeneralizations will be considered, for the sake of computational simplicity.9 .2 The heat kernel. Even though the configuration (3.1) is patently sphericallysymmetric, we choose to postpone the use of spherical coordinates to the next sub-section; here we work in standard Cartesian coordinates ( x i ) i =1 ,...,d , for reasons thatwill soon become apparent.First of all, notice that (3.1) is a configuration of the product type considered insubsection 2.9; in particular, with a trivial generalization of the results therein, wehave for the Hilbert space of the system and for the fundamental operator A := − ∆ + V , respectively, the following representations (compare with Eq. (2.32)): L ( R d ) = d O i =1 L ( R ) , (3.2) A = A ⊗ ⊗ ... ⊗ + ... + ⊗ ... ⊗ ⊗ A , A := − d dx + k x on L ( R ) . (3.3)In this situation, the heat kernel of A is given by (compare with Eq. (2.33)) K ( t ; x , y ) = d Y i =1 K ( t ; x i , y i ) (3.4)where K is the heat kernel of A ; the latter is the familiar Mehler kernel , that is K ( t ; x, y ) = k p π sinh(2 k t ) exp (cid:20) − k (cid:18) x + y k t ) − x y sinh(2 k t ) (cid:19)(cid:21) (3.5)(see [6], using the equations therein with the replacement ( t, x, y ) → ( k t, kx, ky )).The last two relations imply (with x · y := P di =1 x i y i ) K ( t ; x , y ) = k p π sinh(2 k t ) ! d exp (cid:20) − k (cid:18) | x | + | y | k t ) − x · y sinh(2 k t ) (cid:19)(cid:21) . (3.6)A similar analysis can be performed for the heat trace. Firstly note that (comparewith Eq. (2.34)) K ( t ) = d Y i =1 K ( t ) = (cid:16) K ( t ) (cid:17) d (3.7)where K ( t ) indicates the trace for the reduced one-dimensional problem; for the10atter quantity we can easily deduce the expression ( ) K ( t ) = 12 sinh( k t ) . (3.8)Summing up, for the heat trace of the complete d -dimensional problem we have K ( t ) = (cid:18)
12 sinh( k t ) (cid:19) d . (3.9)In the following subsections we explain how to construct the analytic continuationof the zeta-regularized VEV of the stress-energy tensor, using the expression (3.6)for the heat kernel and the integral representation (2.22) for the Dirichlet kernel.Moreover, we show how to determine asymptotic expressions for the renormalizedstress-energy VEV in the limit of small and large | x | , respectively. Moving along thesame lines, we also derive the renormalized VEV of the total energy; to this purpose,we consider the representation (2.23) for the trace Tr A − s and the expression (3.9)for the heat trace. As anticipated in subsection 3.2, we now pass toa set of curvilinear coordinates which best fit the symmetries of the problem (seesubsection 2.11); namely, we introduce the spherical “ k -rescaled” coordinates x q ( x ) ≡ ( r ( x ) , θ ( x ) , ..., θ d − ( x ) , θ d − ( x )) ∈ (0 , + ∞ ) × (0 , π ) × ... × (0 , π ) × (0 , π )(3.10)whose inverse map q x ( q ) is described by the equations k x = r cos( θ ) ,k x = r sin( θ ) cos( θ ) , ... k x d − = r sin( θ ) ... sin( θ d − ) cos( θ d − ) ,k x d = r sin( θ ) ... sin( θ d − ) . (3.11)Needless to say, for d = 1 (when the system is invariant under the parity symmetry x → − x ) we set r := k | x | . (3.12) In order to derive Eq. (3.8) we can proceed, for example, as explained hereafter. The secondequality in Eq. (2.21) and the explicit expression (3.5) for the one-dimensional heat kernel (alongwith some trivial trigonometric identities) allow us to infer K ( t ) = k p π sinh(2 k t ) Z R dx e − k x tanh( k t ) ;then, Eq. (3.8) follows evaluating the above elementary Gaussian integral. d , we have r = k | x | ; (3.13)thus, the coordinate r is an adimensional radius. In order to avoid cumbersomenotations, given a function Ω → Y , x f ( x ) (with Y any set), we indicate thecomposition q f ( x ( q )) as q f ( q ) ; we will use similar conventions for functionson Ω × Ω (see, e.g., what follows).Let us now consider the Dirichlet function D s ( x , y ) and the heat kernel K ( t ; x , y );let q = ( r, θ , ..., θ d − ) ≡ ( r, θ ) , p = ( r ′ , θ ′ , ..., θ ′ d − ) ≡ ( r, θ ′ ) (3.14)and put τ := k t ∈ (0 , + ∞ ) . (3.15)In the sequel we write D s ( q , p ) and K ( τ ; q , p ), respectively, for the Dirichlet andheat kernels at two points x , y of (rescaled) spherical coordinates q , p , and with τ related to t by Eq. (3.15). Eq. (3.6) implies K ( τ ; q , p ) = k p π sinh(2 τ ) ! d exp (cid:20) − (cid:18) r + r ′ τ ) − r r ′ S ( θ ) S ( θ ′ )sinh(2 τ ) (cid:19)(cid:21) (3.16)where S ( θ ) and S ( θ ′ ) are the products of cosines and sines of the angular coordinates( θ , ..., θ d ) and ( θ ′ , ..., θ ′ d ) of Eq. (3.14), corresponding to the scalar product x · y .From Eq. (2.22) it follows that D s ( q , p ) = k − s Γ( s ) Z + ∞ dτ τ s − K ( τ ; q , p ) ; (3.17)substituting this relation into the analogues of Eq.s (2.16-2.18) for curvilinear coor-dinates (see subsection 2.11), we get h | b T uµν ( q ) | i = k d +1 Γ( u +12 ) (cid:16) κk (cid:17) u Z + ∞ dτ τ u − d − H ( u ) µν ( τ ; q ) ( µ, ν ∈ { , r, θ , ..., θ d − } ) (3.18)where the coefficients H ( u ) µν ( τ ; q ) are as follows (for i, j, h, ℓ ∈ { ρ, θ , ..., θ d − } ; here D is the spatial covariant derivative) H ( u )00 ( τ ; q ) := (cid:16) τk (cid:17) d/ (cid:20)(cid:18)
14 + ξ (cid:19)(cid:18) u − (cid:19) + (cid:18) − ξ (cid:19) τ (cid:16) a hℓ ( q ) D q h p ℓ + r (cid:17)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) p = q K ( τ ; q , p ) , (3.19) H ( u )0 i ( τ ; q ) = H ( u ) i ( τ ; q ) := 0 , (3.20)12 ( u ) ij ( τ ; q ) = H ( u ) ji ( τ ; q ) := (cid:16) τk (cid:17) d/ (cid:20)(cid:16) − ξ (cid:17) a ij ( q ) (cid:18) u − − τ (cid:16) a hℓ ( q ) D q h p ℓ + r (cid:17)(cid:19) ++ (cid:16) τk (cid:17)(cid:18)(cid:16) − ξ (cid:17) D q i p j − ξ D q i q j (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) p = q K ( τ ; q , p ) . (3.21)Here and in the reaminder of the paper, we are implicitly understanding the de-pendence on the parameter ξ for simplicity of notation: so, H ( u ) µν ( τ ; q ) stands for H ( u ) µν ( τ ; q ; ξ ).In the following subsection we point out some notable properties of the coefficients H ( u ) µν ( τ ; q ); these properties will be employed in the following subsection 3.5 in orderto evaluate the renormalized stress-energy VEV. H ( u ) µν . Consider the explicit expressions (3.19-3.21) for these coefficients. Using as well Eq. (3.16) for the heat kernel expressed inrescaled, spherical coordinates, one can prove the following facts.i) For fixed τ, q and any µ, ν ∈ { , r, θ , ..., θ d − } , the map u
7→ H ( u ) µν ( τ ; q ) is affine.ii) For fixed q and any u ∈ C , µ, ν ∈ { , r, θ , ..., θ d − } , the map τ
7→ H ( u ) µν ( τ ; q ) issmooth on [0 , + ∞ ) and exponentially vanishing for τ → + ∞ .iii) The final expressions for the coefficients H ( u ) µν do not depend on the parameter k (even though it appears in the right-hand sides of Eq.s (3.19-3.21)).iv) There hold H ( u ) µν = 0 for µ = ν , H ( u ) θ d − θ d − = sin ( θ d − ) H ( u ) θ d − θ d − = ... = sin ( θ d − ) ... sin ( θ ) H ( u ) θ θ . (3.22)v) For µ = ν ∈ { , r, θ } , there hold H ( u ) µν ( τ ; q ) = e − r tanh τ M ( u ) µν ( τ ; r ) (3.23)where M ( u ) µν ( τ ; r ) is a polynomial in r, u of degree 1 in both these variables, withcoefficients depending smoothly on τ .All the above statements i)-v) could be proved for arbitrary d , but we will limitourself to check them in the cases d ∈ { , , } considered hereafter, for which theexplicit expressions of all coefficients H ( u ) µν ( τ ; q ) ( µ = ν ∈ { , r, θ } ) will be given:see the subsequent subsections 4.1, 4.2 and 4.3, respectively.Before moving on, let us make a few remarks. First, consider the integral represen-tation (3.18) for the regularized stress-energy VEV; in consequence of point iii), thelatter VEV only depends on the parameter k through the multiplicative coefficient k d +1 ( κ/k ) u in front of the integral in the cited equation. Next, we note that Eq.133.22) of point iv) indicates that the VEV h | b T uµν ( q ) | i is diagonal and that the onlyindependent components are those with µ = ν ∈ { , r, θ } ; finally, these componentsonly depend on the radial coordinate r (and not on the angular ones { θ , ..., θ d − } ),in accordance with the spherical symmetry of the configuration under analysis. Let us move on to determine the analytic continu-ation of the regularized VEV (3.18). To this purpose, we refer to the framework ofsubsection 2.8; using Eq.s (2.24-2.26) with H = H ( u ) µν ( ; x ), ρ = ( d + 3 − u ) /
2, weobtain h | b T uµν ( q ) | i = k d +1 Γ( u +12 ) ( − n ( u − d − +1) ... ( u − d − + n ) (cid:16) κk (cid:17) u Z + ∞ dτ τ u − d − + n ∂ nτ H ( u ) µν ( τ ; q ) . (3.24)Notice that, due to the features of H ( u ) µν ( τ ; q ) pointed out in the previous subsection,the integral in the above expression converges for ℜ u > d + 1 − n , (3.25)so that Eq. (3.24) yields the required analytic continuation of h | b T uµν ( q ) | i to thevery same region. Since we are interested in evaluating (the regular part of) thisanalytic continuation in u = 0, we choose n so that Eq. (3.25) holds for u = 0, i.e.( ) n > d + 12 . (3.26)Following Eq. (2.10), in general we define h | b T µν ( q ) | i ren := RP (cid:12)(cid:12)(cid:12) u =0 h | b T uµν ( q ) | i . (3.27)For n as in Eq. (3.26), consider the expression on the right-hand side of Eq. (3.24);for any even spatial dimension d this expression is regular in u = 0, so that wecan apply the zeta regularization in its restricted version to obtain the renormalizedstress-energy VEV. On the other hand, for odd d we must resort to the extendedversion of the zeta technique, since the function under analysis has a simple pole in u = 0 .Because of the pole singularity, the procedure of evaluating the regular part of Eq.(3.24) in u = 0 in the case of odd d implies the appearance of a logarithmic term in τ in the integrand ( ). Simple but rather lenghty computations give the following As a matter of fact, for computational simplicity, we will always choose the smaller n ∈{ , , , ... } fulfilling Eq. (3.26). In fact τ u/ = e u ln τ = 1 + u τ + O ( u ) for u → . d either odd or even and µ, ν ∈ { , r, θ } : h | b T µν ( q ) | i ren = k d +1 (cid:16) T (0) µν ( r ) + M κ,k T (1) µν ( r ) (cid:17) , where (3.28) T (0) µν ( r ) := Z + ∞ dτ τ n − d +32 e − r tanh τ h P (0) µν ( τ ; r ) + ln τ P (1) µν ( τ ; r ) i ,T (1) µν ( r ) := Z + ∞ dτ τ n − d +32 e − r tanh τ P (1) µν ( τ ; r ) , M κ,k := γ EM +2 ln (cid:18) κk (cid:19) ( γ EM ≃ .
577 is the Euler-Mascheroni constant). In the above P (0) µν ( τ ; r ) and P (1) µν ( τ ; r ) are suitable functions determined by H ( u ) µν ( τ ; q ); these functions are infact polynomials in r of the form P ( a ) µν ( τ ; r ) = n +1 X i =0 p ( a ) i,µν ( τ ) r i ( a ∈ { , } ) , (3.29)for some smooth functions p (0) i,µν , p (1) i,µν : [0 , + ∞ ) → R ( i ∈ { , ..., n + 1 } ). Let usstress that P (1) µν ( τ ; r ) = 0 , T (1) µν ( r ) = 0 for d even , (3.30)a fact corresponding to the previous comments on the logarithmic terms.Next note that, in consequence of the remarks of subsection 3.4, the renormalizedVEV h | b T µν ( q ) | i ren only depends on the parameter k through the coefficients k d +1 and M κ,k in the first equation of (3.28). In particular, the functions T ( a ) µν ( r ) ( a ∈{ , } ) introduced therein do not depend on k and we can evaluate them computingthe integrals in Eq. (3.28) numerically, for any fixed r ∈ (0 , + ∞ ).To conclude, following the considerations of subsection 2.3 ( ), we define the con-formal and non-conformal parts of the functions r T (0) µν ( r ) , T (1) µν ( r ) respectivelyas T ( a, ♦ ) µν := T ( a ) µν (cid:12)(cid:12)(cid:12) ξ = ξ d , T ( a, (cid:4) ) µν := 1 ξ − ξ d (cid:16) T ( a ) µν − T ( a, ♦ ) µν (cid:17) ( a ∈ { , } ) (3.31) The logarithmic term proportional to u is not relevant in the case of even d , where analyticcontinuation exists up to u = 0 and it is simply obtained setting u = 0 in (3.24). On the contrary,for odd d we must take the regular part at u = 0 of the expression (3.24) and the above term u ln τ contributes to it, since it is multiplied by a term proportional to 1 /u that comes from the u → / ( u − d − +1) ... ( u − d − + n ) in (3.24). Also recall the statement inpoint i) of subsection 3.4. In particular, we use the same convention as in Part I (see Eq. (2.24) therein or Eq. (2.13) inthe present paper): ♦ ≡ conformal , (cid:4) ≡ non-conformal . Compare Eq. (3.31) with Eq.s (2.25) (2.26) in Part I. ξ d is the conformal parameter, see Eq. (2.14)). In subsections 4.1-4.3 we presentthe functions P ( a ) µν ( a ∈ { , } ) and the graphs (obtained via numerical integration)for each one of the functions in Eq. (3.31), for d ∈ { , , } respectively. r = k | x | → + . Hereafter we present a method which wewill use to determine the asymptotic expansion for the renormalized stress-energyVEV in the limit r = k | x | → + .Let us consider a function F : (0 , + ∞ ) → R defined by F ( r ) := Z + ∞ dτ e − r h ( τ ) P ( τ ; r ) , (3.32)where h ( τ ) is a positive bounded function, while P ( τ ; r ) is a polynomial of degree N in r ; more precisely, we assume that P ( τ ; r ) = N X i =0 p i ( τ ) r i , (3.33)for some integrable functions p i : (0 , + ∞ ) → R ( i ∈ { , ..., N } ). We claim that theTaylor expansion of the exponential in Eq. (3.32) implies an expansion F ( r ) = N X i =0 a i r i + R N +1 ( r ) , R N +1 ( r ) = O ( r N +1) ) for r → + , (3.34)where the coefficients a i ∈ R are defined by a i := i X j =0 ( − i − j ( i − j )! Z + ∞ dτ p j ( τ ) ( h ( τ )) i − j ( i ∈ { , ..., N } ) ; (3.35)more precisely, the remainder term is such that | R N +1 ( r ) | C N +1 r N +1) for all r ∈ (0 , + ∞ ) , (3.36) C N +1 := N X i =0 N − i +1)! Z + ∞ dτ | p i ( τ ) | ( h ( τ )) N − i +1 . (3.37)See Appendix A for the derivation of the above results.Consider the expression (3.28) for the independent components of the renormalizedstress-energy VEV; recalling that P (0) µν and P (1) µν are both polynomials in r of degree n +1 (see Eq. (3.29)), we note that each component (fixed µ, ν ) has the form (3.32)(3.33) with h ( τ ) = tanh( τ ) ,P ( τ ; r ) = τ n − d +32 (cid:16) P (0) µν ( τ ; r ) + ( M κ,k +ln τ ) P (1) µν ( τ ; r ) (cid:17) , N = n + 1 . (3.38)16hus, using the above approach, we can infer the expansion for the renormalizedstress-energy VEV in the limit r = k | x | → + ; we defer the explicit results for d ∈ { , , } to Section 4, where the integrals in Eq. (3.35) are evaluated numericallyfor the choices of h and P under analysis. r = k | x | → + ∞ . Let us discuss some general methodsallowing to determine the asymptotic behaviour for certain types of integrals; thesemethods will be employed to determine the asymptotic expansion of renormalizedstress-energy VEV in the limit r = k | x | → + ∞ .Consider a function F : (0 , + ∞ ) → R defined by F ( r ) := Z dv e − r v v λ Q ( v ; r ) ( λ > − , (3.39)where Q ( v ; r ) is a polynomial in r of the form Q ( v ; r ) = Q (0) ( v ; r ) + Q (1) ( v ; r ) ln v ,Q ( a ) ( v ; r ) = N X i =0 q ( a ) i ( v ) r i ( a ∈ { , } ) , (3.40)for some smooth integrable functions q (0) i , q (1) i : [0 , → R ( i ∈ { , ..., N } ). Theasymptotic expansion of F ( r ) for large r can be obtained using the standard theoryof Laplace integrals [7, 13, 21, 22]; in the present subsection we only report the mainresults of this analysis, making reference to Appendix A for the proofs.For any K ∈ { , , , ... } and any v ∈ (0 , F ( r ) = N X i =0 K + i − X m =0 h A i,m ( v , r )+ B i,m ( v , r ) ln r i r i − λ − m − + R K +1 ( v , r ) ,R K +1 ( v , r ) = O ( r − K + λ +1) ln r ) for r → + ∞ . (3.41)In the above expression, the coefficients A i,m ( v , r ), B i,m ( v , r ) (for i ∈ { , ..., N } , m ∈ { , , , ... } ) are defined by A i,m ( v , r ) := q (0) i,m γ ( m + λ +1 , v r ) + q (1) i,m g ( m + λ +1 , v r ) , (3.42) B i,m ( v , r ) := − q (1) i,m γ ( m + λ +1 , v r ) , q ( a ) i,m := d m dv m (cid:12)(cid:12)(cid:12)(cid:12) v =0 q ( a ) i ( v ) ( a ∈ { , } ) , where γ ( , ) denotes the lower incomplete gamma function and we put g ( s, z ) := Z z dw e − w w s − ln w for all z ∈ (0 , + ∞ ), s ∈ C with ℜ s > . (3.43)17oncerning the remainder term R K +1 ( v , r ), we can give quantitative estimates ofthe form | R K +1 ( v , r ) | ( F K +1 + G K +1 ln r ) r − K + λ +1) for all r ∈ (0 , + ∞ ) ; (3.44)these involve some constants F K +1 , G K +1 >
0, that can be determined explicitlyfollowing the indications of Appendix A.Moreover, from the asymptotic behaviour of the incomplete gamma γ ( , ) and ofthe function g ( , ), we infer F ( r ) = N X i =0 K + i − X m =0 h α i,m + β i,m ln r i r i − m − λ − + S K +1 ( v , r ) , S K +1 ( v , r ) = O ( r − K + λ +1) ln r ) for r → + ∞ , (3.45)where we have defined the real coefficients (for i ∈ { , ..., N } , m ∈ { , , , ... } ) α i,m := Γ( m + λ +1) h q (0) i,m + ψ ( m + λ +1) q (1) i,m i ,β i,m := − Γ( m + λ +1) q (1) i,m ; (3.46)in the above definitions, ψ ( s ) := ∂ s ln Γ( s ) denotes the digamma function and the q (0) i,m , q (1) i,m are as in Eq. (3.42). In principle, the remainder term S K +1 ( v , r ) in Eq.(3.45) could be evaluated quantitatively combining Eq. (3.44) with some knownestimates about the incomplete gamma functions; the related computations wouldbe very tedious, and will be avoided here.Let us now consider expression (3.28) for the renormalized VEV of the stress-energytensor and perform inside the integrals appearing therein the change of variable τ = arcth ( v ) , v ∈ (0 ,
1) (3.47)(“arcth” denotes the hyperbolic arctangent); in this way we obtain T (0) µν ( r ) = Z dv e − r v (arcth v ) n − d +32 − v h P (0) µν (arcth v ; r )+ln(arcth v ) P (1) µν (arcth v ; r ) i ,T (1) µν ( r ) = Z dv e − r v (arcth v ) n − d +32 − v P (1) µν (arcth v ; r ) . (3.48)Since P (0) µν and P (1) µν are polynomials in r of degree n + 1 (see Eq. (3.29)), we canre-express each component of the renormalized stress-energy VEV (fixed µ, ν ) in theform (3.39), where Q ( v ; r ) is as in Eq. (3.40) with λ := n − d + 32 , N = n + 1 ; (3.49)18 (0) ( v ; r ) := (3.50)11 − v (cid:18) arcth vv (cid:19) n − d +32 (cid:20) P (0) µν (arcth v ; r ) + (cid:18) ln (cid:16) arcth vv (cid:17) + M κ,k (cid:19) P (1) µν (arcth v ; r ) (cid:21) ; Q (1) ( v ; r ) := 11 − v (cid:18) arcth vv (cid:19) n − d +32 P (1) µν (arcth v ; r ) . (3.51)In this way we can derive asymptotic expressions analogous to (3.41) (and (3.45))for the renormalized stress-energy VEV in the limit r = k | x | → + ∞ ; we report inSection 4 the explicit results of this computation for the cases with d ∈ { , , } ,respectively. Let us recall that the total energy can always be expressedas the sum of a bulk and a boundary contribution as in Eq. (2.36); in the followingwe are going to discuss these two contributions separately.Let us first consider the regularized bulk energy ; according to Eq. (2.37), this is E u = κ u A − u . This is connected through Eq. (2.23) to the heat trace K ( t ) := Tr e − t A that, ac-cording to Eq. (3.9), has the form K ( t ) = 1 t d H ( t ) with H ( t ) := (cid:18) t k t ) (cid:19) d ; (3.52)it is patent that the map t H ( t ) is smooth on [0 , + ∞ ) and exponentially vanishingfor t → + ∞ . Then, following the general considerations of subsection 2.8 (see, inparticular, Eq. (2.28) for the trace Tr A − s ), we obtain for the regularized bulkenergy E u = ( − n κ u u − )( u − − d ) ... ( u − − d + n − Z + ∞ d t t u − − d + n d n d t n H ( t ) . (3.53)The above relation holds for any n ∈ { , , , ... } and the integral appearing thereinconverges for any complex u with ℜ u > d − n ) + 1 ; (3.54)thus, for any integer n > d + 1 /
2, Eq. (3.53) gives the analytic continuation of E u in a neighborhood of u = 0. Since here no singularity appears, we can obtain therenormalized bulk energy simply by setting u = 0 in Eq. (3.53); making again thechange of integration variable τ := k t (see Eq. (3.15)), we infer19 ren = − k d +2 − n √ π n − Y i =0 d − i )+1 ! Z + ∞ dτ τ n − d − d n d t n H ( τ )for any n > d + 12 , with H ( τ ) := (cid:16) τ sinh τ (cid:17) d . (3.55)The above expression is fully explicit and holds for any spatial dimention d ; in thesubsequent Section 4 we are going to evaluate numerically the integral appearing inEq. (3.55) for d ∈ { , , } , making the minimal choice n = d + 1 .Let us point out that in the case under analysis, where the potential is assumed tobe isotropic, we could also derive an alternative representation for the bulk energy E ren in terms of the (analytically continued) Riemann zeta function; we discuss thistopic in detail in Appendix B. Here we prefer to focus the attention on the approach(3.55) for the computation of E ren since it is more general; in fact, it could beemployed straightforwardly also when the background potential is not isotropic ( ).As pointed out in the Introduction, the computation of the renormalzied bulk energywas also performed by Actor and Bender in [2]; these authors employed a global zetaapproach based on ad hoc results on the analytic continuation of some particularkind of “generalized” zeta functions (discussed in detail in [1]), befitting the con-figuration under analysis. Let us stress once more that here, instead, we derive therequired analytic continuation applying a general procedure in a mechanical way.Our method and the one of [2] could be proved to be equivalent in any dimension,yet the general discussion is too involved to be reported here; in subsection 4.3 wecheck by direct comparison that, in the case d = 3 ( ), the numerical result obtainedusing prescription (3.55) agrees with the one derived in [2].Now, let us move on to discuss the boundary energy . In all the cases treated inParts I and II this boundary contribution was easily found to vanish identically inconsequence of the boundary conditions. Here the situation is not so simple: sincethe spatial domain is unbounded (indeed, Ω = R d ), we must resort to the procedurepointed out in subsection 2.10 and intend the integral over ∂ Ω in definition (2.38) asthe limit of integrals over the boundary of suitable, bounded subdomains (Ω ℓ ) ℓ =0 , , ,... (see the comments below Eq. (2.38)). Due to the spherical symmetry and to thecharacteristic scale of the problem under analysis, it is natural to choose for Ω ℓ theballs with center in the origin and radius ℓ/k :Ω ℓ := B ( d ) ℓ/k ( ) ≡ { x ∈ R d | | x | ℓ/k } for ℓ ∈ { , , , ... } . (3.56)Then, Eq. (2.38) for the boundary energy reads B u = κ u (cid:18) − ξ (cid:19) lim ℓ → + ∞ Z S ( d − ℓ/k da ( x ) ∂ r y D u +12 ( x , y ) (cid:12)(cid:12)(cid:12) y = x (3.57) Let us recall that in the present paper we have chosen to omit the analysis of cases withunisotropic background potentials only for simplicity. As a matter of fact, this is the only model discussed both here and in [2]. S ( d − ℓ/k ≡ ∂B ( d ) ℓ/d ( ) indicates the ( d − ∂ r y denotes the derivative in the radial direction with respect to the variable y . In terms of the set of rescaled spherical coordinates (3.11), we have ∂ r y K ( t ; x , y ) (cid:12)(cid:12)(cid:12) y = x = − k p π sinh(2 k t ) ! d ( k r ) tanh( k t ) e − r tanh( k t ) ; (3.58)then, using the integral representation (2.22) for the Dirichlet kernel in terms of theheat kernel (again, with the change of variable τ := k t ) and noting that the aboveexpression for the latter does not depend on the angular variables ( θ , ..., θ d − ) ( ),we infer B u = − k d − Γ( d ) Γ( u +12 ) (cid:16) κk (cid:17) u (cid:18) − ξ (cid:19) B u (3.59)where, for brevity, we have put B u := lim ℓ → + ∞ Z + ∞ dτ τ u − d +12 (cid:18) e − ℓ tanh τ ℓ d τ d/ − tanh τ (sinh(2 τ )) d/ (cid:19) (3.60)For any d ∈ { , , , ... } , ℓ >
0, the expression within the round brackets in theabove equation is an analytic function of τ on [0 , + ∞ ) and vanishes exponentiallyfor τ → + ∞ . In consequence of this, we easily infer that the integral in Eq. (3.60)converges for any complex u with ℜ u > d − . (3.61)Within this region, by Lebesgue’s dominated convergence theorem ( ) we can takethe limit under the integral sign in Eq. (3.60), and the conclusion is B u = 0 for ℜ u > d − . (3.62) We also use the fact that the area of the ( d − r is Area (cid:16) S ( d − r (cid:17) = 2 π d/ Γ( d/ r d − . Indeed, notice that, for any ℓ >
0, there holds (cid:12)(cid:12)(cid:12)(cid:12) τ u − d +12 (cid:18) e − ℓ tanh τ ℓ d τ d/ − tanh τ (sinh(2 τ )) d/ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) τ u − d +12 (cid:18) τ d/ − tanh τ (sinh(2 τ )) d/ (cid:19) (cid:18) d e tanh( τ ) (cid:19) d/ where the right-hand side is Lebesgue-integrable over (0 , + ∞ ) ; besides,lim ℓ → + ∞ τ u − d +12 (cid:18) e − ℓ tanh τ ℓ d τ d/ − tanh τ (sinh(2 τ )) d/ (cid:19) = 0 for all τ ∈ (0 , + ∞ ) . B ren := B u (cid:12)(cid:12)(cid:12) u =0 = 0 . (3.63) d ∈ { , , } d = 1. Let us first recall that in this case the configuration is symmetri-cal under the reflection x → − x ; on any one of the intervals (0 , + ∞ ) and ( −∞ , q ≡ r := k | x | ∈ (0 , + ∞ ) (see Eq. (3.12)).After lenghty computations, based on Eq.s (2.16-2.18) and on the general consid-erations of subsection 3.5, we obtain for the regularized VEV of the stress-energytensor an expression of the form (3.18). More precisely, we have h | b T uµν ( q ) | i = k Γ( u +12 ) (cid:16) κk (cid:17) u Z + ∞ dτ τ − u H ( u ) µν ( τ ; q ) ( µ, ν ∈ { , r } ) , (4.1)where (recall that dependence on ξ is understood) H ( u )00 ( τ ; q ) := A ( τ, r ) (cid:20) − (1 − u )(1+4 ξ ) + (1 − ξ ) (cid:18) τ sinh 2 τ (cid:19)(cid:18) τ τ r (cid:19)(cid:21) , (4.2) H ( u ) rr ( τ ; q ) := A ( τ, r ) (cid:20) τ (1+4 ξ cosh 2 τ )sinh 2 τ − (1 − ξ ) (cid:18) − u + 2 τ r τ (cid:19)(cid:21) , (4.3)and the off-diagonal components H ( u ) µν ( τ ; q ) ( µ = ν ) vanish identically; in the above,for simplicity of notation we have set A ( τ, r ) := 116 √ π e − r tanh τ s τ sinh(2 τ ) . (4.4)The above expressions for the components of the tensor H ( u ) µν ( τ ; q ) are easily seen topossess the features anticipated in Eqs. (3.22) (3.23) and in the related comments.Thus, according to the general framework developed in subsection 3.5, we can obtainthe analytic continuation of h | b T uµν ( q ) | i given in Eq. (4.1) integrating by parts n times the integral therein, with n > n = 2 , (4.5)so that Eq. (3.24) gives h | b T uµν ( q ) | i = k Γ( u +12 ) 1( u − u (cid:16) κk (cid:17) u Z + ∞ dτ τ u ∂ τ H ( u ) µν ( τ ; q ) . (4.6)22t is evident that h | b T uµν ( q ) | i has a simple pole at u = 0 (this reflects a generalfeature of the cases with odd spatial dimension, already indicated in subsection 3.5).Thus, the renormalized stress-energy VEV is given by the regular part in u = 0 ofthe function in Eq. (4.6) (see Eq. (3.27)); with some effort, we can re-express thisquantity as in Eq. (3.28), i.e. h | b T µν ( q ) | i ren = k (cid:16) T (0) µν ( r ) + M κ,k T (1) µν ( r ) (cid:17) , (4.7) T (0) µν ( r ) := Z + ∞ dτ e − r tanh τ h P (0) µν ( τ ; r ) + ln τ P (1) µν ( τ ; r ) i ,T (1) µν ( r ) := Z + ∞ dτ e − r tanh τ P (1) µν ( τ ; r ) , M κ,k := γ EM +2 ln (cid:18) κk (cid:19) , where P (0) µν ( τ ; r ) := − √ π e r tanh τ h ∂ τ H (0) µν ( τ ; r ) + 2 ∂ u (cid:12)(cid:12)(cid:12) u =0 ∂ τ H ( u ) µν ( τ ; r ) i , P (1) µν ( τ ; r ) := − √ π e r tanh τ ∂ τ H (0) µν ( τ ; r ) . (4.8)It is readily found that P (0) µν , P (1) µν are polynomials of degree N = 3 in r . We canevaluate numerically the integrals in Eq. (4.7), distinguishing between the conformaland non-conformal parts ♦ , (cid:4) of each component: see Eq. (3.31), recalling that for d = 1 we have (see Eq. (2.14)) ξ = 0 . (4.9)The forthcoming Fig.s 1-4 show the graphs of the functions r T ( a, × ) µν ( r ) for µ = ν ∈ { , r } , a ∈ { , } , × ∈ { ♦ , (cid:4) } . (4.10) - - - - - T H í L T H í L Figure 1: d = 1: graphs of T (0 , ♦ )00 and T (1 , ♦ )00 .23 - T H à L - - T H à L Figure 2: d = 1: graphs of T (0 , (cid:4) )00 and T (1 , (cid:4) )00 . T rr H í L - - - - T rr H í L Figure 3: d = 1: graphs of T (0 , ♦ ) rr and T (1 , ♦ ) rr . - - - - T rr H à L - - T rr H à L Figure 4: d = 1: graphs of T (0 , (cid:4) ) rr and T (1 , (cid:4) ) rr .24et us pass to evaluate the small and large r asymptotics of the functions inEq. (4.10). On the one hand, using Eq.s (3.32-3.35) (with N = 3) we obtain,for r = k | x | → + , T (0 , ♦ )00 ( r ) = − . . r − . r + 0 . r + O ( r ) ,T (1 , ♦ )00 ( r ) = 0 . r + O ( r ) ,T (0 , (cid:4) )00 ( r ) = 0 . − . r + 0 . r − . r + O ( r ) ,T (1 , (cid:4) )00 ( r ) = O ( r ) ; (4.11) T (0 , ♦ ) rr ( r ) = − . − . r + 0 . r − . r + O ( r ) ,T (1 , ♦ ) rr ( r ) = − . r + O ( r ) ,T (0 , (cid:4) ) rr ( r ) = − . − . r + O ( r ) ,T (1 , (cid:4) ) rr ( r ) = O ( r ) . (4.12)The numerical coefficients appearing here and in other small r expansions are ob-tained from numerical computation of the integrals in (3.35).On the other hand, using Eq.s (3.39) (3.45) (3.46) (with K = 3, λ = 0) we obtainthe following asymptotic expansions, for r = k | x | → + ∞ : T (0 , ♦ )00 ( r ) = − r π (cid:16) ln r + γ EM + 1 (cid:17) + 18 π r + 49120 π r + O ( r − ln r ) ,T (1 , ♦ )00 ( r ) = r π + O ( r − ln r ) ,T (0 , (cid:4) )00 ( r ) = − π r − π r + O ( r − ln r ) ,T (1 , (cid:4) )00 ( r ) = O ( r − ln r ) ; (4.13) T (0 , ♦ ) rr ( r ) = r π (cid:16) ln r + γ EM − (cid:17) + 124 π r + 7120 π r + O ( r − ln r ) ,T (1 , ♦ ) rr ( r ) = − r π + O ( r − ln r ) ,T (0 , (cid:4) ) rr ( r ) = O ( r − ln r ) ,T (1 , (cid:4) ) rr ( r ) = O ( r − ln r ) . (4.14)Let us now discuss the bulk energy; using the expression (3.55) with n = 2 andevaluating numerically the integral appearing therein, we infer E ren = (0 . ± − ) k . (4.15)25 .2 Case d = 2. Consider the general framework of subsection 3.5; in this casewe use the (rescaled) polar coordinates q = ( r, θ ) ∈ (0 , + ∞ ) × [0 , π ), fulfilling (seeEq. (3.11)) ( ) k x = r cos θ , k x = r sin θ . (4.16)Proceeding as in the one dimensional case, with some effort we obtain the followingintegral representation for the zeta-regularized stress-energy VEV (compare withEq. (3.18) and recall that dependence on ξ is understood): h | b T uµν ( q ) | i = k Γ( u +12 ) (cid:16) κk (cid:17) u Z + ∞ dτ τ − + u H ( u ) µν ( τ ; q ) ( µ, ν ∈ { , r, θ } ) , (4.17)where the tensor H ( u ) µν is diagonal and, concerning the diagonal components, we have H ( u )00 ( τ ; q ) := A ( τ, r ) (cid:20) − (1 − u )(1+4 ξ ) + (1 − ξ ) (cid:18) τ sinh 2 τ (cid:19)(cid:18) r sinh 4 τ τ (cid:19)(cid:21) , (4.18) H ( u ) rr ( τ ; q ) := A ( τ, r ) (cid:20) ξ τ tanh τ − (1 − ξ ) (cid:18) − u + r τ cosh τ (cid:19)(cid:21) , (4.19) H ( u ) θ θ ( τ ; q ) := (cid:16) rk (cid:17) A ( τ, r ) (cid:20) ξ τ tanh τ − (1 − ξ ) (cid:18) − u + r τ cosh 2 τ cosh τ (cid:19)(cid:21) . (4.20)In the above, for simplicity of notation we have put A ( τ, r ) := 164 π e − r tanh τ (cid:18) τ sinh 2 τ (cid:19) . (4.21)Again, the features indicated by Eq.s (3.22) (3.23) (and related comments) are allpossessed by the expressions (4.18-4.21) for H ( u ) µν . So, we can analytically continuein u the expression in Eq. (4.17) integrating by parts n times, with n > / n = 2 , (4.22)giving h | b T uµν ( q ) | i = k Γ( u +12 ) 1( u − )( u − ) (cid:16) κk (cid:17) u Z + ∞ dτ τ u − ∂ τ H ( u ) µν ( τ ; q ) . (4.23)Patently, each component of h | b T uµν | i is regular at u = 0; thus, we can obtainthe renormalized version of the stress-energy VEV by simply putting u = 0 in Eq. Of course, the spatial line element in this coordinate system reads dℓ = k − ( dr + r dθ ); thisdetermines the Christoffel symbols in Eq. (2.42) for the derivatives D ij . d ; see below Eq. (3.26)). In conclusion, h | b T µν ( q ) | i ren = k T (0) µν ( r ) , (4.24) T (0) µν ( r ) := Z + ∞ dτ e − r tanh τ P (0) µν ( τ ; r ) , P (0) µν ( τ ; r ) := 43 √ π τ e r tanh τ ∂ τ H (0) µν ( τ ; q )(compare with Eq.s (3.28-3.30)); also in this case, we easily check that P (0) µν is apolynomial of degree N = 3 in r . Next, we proceed as we did in the previoussubsection for the case of spatial dimension d = 1: we evaluate numerically theintegrals in Eq. (4.24) and separate the conformal and non-conformal parts ♦ , (cid:4) ofeach component (see Eq. (3.31)), noting that Eq. (2.14) gives ξ = 18 . (4.25)The forthcoming Fig.s 5-7 show the graphs of the functions r T (0 , × ) µν ( r ) ( µ = ν ∈ { , r } ) , ( k/r ) T (0 , × ) θ θ ( r ) for × ∈ { ♦ , (cid:4) } . (4.26)27 - - - - - T H í L T H à L Figure 5: d = 2: graphs of T (0 , ♦ )00 and T (0 , (cid:4) )00 . T rr H í L - - - - - - - T rr H à L Figure 6: d = 2: graphs of T (0 , ♦ ) rr and T (0 , (cid:4) ) rr . H k (cid:144) r L T ΘΘ H í L - - - - H k (cid:144) r L T ΘΘ H à L Figure 7: d = 2: graphs of ( k/r ) T (0 , ♦ ) θ θ and ( k/r ) T (0 , (cid:4) ) θ θ .28ow, let us consider the small and large r asymptotics of the functions in (4.26).On the one hand, Eq.s (3.32-3.35) (with N = 3) give, for r = k | x | → + , T (0 , ♦ )00 ( r ) = − . − . r − . r + 0 . r + O ( r ) ,T (0 , (cid:4) )00 ( r ) = 0 . − . r + 0 . r − . r + O ( r ) ; (4.27) T (0 , ♦ ) rr ( r ) = − . . r + 0 . r − . r + O ( r ) ,T (0 , (cid:4) ) rr ( r ) = − . . r − . r + 0 . r + O ( r ) ; (4.28)( k/r ) T (0 , ♦ ) θ θ ( r ) = − . . r + 0 . r − . r + O ( r ) , ( k/r ) T (0 , (cid:4) ) θ θ ( r ) = − . . r − . r + 0 . r + O ( r ) . (4.29)Again, the above coefficients were obtained calculating numerically the integrals inEq. (3.35).On the other hand, using Eq.s (3.39) (3.45) (3.46) (with K = 5, λ = − /
2; note thatno logarithmic term ln r appears in this case) we obtain the following asymptoticexpansions for r = k | x | → + ∞ : T (0 , ♦ )00 ( r ) = − r π − π r + O ( r − ) ,T (0 , (cid:4) )00 ( r ) = 14 π r + 332 π r + O ( r − ) ; (4.30) T (0 , ♦ ) rr ( r ) = r π + 148 π r − π r + O ( r − ) ,T (0 , (cid:4) ) rr ( r ) = − π r + 132 π r + O ( r − ) ; (4.31)( k/r ) T (0 , ♦ ) θ θ ( r ) = r π − π r + + 332560 π r + O ( r − ) , ( k/r ) T (0 , (cid:4) ) θ θ ( r ) = − π r + O ( r − ) . (4.32)As for the bulk energy, using again the expression (3.55) with n = 3 and evaluatingnumerically the integral appearing therein, we obtain E ren = − (0 . ± − ) k . (4.33) d = 3. In this case we use the coordinates q = ( r, θ , θ ) ∈ (0 , + ∞ ) × (0 , π ) × [0 , π ), which are related to the Cartesian coordinates x ≡ ( x , x , x ) via(see Eq. (3.11)) ( ) k x = r cos θ , k x = r sin θ cos θ , k x = r sin θ sin θ . (4.34) In this case the framework of subsection 2.11 must be employed using the spatial line element dℓ = k − ( dr + r ( dθ +sin θ dθ )) and the corresponding Christoffel symbols. ξ is understood) h | b T uµν ( q ) | i = k Γ( u +12 ) (cid:16) κk (cid:17) u Z + ∞ dτ τ − u H ( u ) µν ( τ ; q ) ( µ, ν ∈ { , r, θ , θ } ) , (4.35)where H ( u ) µν is diagonal and we only have to consider the independent components H ( u )00 ( τ ; q ) := A ( τ, r ) (cid:20) − (1 − u )(1+4 ξ ) + (1 − ξ ) (cid:18) τ sinh 2 τ (cid:19)(cid:18) r sinh 3 τ − sinh τ cosh τ (cid:19)(cid:21) , (4.36) H ( u ) rr ( τ ; q ) := A ( τ, r ) (cid:20) ξ τ tanh τ − (1 − ξ ) (cid:18) − u + (cid:18) τ sinh 2 τ (cid:19)(cid:16) r tanh τ (cid:17)(cid:19)(cid:21) , (4.37) H ( u ) θ θ ( τ ; q ) := (cid:16) rk (cid:17) A ( τ, r ) (cid:20) ξ τ tanh τ − (1 − ξ ) (cid:18) − u + (cid:18) τ sinh 2 τ (cid:19)(cid:16) r cosh τ (cid:17)(cid:19)(cid:21) (4.38)(for the remaining diagonal component, i.e. H ( u ) θ θ ( τ ; q ), see Eq. (3.22)). In theabove, for simplicity of notation we have put A ( τ, r ) := 164 π / e − r tanh τ (cid:18) τ sinh 2 τ (cid:19) / . (4.39)Also this time, the expressions (4.36-4.39) for H ( u ) µν possess the properties indicatedin Eq.s (3.22) (3.23) (and in the related comments); thus, we can obtain the analyticcontinuation in u of the expression in Eq. (4.35) integrating by parts n times, forany n > n = 3 , (4.40)so that Eq. (3.24) reads h | b T uµν ( q ) | i = − k Γ( u +12 ) 1( u − u − u (cid:16) κk (cid:17) u Z + ∞ dτ τ u ∂ τ H ( u ) µν ( τ ; q ) . (4.41)As in all cases with odd spatial dimension, the analytic continuation of the reg-ularized stress-energy VEV given in Eq. (4.41) has a simple pole in u = 0 (recallsubsection 3.5). In consequence of this, we have to adopt the extended version of the30eta approach to define the renormalized VEV h | b T µν ( q ) | i ren , taking the regularpart in u = 0 of Eq. (4.41) (see Eq. (3.27)); with some effort, we obtain h | b T µν ( q ) | i ren = k (cid:16) T (0) µν ( r ) + M κ,k T (1) µν ( r ) (cid:17) , (4.42) T (0) µν ( r ) := Z + ∞ dτ e − r tanh τ h P (0) µν ( τ ; r ) + ln τ P (1) µν ( τ ; r ) i ,T (1) µν ( r ) := Z + ∞ dτ e − r tanh τ P (1) µν ( τ ; r ) , M κ,k := γ EM +2 ln (cid:18) µk (cid:19) , where P (0) µν ( τ ; r ) := − √ π e r tanh τ h ∂ τ H (0) µν ( τ ; q ) + 4 ∂ u (cid:12)(cid:12)(cid:12) u =0 ∂ τ H ( u ) µν ( τ ; q ) i , P (1) µν ( τ ; r ) := − √ π e r tanh τ ∂ τ H (0) µν ( τ ; q ) ; (4.43)this time, we readily infer that P (0) µν , P (1) µν are polynomials of degree N = 4 in r .Now, we evaluate numerically the integrals in Eq. (4.42) and distinguish betweenthe conformal and non-conformal parts ♦ , (cid:4) of each component; once more we referto Eq. (3.31), recalling that for d = 3 we have (see Eq. (2.14)) ξ = 16 . (4.44)The forthcoming Fig.s 8-13 show the graphs of the functions r T ( a, × )00 ( r ) , T ( a, × ) rr ( r ) , ( k/r ) T ( a, × ) θ θ ( r ) for a ∈ { , } , × ∈ { ♦ , (cid:4) } . (4.45)31 T H í L - - - T H í L Figure 8: d = 3: graphs of T (0 , ♦ )00 and T (1 , ♦ )00 . - - - - - T H à L T H à L Figure 9: d = 3: graphs of T (0 , (cid:4) )00 and T (1 , (cid:4) )00 . - - - - - - T rr H í L T rr H í L Figure 10: d = 3: graphs of T (0 , ♦ ) rr and T (1 , ♦ ) rr .32 T rr H à L - - - - - T rr H à L Figure 11: d = 3: graphs of T (0 , (cid:4) ) rr and T (1 , (cid:4) ) rr . - - - - - - H k (cid:144) r L T ΘΘ H í L H k (cid:144) r L T ΘΘ H í L Figure 12: d = 3: graphs of ( k/r ) T (0 , ♦ ) θ θ and ( k/r ) T (1 , ♦ ) θ θ . H k (cid:144) r L T ΘΘ H à L - - - - - H k (cid:144) r L T ΘΘ H à L Figure 13: d = 3: graphs of ( k/r ) T (0 , (cid:4) ) θ θ and ( k/r ) T (1 , (cid:4) ) θ θ .33n conclusion, let us discuss the small and large r asymptotics of the functions in(4.45). On the one hand, Eq.s (3.32-3.35) (with N = 4) yield, for r = k | x | → + , T (0 , ♦ )00 ( r ) = − . − . r + 0 . r + 0 . r − . r + O ( r ) ,T (1 , ♦ )00 ( r ) = − . r + O ( r ) ,T (0 , (cid:4) )00 ( r ) = − . − . r + 0 . r − . r + 0 . r + O ( r ) ,T (1 , (cid:4) )00 ( r ) = 0 . O ( r ) ; (4.46) T (0 , ♦ ) rr ( r ) = − . . r − . r − . r + O ( r ) ,T (1 , ♦ ) rr ( r ) = 0 . r + O ( r ) ,T (0 , (cid:4) ) rr ( r ) = 0 . . r − . r + 0 . r − . r + O ( r ) ,T (1 , (cid:4) ) rr ( r ) = − . O ( r ) ; (4.47)( k/r ) T (0 , ♦ ) θ θ ( r ) = − . . r + 0 . r − . r + 0 . r + O ( r ) , ( k/r ) T (1 , ♦ ) θ θ ( r ) = 0 . r + O ( r ) , ( k/r ) T (0 , (cid:4) ) θ θ ( r ) = 0 . . r − . r + 0 . r − . r + O ( r ) , ( k/r ) T (1 , (cid:4) ) θ θ ( r ) = − . O ( r ) . (4.48)On the other hand, Eq.s (3.39) (3.45) (3.46) (with K = 4, λ = 0) allow us to inferthe following asymptotic expansion, for r = k | x | → + ∞ : T (0 , ♦ )00 ( r ) = r π (cid:16) ln r + γ EM + 12 (cid:17) − π − π r + O ( r − ln r ) ,T (1 , ♦ )00 ( r ) = − r π + O ( r − ln r ) ,T (0 , (cid:4) )00 ( r ) = − π (cid:16) ln r + γ EM + 23 (cid:17) + 112 π r + O ( r − ln r ) ,T (1 , (cid:4) )00 ( r ) = 38 π + O ( r − ln r ) ; (4.49) T (0 , ♦ ) rr ( r ) = − r π (cid:18) ln r + γ EM − (cid:19) + 196 π − π r + O ( r − ln r ) ,T (1 , ♦ ) rr ( r ) = r π + O ( r − ln r ) ,T (0 , (cid:4) ) rr ( r ) = 14 π (cid:16) ln r + γ EM (cid:17) + 16 π r + O ( r − ln r ) ,T (1 , (cid:4) ) rr ( r ) = − π + O ( r − ln r ) ; (4.50)34 k/r ) T (0 , ♦ ) θ θ ( r ) = − r π (cid:18) ln r + γ EM − (cid:19) − π + 312880 π r + O ( r − ln r ) , ( k/r ) T (1 , ♦ ) θ θ ( r ) = r π + O ( r − ln r ) , ( k/r ) T (0 , (cid:4) ) θ θ ( r ) = 14 π (cid:16) ln r + γ EM +1 (cid:17) − π r + O ( r − ln r ) , ( k/r ) T (1 , (cid:4) ) θ θ ( r ) = − π + O ( r − ln r ) . (4.51)Finally, Eq. (3.55) with n = 4 and numerical evaluation of the corresponding integralallow us to derive the bulk energy E ren = − (0 . ± − ) k . (4.52)This result agrees with the one obtained by Actor and Bender [2] using a differentmethod ( ). Acknowledgments.
This work was partly supported by INdAM, INFN and byMIUR, PRIN 2010 Research Project “Geometric and analytic theory of Hamiltoniansystems in finite and infinite dimensions”. To check this, one must compare the numerical value reported in the above Eq. (4.52) withthe one reported in Eq. (4.4) of [2]. Let us stress that conventions different from ours are usedtherein. In fact, using our language, the bulk energy is formally defined in [2] as E := P k ω k ,while our general prescription (2.37) gives E = P k ω k ; moreover the parameter α of [2] and ourparameter k are related by α = √ k . Summing up, the “total energy” derived in [2] has to bemultiplied by 1 / E ren . Appendix. Asymptotic expansions for certainintegrals
Hereafter we show how to obtain the expansions (3.34) (3.41), holding respectivelyfor the functions defined via the integral representations (3.32) (3.39). The proofsare given in subsections A.1 and A.3; subsection A.2 is an interlude on gamma-typefunctions, useful in view of subsection A.3.
A.1 Derivation of Eq.s (3.34-3.37).
Let us consider the framework of subsec-tion 3.6, where F ( r ) := Z + ∞ dτ e − r h ( τ ) P ( τ ; r ) , P ( τ ; r ) = N X i =0 p i ( τ ) r i ( r ∈ (0 , + ∞ )) , (A.1)( h a positive bounded function, p i ( i ∈ { , ..., N } ) some integrable functions).To evaluate F ( r ) we start from Taylor’s formula with Lagrange remainder for theexponential; this ensures that, for any z ∈ (0 , + ∞ ) and any M ∈ { , , , ... } , e − z = M X m =0 ( − z ) m m ! + ρ M +1 ( z ⋆ ) z M +1 ,ρ M +1 ( z ⋆ ) := 1( M +1)! d M +1 dz M +1 e − z (cid:12)(cid:12)(cid:12)(cid:12) z = z ⋆ = ( − M +1 ( M +1)! e − z ⋆ , (A.2)for some z ⋆ = z ∗ ( z ) ∈ (0 , z ); in particular, we have | ρ M +1 ( z ⋆ ) | M +1)! . (A.3)Substituting expansion (A.2) with z = r h ( τ ) and M = N − i into Eq. (A.1), wereadily infer F ( r ) = N X i =0 " N − i X m =0 (cid:18) ( − m m ! Z + ∞ dτ p i ( τ ) ( h ( τ )) m (cid:19) r m + i ) ++ (cid:18)Z + ∞ dτ p i ( τ ) ( h ( τ )) N − i +1 ρ N − i +1 ( z ⋆ ( τ, r )) (cid:19) r N +1) (cid:21) , (A.4)with z ⋆ ( τ, r ) ∈ (0 , r h ( τ )). Introducing a new summation index i such that m = j − i ,36nd performing the exchange i ↔ j ( ), the above relation yields F ( r ) = N X i =0 a i r i + R N +1 ( r ) for r ∈ (0 , + ∞ ) , (A.5)where the coefficients a i ∈ R are defined as in Eq. (3.35), while the remainder term R N +1 ( r ) is R N +1 ( r ) := N X i =0 Z + ∞ dτ p i ( τ ) ( h ( τ )) N − i +1 ρ N − i +1 ( z ⋆ ( τ, r )) ! r N +1) . (A.6)Using the estimate (A.3), we readily infer the uniform bound, holding for all r ∈ (0 , + ∞ ), | R N +1 ( r ) | N X i =0 N − i +1)! Z + ∞ dτ | p i ( τ ) | ( h ( τ )) N − i +1 ! r N +1) ; (A.7)this proves Eq. (3.36), with the expression (3.37) for the constant C N +1 therein. A.2 The incomplete gamma functions and the integral g ( s, x ) ; asymp-totics and bounds. Let us first consider the “lower” and “upper” incompletegamma functions, respectively defined as γ ( s, z ) := Z z dw e − w w s − for all s ∈ C with ℜ s > z ∈ (0 , + ∞ ) , (A.8)Γ( s, z ) := Z + ∞ z dw e − w w s − for all s ∈ C , z ∈ (0 , + ∞ ) . (A.9)Hereafter we report some well-known properties of these functions (see [20], Chapter8), to be used in the following subsection. First of all, there holds γ ( s, z ) + Γ( s, z ) = Γ( s ) ( s / ∈ { , − , − , ... } ) , (A.10)where Γ( ) is the Euler gamma function. Concerning the lower incomplete gamma,there hold the relations γ ( s +1 , z ) = s γ ( s, z ) − e − z z s , γ ( s, z ) = Γ( s ) + O ( z −∞ ) for z → + ∞ ;0 γ ( s, z ) Γ( s ) for s ∈ (0 , + ∞ ) . (A.11) Notice as well that, for any family ( b ij ) i,j =1 ,...,N , it is N X j =0 N X i = j b ij = N X i =0 i X j =0 b ij .
37n the other hand, the upper incomplete gamma fulfillsΓ( s +1 , z ) = s Γ( s, z ) + e − z z s , Γ( s, z ) = O ( z −∞ ) for z → + ∞ ;0 Γ( s, z ) Γ( z ) for s ∈ (0 , + ∞ ) . (A.12)(both here and in Eq. (A.11), the remainder O ( z −∞ ) indicates a quantity which is O ( z − N ) for all N ∈ { , , , ... } ).Let us move on to discuss the properties of the function g ( s, x ) defined via theintegral representation (see Eq. (3.43)) g ( s, z ) := Z z dw e − w w s − ln w for all s ∈ C with ℜ s > z ∈ (0 , + ∞ ) . First of all, since ∂ s ( w s − ) = w s − ln w , from the above definition and Eq. (A.8) ittrivially follows that g ( s, z ) = ∂ s γ ( s, z ) ; (A.13)thus, using the recursive relation in Eq. (A.11), we can easily infer g ( s +1 , z ) = s g ( s, z ) + γ ( s, z ) − e − z z s ln z ( ℜ s > . (A.14)Now, let us show that g (1 , z ) = − e − z ln z − γ EM − Γ(0 , z ) (A.15)( γ EM denotes the Euler-Mascheroni constant). To this purpose, let us write g (1 , z ) = lim δ → + g δ (1 , z ) , g δ (1 , z ) := Z zδ dw e − w ln w ( δ > . (A.16)Consider the regularized function g δ (1 , z ) and integrate by parts to obtain g δ (1 , z ) = − e − x ln x + e − δ ln δ + Z zδ dw e − w w − ; (A.17)now, note that (for 0 < δ < z ) Z zδ dw e − w w − = Z + ∞ δ dw e − w w − − Z + ∞ z dw e − w w − = Γ(0 , δ ) − Γ(0 , z ) . (A.18)Since Γ(0 , δ ) = − ln δ − γ EM + O ( δ ) for δ → + (see [20], p.177, Eq. 8.4.15), we have g δ (1 , z ) = − e − x ln x − γ EM − Γ(0 , z ) − (1 − e − δ ) ln δ + O ( δ ) for δ → + , (A.19)which, along with Eq. (A.16), yields Eq. (A.15).38et us proceed to prove that there holds the bound g ( s, z ) > − s − for all s, z ∈ (0 , + ∞ ) ; (A.20)we are going to give the proof for z < z > z < s > g ( s, z ) > Z dw e − w w s − ln w > Z dw w s − ln w = − s − , (A.21)where the first passage follows from the negativity of the integrand (ln w <
0, for w ∈ (0 , e − w w ∈ (0 , z > s > g ( s, z ) = Z dw e − w w s − ln w + Z z dw e − w w s − ln w > Z dw e − w w s − ln w (A.22)since the integral over (1 , z ) is positive; then, the thesis (A.20) for z > g ( s, z ) = Γ( s ) ψ ( s ) + O ( z −∞ ) for all s ∈ C with ℜ s > z → + ∞ , (A.23)where s ψ ( s ) := ∂ s ln Γ( s ) is the digamma function. To this purpose, write g ( s, z ) = Z + ∞ dw e − w w s − ln w − Z + ∞ z dw e − w w s − ln w . (A.24)Concerning the first integral, we have (see [14], Eq. 4.352.1) Z + ∞ dw e − w w s − ln w = Γ( s ) ψ ( s ) ( s ∈ C , ℜ s > , (A.25)so that Eq. (A.23) follows if we can prove that Z + ∞ z dw e − w w s − ln w = O ( z −∞ ) for s ∈ C with ℜ s > z → + ∞ . (A.26)Indeed, for any z ∈ (1 , + ∞ ), we have0 (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ z dw e − w w s − ln w (cid:12)(cid:12)(cid:12)(cid:12) Z + ∞ z dw e − w w ℜ s = Γ( ℜ s +1 , z ) , (A.27)where the second inequality follows from the fact that | ln w | w (for w ∈ ( z, + ∞ ) ⊂ (1 , + ∞ )), while in the last equality we used the definition (A.9) of the upper incom-plete gamma function (with ℜ s > .3 Derivation of Eq.s (3.45) (3.46). Consider the framework of subsection3.7, where F ( r ) = Z dv e − r v v λ Q ( v ; r ) ( λ > − ,Q ( v ; r ) = Q (0) ( v ; r ) + Q (1) ( v ; r ) ln v , Q ( a ) ( v ; r ) = N X i =0 q ( a ) i ( v ) r i (A.28)for some smooth integrable functions q (0) i , q (1) i : [0 , → R ( i ∈ { , ..., N } ). Let usbegin fixing v ∈ (0 ,
1) and re-expressing F ( r ) as F ( r ) = N X i =0 (cid:16) I ( < ) i ( v , r ) + I ( > ) i ( v , r ) + J ( < ) i ( v , r ) + J ( > ) i ( v , r ) (cid:17) r i , (A.29)where we set, for i ∈ { , ..., N } , I ( < ) i ( v , r ) := Z v dv e − r v v λ q (0) i ( v ) , I ( > ) i ( v , r ) := Z v dv e − r v v λ q (0) i ( v ) ; (A.30) J ( < ) i ( v , r ) := Z v dv e − r v v λ q (1) i ( v ) ln v , J ( > ) i ( v , r ) := Z v dv e − r v v λ q (1) i ( v ) ln v . Concerning the integrals on ( v , (cid:12)(cid:12)(cid:12) I ( > ) i ( v , r ) (cid:12)(cid:12)(cid:12) e − v r Z v dv | v λ q (0) i ( v ) | , (cid:12)(cid:12)(cid:12) J ( > ) i ( v , r ) (cid:12)(cid:12)(cid:12) e − v r Z v dv | v λ ln( v ) q (1) i ( v ) | . (A.31)Let us move on to discuss the integrals I ( < ) i , J ( < ) i . For any fixed i ∈ { , ..., N } andany M ∈ { , , , ... } , consider the Taylor expansions near v = 0 with Lagrangeremainder q ( a ) i ( v ) = M X m =0 q ( a ) i,m v m + ρ ( a ) i,M +1 ( v ⋆ ) v M +1 ; (A.32) ρ ( a ) i,M +1 ( v ⋆ ) := 1( M +1)! d M +1 dz M +1 q ( a ) i ( v ) (cid:12)(cid:12)(cid:12)(cid:12) v = v ⋆ , for some v ⋆ = v ⋆ ( v ) ∈ (0 , v ) . Of course,sup v ∈ (0 ,v ) | ρ ( a ) i,M +1 ( v ⋆ ) | s ( a ) i,M +1 ( v )( M +1)! , with s ( a ) i,M +1 ( v ) := sup v ∈ (0 ,v ) (cid:12)(cid:12)(cid:12)(cid:12) d M +1 dv M +1 q ( a ) i ( v ) (cid:12)(cid:12)(cid:12)(cid:12) . (A.33)40ow, substitute the expansion (A.32) into Eq. (A.30) and make the change of vari-able v = w/r (with w ∈ (0 , v r )) in the ( < ) integrals appearing therein. Recallingthe definitions (A.8) and (3.43) respectively of the lower incomplete gamma γ ( , )and of the function g ( , ), we obtain ( ) I ( < ) i ( v , r ) = M X m =0 q (0) i,m γ ( m + λ +1 , v r ) r − m + λ +1) + R (0) i,M +1 ( v , r ) , (A.34) J ( < ) i ( v , r ) = (A.35)= M X m =0 q (1) i,m h g ( m + λ +1 , v r ) − γ ( m + λ +1 , v r ) ln r i r − m + λ +1) + R (1) i,M +1 ( v , r ) , where the remainder functions are defined as R (0) i,M +1 ( v , r ) := Z v dv e − r v ρ (0) i,M +1 ( v ⋆ ) v M + λ +1 ,R (1) i,M +1 ( v , r ) := Z v dv e − r v ρ (1) i,M +1 ( v ⋆ ) v M + λ +1 ln v . (A.36)Then, using the bound (A.33) and the definitions (3.43) (A.8), we readily infer theestimates | R (0) i,M +1 ( v , r ) | s (0) i,M +1 ( v )( M +1)! γ ( M + λ +2 , v r ) r − M + λ +2) , (A.37) | R (1) i,M +1 ( v , r ) | s (1) i,M +1 ( v )( M +1)! h γ ( M + λ +2 , v r ) ln r − g ( M + λ +2 , v r ) i r − M + λ +2) ;these, along with the bounds in Eq.s (A.11) (A.20), imply in turn | R (0) i,M +1 ( v , r ) | (cid:18) Γ( M + λ +2)( M +1)! s (0) i,M +1 ( v ) (cid:19) r − M + λ +2) , (A.38) | R (1) i,M +1 ( v , r ) | (cid:18) Γ( M + λ +2)( M +1)! s (1) i,M +1 ( v ) (cid:19)(cid:20) ln r + ( M +1)!( M + λ +2) (cid:21) r − M + λ +2) . For example, we have the following chain of equalities: Z v dv e − r v v m + λ ln v = r − m + λ +1) Z v r dw e − w w m + λ ln( w/r ) == r − m + α +1) Z v r dw e − w w m + λ h ln w − ln r i = r − m + λ +1) h g ( m + λ +1 , v r ) − γ ( m + λ +1 , v r ) ln r i . M = K + i −
1; the remainder estimate (3.44) descend easilyfrom Eq.s (A.31) (A.38), which also give explicit expressions for the constants in(3.44). Finally, using the asymptotic expansions in Eq.s (A.11) (A.23) (respectivelyholding for the lower incomplete gamma and for the function g ( , )), Eq.s (A.34)(A.38) imply Eq.s (3.45) (3.46). B Appendix. An alternative representation forthe bulk energy
In subsection 3.8 we obtained an integral representation for the renormalized bulk en-ergy E ren (see Eq. (3.55)) to be evaluated numerically ( ). In the present appendixwe derive an alternative representation for the regularized energy E u , allowing toexpress it in terms of the Riemann zeta function (see [20], p. 602, Eq. 25.2.1) ζ ( s ) := + ∞ X n =1 n s for s ∈ C with ℜ s > . (B.1)Next, the renormalized energy E ren is computed via the zeta approach, using thewell-known analytic continuation of ζ to the whole complex plane.Let us stress that the methods discussed here only works if the background potentialis isotropic. Nonetheless, they can be of some interest since, for example, they allowus to perform a direct comparison with the results of Actor and Bender [2] in thecase with d = 3 (see subsection B.3).In the forthcoming subsection B.1 we introduce a family of integrals and discuss somerelations allowing to express them in terms of the Riemann zeta; in the followingsubsection B.2 the regularized bulk energy is represented in terms of these integralsand its analytic continuation is discussed. In the concluding subsection B.3 we usethe results of subsections B.1 B.2 to derive, as examples, the renormalized energy E ren in the cases with d ∈ { , , } . B.1 The integrals I n ( s ) and a recursive relation. Let us consider the familyof integrals I n ( s ) := Z + ∞ dτ τ s − sinh n τ for n ∈ { , , , ... } , s ∈ C with ℜ s > n (B.2)(of course, the restriction on ℜ s arises in order to guarantee the convergence of theintegral I n ). We pointed out in the cited subsection that the main advantage of this approach is that itwork also for configurations where the background harmonic potential is not isotropic. n = 1 and n = 2, the above integrals are known to be related tothe Riemann zeta function ζ ; more precisely, we have (see [20], p.604, Eq.s 25.5.8and 25.5.9) I ( s ) = 2(1 − − s ) Γ( s ) ζ ( s ) , (B.3) I ( s ) = 2 − ( s − Γ( s ) ζ ( s − . (B.4)Next, let us fix n ∈ { , , , ... } and consider the elementary identity d dτ (cid:18) n τ (cid:19) = n ( n +1)sinh n +2 τ + n sinh n τ ; (B.5)using this result and the definition (B.2), we infer (for s ∈ C with ℜ s > n + 2) I n +2 ( s ) = − nn +1 I n ( s ) + 1 n ( n +1) Z + ∞ dτ τ s − d dτ (cid:18) n τ (cid:19) . (B.6)Concerning the integral appearing in the right-hand side of the above equation,integrating by parts two times we obtain Z + ∞ dτ τ s − d dτ (cid:18) n τ (cid:19) =( s − s − Z + ∞ dτ τ s − sinh n τ = ( s − s − I n ( s − . (B.7)Summing up, Eq.s (B.6) (B.7) give the recursive relation I n +2 ( s ) = − nn +1 I n ( s ) + ( s − s − n ( n +1) I n ( s −
2) ( s ∈ C , ℜ s > n + 2) . (B.8)This result and the explicit expressions (B.3) (B.4) allow to express any integral I n ( s ) in terms of (suitable linear combinations of) the Riemann zeta ζ ; needless tosay, the analytic continuation of I n is then determined by the well-known analyticcontinuation of ζ . B.2 The bulk energy in terms of the integrals I n ( s ). Using Eq. (3.53) forthe regularized energy E u with n = 0 and the explicit representation (3.52) for theheat trace K ( t ), we obtain E u = κ u d +1 Γ( u − ) Z + ∞ d t t u − sinh d ( k t ) ; (B.9)making the change of variable τ := k t ∈ (0 , + ∞ ) and recalling the definition (B.2),we infer E u = k d +1 Γ( u − ) (cid:16) κk (cid:17) u I d (cid:18) u − (cid:19) . (B.10)43he recursive relation (B.8) for I d and the identities (B.3) (B.4) allow us to deter-mine the analytic continuation of E u to the whole complex plane via Eq. (B.10).The renormalized bulk energy E ren is defined according to Eq. (2.39), setting E ren := RP (cid:12)(cid:12)(cid:12) u =0 E u (cid:16) or E ren := E u (cid:12)(cid:12)(cid:12) u =0 (cid:17) . The final expression of E ren for arbitrary spatial dimension d is too complicate tobe reported here; in the next subsection we consider, as examples, the cases with d ∈ { , , } . B.3 The bulk energy in terms of the Riemann zeta function for d ∈ { , , } . As mentioned at the end of the previous subsection, here we exemplify the generalmethods developed in the previous subsections B.1 B.2 for the computation of E ren in the cases with d ∈ { , , } ; we also show that the results found are in agreementwith those derived in Section 4 using another approach. The case d = 1. Using Eq.s (B.3) (B.10) and setting u = 0, we infer E ren = − k √ − ! ζ (cid:18) − (cid:19) . (B.11)Evaluating numerically this expression, we have (in agreement with Eq. (4.15)) E ren = k (0 . ... ) . (B.12) The case d = 2. Eq.s (B.4) and (B.10) with u = 0 imply E ren = k √ ζ (cid:18) − (cid:19) . (B.13)Numerical evaluation of the above result gives E ren = − k (0 . ... ) , (B.14)which agrees with the one of Eq. (4.33) (the last digit being different due to trun-cation approximation). The case d = 3. This time we have to resort to the recursive relation (B.8) (hereemployed with n = 1) as well as to the identity (B.3). Then, using once more Eq.(B.10) with u = 0, we infer E ren = k " √ − ! ζ (cid:18) − (cid:19) − √ − ! ζ (cid:18) − (cid:19) . (B.15)We state that the above result coincides with the one of Eq. (4.4) in [2], which inour language reads (see footnote 12) E ren = k (cid:20) √ ζ (cid:18) − , (cid:19) − √ ζ (cid:18) − , (cid:19)(cid:21) (B.16)44here ζ ( , ) indicates the analytic continuation of the Hurwitz zeta function; thisstatement follows is easily proved using the known identities (see [20], p. 607, Eq.25.11.3 and p. 608, Eq. 25.11.11) ζ ( s, a + 1) = ζ ( s, a ) − a − s , (B.17) ζ (cid:18) s, (cid:19) = (2 s − ζ ( s ) . (B.18)Finally, let us note that evaluating numerically the expression in the right-hand sideof Eq. (B.15) we have E ren = − k (0 . ... ) , (B.19)in agreement with Eq. (4.52). 45 eferences [1] A.A. Actor, Multiple harmonic oscillator zeta functions , J.Phys.A: Math.Gen. , 927–936, (1987).[2] A.A. Actor, I. Bender, Casimir effect for soft boundaries , Phys.Rev.D (6),3581–3590 (1995).[3] M. Beauregard, G. Fucci, K. Kirsten, P. Morales, Casimir Effect in the Pres-ence of External Fields , J.Phys.A:Math.Theor. , 115401 (2013).[4] N. Berline, E. Getzler, M. Vergne, “Heat Kernels and Dirac Operators”,Springer Science & Business Media (1992).[5] M. Bordag, D. Hennig, D. Robaschik, Vacuum energy in quantum field theorywith external potentials concentrated on planes , J.Phys.A: Math.Gen. (16),4483 (1992).[6] O. Calin, D.-C. Chang, K. Furutani, C. Iwasaki, “Heat Kernels for Ellipticand Sub-elliptic Operators: Methods and Techniques”, Springer Science &Business Media (2010).[7] E.T. Copson, “Asymptotic Expansions”, Cambridge University Press (2004).[8] E.B. Davies, “Heat Kernels and Spectral Theory”, Cambridge University Press(1990).[9] D. Fermi, L. Pizzocchero, Local zeta regularization and the Casimir effect ,Prog.Theor.Phys. (3), 419–434 (2011); see also arXiv:1104.4330.[10] D. Fermi, L. Pizzocchero,
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