9 May 2013 of the ﬁelds and Their Derivatives [2]

Quantum vacuum energies and Casimir forces between partially transparent δ-function plates

J. M. MuñozCastañeda,a, J. Mateos Guilarteb, and A. Moreno Mosquerab,c aInstitut fürTheoretische Physik, UniversitätLeipzig, Germany. bDepartamento de F´ısica Fundamental and IUFFyM, Universidad de Salamanca, Spain. cFacultad Tecnologica. Universidad Distrital Francisco J. de Caldas, Colombia. (Dated: May 10, 2013) In this paper the quantum vacuum energies induced by massive fluctuations of one real scalar field on a configuration of two partially transparent plates are analysed. The physical properties of the infinitely thin plates are characterized by two Dirac-δ potentials. We find that an attractive/repulsive Casimir force arises between the plates when the weights of the δ’s have equal/different sign. If some of the plates absorbs fluctuations below some threshold of energy (the corresponding weight is negative) there is the need to set a minimum mass to the scalar field fluctuations to preserve unitarity in the corresponding quantum field theory. Two repulsive δ-interactions are compatible with massless fluctuations. The effect of Dirichlet boundary conditions at the endpoints of the interval (−a, a) on a massless scalar quantum field theory defined on this interval is tantamount letting the weights of the repulsive δ-interactions to +∞.

PACS numbers: 11.27.+d, 11.25.-w

I. INTRODUCTION tion potential concentrated on an infinitely thin closed surface. The spectacular success of quantum field theory ac- More recently Fosco et al [6] derived the Casimir en- curately describing scattering processes between leptons ergy induced by the scalar field fluctuations between two and photons in QED brought with it a striking para- finite-width mirrors by using a field derivative local ex- dox. The theoretical framework required that particles pansion. The effective field theory replaces the mirrors lived in open spaces of infinite volume and hence bound- by Dirac δ potentials, a set up that the authors rein- ary conditions played no role at all [1].This fact was terpret as imposing imperfect Dirichlet conditions. In in sharp contrast with what happened in classical field reference [7] Milton and collaborators extend this idea theory where boundary conditions at the boundaries of to a fully electromagnetic context describing the elec- closed (finite volume) manifolds were a central part of the tric permittivity and magnetic permeability in terms of theory. The situation started to change in the nineteen δ-functions with appropriate coefficients related to the seventies when infrared phenomena, e.g., the quark con- plasma frequency in Barton’s model for the Casimir effect finement mystery, entered into the scene of quantum field on spherical shells (see ref. [8]). The authors showed that theory (QFT). It was recognised very quickly, see refer- the Casimir energies derived in Barton’s plasma model [8] ence [2], that boundary conditions are implemented in are in perfect agreement with the results that they found QFT by surface interactions. More precisely, assuming in [7]. Moreover, in reference [7], it was pointed out that that the quantum fields live on a 3 dimensional mani- the thin and thick boundary conditions considered by fold with a surface boundary, the surface interactions are Bordag in references [9] and [10] respectively lead to the determined by adding a term proportional to a Dirac δ same Casimir-Polder forces. In summary, it is suggested function to the Lagrangian on the surface times functions that the δ-function interactions, admittedly an idealisa- arXiv:1305.2054v1 [hep-th] 9 May 2013 of the fields and their derivatives [2]. tion, capture the essential features of the quantum vac- A profound phenomenon fitting within this theoretical uum energy Casimir phenomenon. This statement is also framework is the Casimir effect [3]. In reference [4] Bor- confirmed by Fosco et al in reference [11] where the au- dag et al described analytically the two parallel plates of thors extend the procedures proposed in reference [6] to the ideal Casimir set-up by means of two Dirac δ poten- analyse the vacuum fluctuations of the electromagnetic tials concentrated on a point at the centre of the plates. field disturbed by the presence of two Dirac δ-plates. The The Casimir energy, due to massive and massless spinor boundary conditions chosen in Fosco et al work respond and scalar fields, was subsequently calculated setting two to a modelization of the plate in the context of relativis- δ-functions on the plate surfaces as electrostatic poten- tic macroscopic electrodynamics. In this sense they dif- tial. These ideas were subsequently applied in the anal- fer from those used by Milton et al in [7], [12] where ysis of a similar effect due to field fluctuations in the the boundary conditions are adapted to non-relativistic presence of a penetrable spherical shell, see [5]. The ul- magneto-electric δ-plates. traviolet divergences appear as infinite factors multiply- The aim of this work is to delve into the deepest an- ing the first heat trace coefficients coming from the heat alytic aspects arising from this approach to the Casimir kernel expansion of the free Laplacian plus a delta func- effect by focusing our investigation on the simplest phys- 2 ical context. We shall thus study the quantum vacuum repulsive Casimir forces, occur when the signs of the δ- energy induced by the fluctuations over the real line of couplings are different. If one or both of the δ-potentials, one real scalar massive field under the influence of two mimicking the plates, are attractive, unitarity of the Dirac δ-potentials located at two different points. Fol- quantum field theory sets a lower bound on the mass lowing section 2.1 of reference [13] we treat the quantum of the quantum fluctuations such that the total energy mechanical spectral problem posed by a double delta po- of the lowest energy state is zero. In the computation of tential from a (1 + 1)-dimensional field theoretical per- the double-δ Casimir energy one must subtract the indi- spective. Papers [14] and [15] by Jaffe and collaborators vidual single-δ’s vacuum energies. If they are attractive, provide good evidence of interest in this task. the individual δ-plates can absorb fluctuations of mass Specifically, we compute the Casimir energy induced below the modulus of the bound state energy for a single by the quantum fluctuations of one massive real scalar δ-potential. Therefore, allowing these light fluctuations field between two partially transparent plates using three the subtraction process renders the T GT G Casimir en- different methods. ergy of the double-δ system complex. A lower bound, • 1. We first analyse this problem by computing the equal to the sum of the two individual bound state ener- “reduced” Green function and, subsequently, the 00- gies on the mass of the fluctuations for the double delta component of the energy-momentum tensor. Casimir en- system, is necessary to avoid the appearance of an imag- ergies and forces then follow through spatial integration inary part in the double-δ Casimir energy coming from and “sum” over the frequencies. The method is identical the subtraction of single-δ’s vacuum energies. This lower to the procedure used in refs. [4] and [13] and our results bound, however, is generically insufficient to ensure the agree with the outcomes of these two papers. Never- unitarity of the quantum field theory. When both δ- theless, we tackle a slightly more general situation. As couplings are negative there is a small region in the plane in reference [13] we allow different couplings for the two of couplings where the lowest energy level of the double-δ Dirac δ-potentials, but we also consider the possibility of system is lower than the mass lower bound. In this re- having negative couplings. In this case, the δ-potentials gion the theory becomes non-unitary and a higher lower (wells) are attractive having bound states that can trap bound on the mass must be imposed to ensure the uni- the quantum field fluctuations, i.e., we also envisage par- tarity of the quantum field theory. tially absorbing plates. To avoid the problems of a lack If the two δ-interactions are repulsive there are no of unitarity of the quantum field theory arising in this bound states and even the massless fluctuations give rise situation we balance the mass of the fluctuations with to a real Casimir energy and a unitary quantum field the- the bound state eigenvalues in order to make fluctuation ory. In the infinitely repulsive limit of the two δ-potential absorption impossible. plates the Casimir energy induced by massless fluctua- • 2. We apply, in a second development, the T GT G pre- tions is exactly the Casimir energy produced by two per- scription to calculate the Casimir energy between two fectly conducting plates (Dirichlet boundary conditions compact objects from the knowledge of the “transfer” on the quantum fields over the plates), see references [18– T -matrix of a single object defined by the Lipmann- 20]. Thus we interpret the double-δ Casimir energy as Schwinger equation (see refs. [16, 17]). The simplest the Casimir energy due to a family of generalised Dirich- system of two disjoint objects merely includes two points, let boundary conditions set on the quantum fluctuations so it is suitable to use the T GT G formalism in this case of one scalar field produced by semi-transparent plates, to find a clarification of the structure of the Casimir en- rather than impenetrable. ergies and forces in this way. In particular, the Casimir • 3. Our third way to deal with δ-potential Casimir ener- energy of fluctuations around one single δ-plate is ultra- gies is inspired by the calculation of one-loop kink mass violet divergent even after the subtraction of the contri- shift (see refs. [21, 22]). In the kink “string” limit of bution of quantum fluctuations around a constant back- the sine-Gordon model, see ref. [23], the one-loop fluc- ground. Suitable regularisation of one-δ Casimir energy tuations of the scalar field in the kink background are allows to show that the results obtained from the stress governed by the one-δ well Hamiltonian with the thresh- tensor and the TG-formula only differ in a finite con- old of the continuous spectrum displaced in such a way stant. The double-δ Casimir energy, however, is finite that the energy of the bound state is zero. Thus the after the subtraction of the two single-δ and empty space Casimir energy of an attractive δ-plate can be understood vacuum energies. A partial integration shows that the as the sine-Gordon kink Casimir energy in the “string” stress tensor result and the T GT G-formula provide iden- limit excluding the contribution of the mass renormalisa- tical results for the quantum vacuum interaction between tion counterterm. The Casimir energy is accordingly ex- two δ-plates. pressed as a sum over the discrete spectrum of the square The T GT G formalism offers a very good understand- root of the eigenvalues plus the integration over the con- ing of the double-δ Casimir energy as a function from tinuous spectrum of the square root of the eigenvalues the (α, β)-plane of couplings (weights) into the complex weighted with the spectral density. Because the phase plane. The real part of the Casimir energy is negative, shifts are analytically known the calculation is completely thus producing attractive Casimir force, when both cou- feasible and offers a more disclosed physical information plings have equal sign. Positive Casimir energies, hence to that encoded in either the energy-momentum tensor 3 or the TG calculation. Knowledge of the spectral data of B. The stress tensor and Casimir energies the double δ-Hamiltonian is also completely accessible allowing to explicitly write the DHN formula (see ref. [21]) The energy-momentum tensor arising from the action for the two-δ Casimir energy. Bearing in mind that the functional (1) is given by double δ-wells arise in the string limit of two sine-Gordon kinks it is interesting to compare this result with the Tµν = ∂µΦ∂ν Φ − gµν L (8) quantum vacuum interaction between two sine-Gordon 1 1 L = ∂ Φ∂µΦ − U(x)Φ2(x, t) kinks computed by Bordag and Muñoz-Castañedain ref- 2 µ 2 erence [24]. The (0, 0) component of the energy momentum tensor gives the energy density for any field configuration ! II. QUANTUM FLUCTUATIONS OF 1 + 1 1 ∂Φ2 ∂Φ2 DIMENSIONAL SCALAR FIELDS T (x, t) = + + U(x)Φ2(x, t) 00 2 ∂t ∂x A. The field equation and the Green function 1 ∂2Φ ∂2Φ = −Φ(x, t) − Φ(x, t) + U(x)Φ2(x, t) .(9) 2 ∂t2 ∂x2 The fluctuations of 1D scalar fields on static classical For those field configurations that are solutions of the backgrounds modelled by the function U(x) are governed equations of motion arising from (1) the partial integra- by the action: tion shown in (9) tells us that the energy density is given Z by 2 1 µ 1 2 S[Φ] = d x ∂µΦ∂ Φ − U(x)Φ (x, t) (1) 2 2 ∂Φ2 T (x, t) = . (10) 00 ∂t In order to have a well defined scattering problem we must impose the finite area condition over the U(x) clas- The Green function is also the vacuum expectation value sical background [25]: of a time ordered product of quantum fields Z ∞ G(x, t; x0, t0) = ih0|T (Φ(ˆ x, t)Φ(ˆ x0, t0))|0i. (11) lim U(x) = m2 , dx (U(x) − m2) < +∞. (2) x±∞ −∞ Therefore the vacuum expectation value of the energy density is given in terms of the Green function as [17] The classical field equation and the Green’s function equation arising from (1) are 1 0 0 h0|Tˆ (x)|0i = ∂ ∂ 0 G(x, t; x , t )| . (12) 00 i t t x=x0,t=t0 ∂2 − ∂2 + U(x) Φ(x, t) = 0 (3) t x This is the basic formula relating the Green function to 2 2 0 0 0 0 ∂t − ∂x + U(x) G(x, t; x , t ) = δ(x − x )δ(t − t ). the (0, 0) component energy-momentum tensor. The vac- ˆ Performing a Fourier decomposition in the time coordi- uum energy is given as the integral of h0|T00(x)|0i over nate of the fluctuating field the real line.

Z ∞ dω Φ(t, x) = eiωtφ (x), (4) 2π ω C. The T GT G method for (1 + 1)-dimensional −∞ theories and using the field equation we obtain the static fluctuation Scrödingeroperator: Following references [13, 16, 17, 24] we summarise the main formulas and results that lead to the T GT G for- 00 2 − φω(x) + U(x)φω(x) = ω φω(x). (5) mula for the Casimir energy and the quantum vacuum interaction between two compact/topological disjoint ob- The same Fourier decomposition leads to the reduced jects in (1 + 1)-dimensional scalar quantum field theo- 0 Green’s function Gω(x, x ) and its corresponding differ- ries. The Lipmann-Schwinger equation arising in quan- ential equation: tum mechanical scattering theory defines the transfer ∞ matrix [26] (see references [25, 27]) as Z dω 0 G(x, t; x0, t0) = eiω(t −t) G (x, x0) (6) 2π ω G(0) · U · G(0) −∞ G = G(0) − ≡ G(0) · (I − T · G(0)) (13) 2 2 2 0 0 I + U · G(0) −ω − d /dx + U(x) Gω(x, x ) = δ(x − x ). (7) where G(0) is the Green’s function for the free particle The reduced Green function plays a central role in the operator paper. We use the reduced Green function in the calculation of the Casimir energy using the T GT G formalism d2 K0 = − + m2 . (14) developed in references [13, 16, 17, 24]. dx2 4

The last equality in equation (13) is written in terms of Formula (16) does not account explicitly for dissipa- the corresponding integral kernels as tive eﬀects. However there are some dissipative phe- (U) (0) nomena that are implicit in formula (16). Whenever the Gω (x, y) = Gω (x, y) − 2 Z L norm of the operator Mω becomes greater than one (0) (U) (0) (k Mω kL2 > 1) we have that − dz1dz2Gω (x, z1)Tω (z1, z2)Gω (z2, y).

The general expression for the kernel of the T operator TrL2 ln (1 − Mω) ∈ C. (21) can then be obtained (see reference [24] for a detailed Therefore the vacuum Casimir interaction acquires an demonstration): imaginary part. Such kind of dissipative phenomena (0) arise, for instance, when negative energy bound states Tω(x, y) = U(x)δ(x − y) + U(x)G (x, y)U(y). (15) ω appear in the one-particle states on which the quantum Compact disjoint objects in one dimension are mod- field theory is built. As long as unitarity is preserved all elled by potentials of the form the conservation laws that are consequence of the quantum version of the Nöthertheorem remain valid. U(x) = U1(x) + U2(x), where the smooth functions Ui(x), i = 1, 2, have disjoint compact supports on the real line. Under this assumption D. Casimir energy from the spectral heat trace the T GT G formula for the vacuum interaction energy is and zeta function [16] The one-particle states of the quantum field theory are Z ∞ int i dω given by the eigenfunctions of the Schrödingeroperator E0 = − TrL2 ln (1 − Mω) (16) 2 0 π K = K0 + U(x),. (22) where the operator Mω and its kernel are defined as (0) (1) (0) (2) In general this operator has both continuous and discrete Mω = Gω Tω Gω Tω (17) Z spectrum h (0) (1) Mω(x, y) = dz1dz2dz3 Gω (x, z1)Tω (z1, z2)× 2 Kψj(x) = ωj ψj(x), j = 1, 2, ..., l, l ∈ N (23) (0) (2) i 2 2 2 2 × Gω (z2, z3)Tω (z3, y) (18) Kψk(x) = ω(k) ψk(x; k), ω(k) = k + m , k ∈ R (24)

In these last expressions T(i), i = 1, 2, is the T operator For each k ∈ R the differential equation (24) has two ω linear independent solutions: the left-to-right incom- associated to the object represented by Ui(x), i = 1, 2. ing waves (ψ(R)(x)) and right-to-left incoming waves The potentials Ui(x), i = 1, 2, representing the two ob- k (L) jects define separately two Schrödingerproblems given (ψk (x)). The asymptotic behaviour of these solutions is by the operators determined by the scattering amplitudes (see references 2 [25, 27]): (i) d Kω = − 2 + Ui(x), i = 1, 2. (19) ikx −ikx dx (R) e + rR (k)e , x → −∞ ψk (x) ' ikx (25) (i) t(k)e , x → ∞ In general the operators Kω , i = 1, 2, are defined over a −ikx Hilbert space that is not isomorphic to the Hilbert space (L) t(k)e , x → −∞ ψk (x) ' −ikx ikx . (26) of the free quantum particle spanned by the eigenstates e + rL (k)e , x → ∞ of the operator K0 (see [28]). When this happens the (0) The Wronskian of the two independent scattering solu- operator Gω is defined over a different Hilbert space tions is given in terms of the transmission amplitude t(k) (i) than the operators Tω , i = 1, 2. Therefore the product (0) (i) (R) (L) Gω · Tω , i = 1, 2, is not defined and the formula (16) W [ψk (x), ψk (x)] = −2ik t(k) ≡ WRL(k). (27) is not valid. To avoid this problem we must perform a Wick rotation ω → iξ: in the corresponding euclidean The reduced Green function defined above in (6) can be rotated quantum theories all the operators act over the obtained from the two independent scattering solutions same Hilbert space. When performing this transforma- using the following expression (see [17]): tion the T GT G formula reads 0 1 0 (R) (L) 0 Z ∞ Gω(x, x ) = θ(x − x )ψk (x)ψK (x ) int 1 dξ WRL(k) E0 = TrL2 ln (1 − Miξ) . (20) 2 0 π 0 (R) 0 (L) +θ(x − x)ψk (x )ψK (x) , (28) Now all the operators appearing in the T GT G formula are the euclidean rotated versions of the original defini- where θ(x) is the Heaviside step function. tion. Hence all the operators appearing act over the same Confining the whole system in a very long interval of Hilbert space (for more details see refs. [17, 24]). length L and imposing periodic boundary conditions over 5 the eigenfunctions of the operator K, the spectral density the soliton quantisation framework that the correct reg- of the continuous spectrum is defined as ularisation, when the operator K has a finite number of bound states, is to set a cutoff in the number of fluctu- L dδ(k) 0 %(k) = + , (29) ation modes of both K and K . In the limit of infinite 2π dk length L this prescription requires an integration by parts in the integral over the momenta, from which is obtained where δ(k) = δ+(k) + δ−(k) is the total phase shift; the sum of the arguments of the eigenvalues of the unitary the quantum energy of the extended object in the form scattering matrix [27]: of DHN, [21]:

l ∞ 2iδ±(k) p 1 m 1 Z dk dω e = t(k) ± rR (k)rL (k). (30) DHN X EC = ωj − − (k) δ(k). (34) 2 4 2 −∞ 2π dk The discrete spectrum of the operator K arises at the j=1 poles in the complex k-plane of the transmission ampli- For a detailed demonstration and discussion about this tude t(k) located in the positive imaginary momentum formula see also reference [31]. line kj = iκj (κj > 0): ( eκj x , x → −∞ III. DIRAC DELTA BACKGROUNDS. CASIMIR ψj(x) ' t(iκj ) −κ x . (31) )e j , x → ∞ ENERGY FROM THE ENERGY-MOMENTUM rR (iκj TENSOR. The eigenvalues for the discrete spectrum are given by 2 2 ωj = −κj + m . Note that at kj = iκj the Wronskian The Dirac delta potentials have been thoroughly stud- has poles and the doubly degeneracy of the scattering ied in standard textbooks on Quantum Mechanics (see eigenfunctions disappears. for example [25]). In this section we introduce the re- The heat trace and the spectral zeta function collect sults and establish the notation for the single delta and this spectral information through the definitions: the double delta potential. l Z ∞ X −τω2 dk dδ −τω2(k) hK(τ) = e j + L + (k) e 2π dk A. One Dirac delta configuration background j=1 −∞ l X Z ∞ dk dδ ζ (s) = ω−2s + L + (k) ω−2s(k) , The potential governing massive fluctuations in the K j 2π dk j=1 −∞ delta background is: where τ ∈ + and s ∈ are respectively a positive real α2 R C U (1δ)(x) = m2 + αδ(x) , m2 = µ2 + . (35) parameter and a complex one [29]. These two spectral 4 functions are related by means of the Mellin transform: 2 The term α in the choice of the fluctuation mass is added Z ∞ 4 1 s−1 to guarantee that even the possible bound state has non- ζK(s) = dτ τ hK(τ) . (32) Γ(s) 0 negative energy. Even if we choose µ = 0 the vacuum energy will always be real and the corresponding quan- Accordingly the energy induced by the quantum fluctua- tum field theory is unitary. The one-particle states of tions around the object described by the U(x) potentials the quantum field theory are the eigenfunctions of the measured with respect to the vacuum fluctuation energy Schrödingeroperator is: 2 1 1 1 1δ 0 0 d 2 E = ζ (− ) − ζ 0 (− ) K = K + αδ(x) , K = − + m . (36) C 2 K 2 K 2 dx2 l 1 X m 1 Z ∞ dk dδ The self-adjoint extension of the operator K1δ is defined = ω − + (k) ω(k). (33) 2 j 4 2 2π dk by the Dirac delta matching conditions: j=1 −∞ − + 0 + 0 − A subtle point is the existence of a half-bound state of ψ(0 ) = ψ(0 ), ψ (0 ) − ψ (0 ) = αψ(0). (37) 0 eigenvalue ω1/2 = m in the spectrum of K , the constant function. According to the 1D Levinson theorem it must The continuity/discontinuity conditions (37) determine be accounted for with a weight of one-half, see [30]; this subsequently the eigenstates in both the continuous and is the reason for subtracting the factor m/4 in formula the discrete spectrum: (33). Another even more subtle point is the regularisa- • The scattering states (continuous spectrum) for the tion implicitly used in deriving formula (33). A cutoff in single delta Schrödingerproblem the energy allows us to manage a finite integration do- R Λ 1δ 2 2 main, −Λ dk, at intermediate stages. It is known from K ψk(x) = (k + m )ψk(x) , k ∈ R. (38) 6

are of the general form (25)-(26) but this generic vacuum energy. By doing this, even the possible bound asymptotic behaviour extends to the whole real line state energies of the isolated δ-wells will always be pos- except at the origin [25]: itive. This selection of the mass ensures that the corresponding quantum field theory is unitary and therefore eikx + r(k)e−ikx x < 0 the vacuum energy is real, except in the case that the ψ(R)(x) = k t(k)eikx x > 0 ground state energy of the two-δ configuration is lower than the addition of the bound state energies of each δ- t(k)e−ikx x < 0 ψ(L)(x) = . well separated infinitely apart. The analytical problem k e−ikx + r(k)eikx x > 0 posed by the potential (40) is suitable for the physical description of two parallel infinitely thin partially trans- Moreover, since the Dirac delta is symmetric under parent plates. In this physical picture the parameters parity (δ(−x) = δ(x)) the reflection amplitudes for α and β play the role of the plasma model frequencies the R and the L states are equal: r (k) = r (k) ≡ R L mimicking the plates (see refs. [7, 13, 32]). r(k). Hence for a fixed k ∈ R the wave func- The one-particle states of the quantum field theory are tions ψ(R)(x) and ψ(L)(x) are fully characterised by k k the eigenfunctions of the Schrödingeroperator the transmission t(k) and the reflection r(k) amplitudes. Solving the linear system of equations required by the Dirac delta matching conditions (37) K2δ = K0 + αδ(x − a) + βδ(x + a). (42) on the unknowns t and r one finds: 2δ 2ik α The self-adjoint extension of the operator K to the t(k) = , r(k) = . (39) points x = ±a is fixed by the following Dirac delta match- 2ik − α 2ik − α ing conditions at these points: • When the coupling is negative α < 0 and the δ- potential is attractive, the scattering amplitude t(k) ψ(±a−) = ψ(±a+), (43) has a pole in the positive imaginary axis. In this ψ0(−a+) − ψ0(−a−) = αψ(−a), (44) case the single delta well has a bound state: ψ0(a+) − ψ0(a−) = βψ(a). (45)

α 2 2 2 2 kb = −iκb = −i ; ωb = −κb + m = µ 2 This operator has a continuous spectrum and for certain r α α values of the parameters α and β also exhibits discrete ψ (x) = − exp[ |x|] . b 2 2 spectrum. Both continuous and discrete spectrum eigenstates are determined from the Dirac delta matching con- Using formula (28) the reduced Green function for the ditions along similar lines to those described for the single single delta potential is written as delta potential:

( ik(|x|+|x0|) (0) α e 0 • The double delta potential (40) divides the real line (1δ) 0 Gω − 2ik−α 2ik , sgn(xx ) = +1 G (x, x ) = 0 , ω ik|x−x | into three different zones: x < −a, −a < x < a − e sgn(xx0) = −1 2ik−α and x > a. Therefore, the asymptotic behaviour of whereas the R and L scattering states only extends to the x < −a and x > a zones: 0 eik|x−x | G(0)(x, x0) = − ω 2ik  eikx + r (k)e−ikx x < −a  R is the reduced Green function for the operator K0. (R) ikx −ikx ψk (x) = AR (k)e + BR (k)e −a < x < a  t(k)eikx a < x  t(k)e−ikx x < −a B. Two Dirac deltas configuration background (L)  ikx −ikx ψk (x) = AL (k)e + BL (k)e −a < x < a  e−ikx + r (k)eikx a < x The potential governing massive fluctuations in two- L delta configuration backgrounds is: U (2δ)(x) = m2 + αδ(x − a) + βδ(x + a), (40) Note that in the intermediate zone the solutions are merely superposition of plane waves with op- α2 β2 a > 0 , m2 = µ2 + + . (41) posite wave vector. The Dirac delta matching con- 4 4 ditions on these scattering waves give rise to a lin- We choose the mass of the fluctuations in such a way ear system of four algebraic equations in the four that the subtraction of the energy of the individual δ’s unknowns t, r, A, and B. The solution is eas- will not induce spurious imaginary contributions to the ily obtained through Cramer’s rule implemented in 7

Mathematica: α(2k + iβ)e−2iak + β(2k − iα)e2iak r (k) = −i (46) R ∆(k) 2k(2k + iβ) 2kβe2iak A (k) = ,B (k) = −i (47) R ∆(k) R ∆(k) α(2k − iβ)e2iak + β(2k + iα)e−2iak r (k) = −i (48) L ∆(k) 2k(2k + iα) 2kαe2iak B (k) = ,A (k) = −i (49) L ∆(k) L ∆(k) 4k2 t(k) = (50) ∆(k)

The common denominator ∆(k) for all the amplitudes is ∆(k) = 4k2 + 2ik(α + β) + e4iak − 1 αβ. (51) FIG. 1. Bound states distribution in the (αa) − (βa) plane. The 1 • The existence of bound states in this system is diﬀerent zones are limited by the branches of the hyperbola − 2α − 1 = a [33]. determined by the poles of t(k) over the positive 2β imaginary axis in the complex k-plane. Note that the poles of t(k) are the zeroes of the denominator ∆(k). Thus, k = −iκ with κ > 0, and these ze-

roes are the positive solutions of the transcendent 0 e−ik(x+x ) equation δG(2δ)(x, x0) = α(2k + iβ)e−2iak ω,2 2k∆(k) 2 2 e−4aκ = (1 + κ)((1 + κ). (52) 2iak α β + β(2k − iα)e (54) 2iak (2δ) 0 e h ik(x+x0) The solutions are the intersections of the quadric δGω,1 (x, x ) = α(2k + iβ)e 4 2 2(α+β) 2k∆(k) f(κ) = αβ κ + αβ κ + 1 and the exponential 0 i g(κ) = exp[−4aκ]. There are always two intersec- + β(2k + iα)e−ik(x+x ) tions for positive κ if |g0(0)| > |f 0(0)| and only one iαβ if this inequality is not satisfied. There are two − e4iak cos[k(x − x0)] (55) bound states if the separation between the wells k∆(k) 1 1 ik(x+x0) (α, β < 0) 2a is such that a > − − = a0 e 2α 2β δG(2δ)(x, x0) = α(2k − iβ)e2iak but only one if it is shorter than this characteristic ω,3 2k∆(k) length a of the system. The energy of the bound 0 + β(2k + iα)e−2iak . (56) states becomes more and more negative with longer a. Figure 1 shows the distribution of the number of bound states in the α-β plane of couplings (see ref- C. One-delta stress tensor and Casimir energy erence [33] for more details about the double delta system). If κj is a positive solution of (52) the form of the bound state wave function is Proper zero point renormalization of the vacuum expectation value of the energy momentum tensor requires  κj x  A(κj)e x < −a the subtraction of the vacuum expectation value of the −κj x κj x ψj(x) = B(κj)e + C(κj)e −a < x < a . (53) energy momentum tensor of the free theory [17]. There- −κj x  D(κj)e a < x fore the vacuum energy density for the scalar quantum fluctuations around a classical single delta background The two-delta reduced Green function in the zones normalized with respect to the vacuum energy density of where the two points may coincide has the structure free field fluctuations is  (2δ) 0 0 n o δGω,2 (x, x ) x, x < −a ˆ1δ ˆ0  h0| T00 (x) − T00(x) |0i = G(2δ)(x, x0) = G(0)(x − x0) + δG(2δ)(x, x0) |x|, |x0| < a . ω ω ω,1 1 n o  (2δ) 0 0 (1δ) 0 0 (0) 0 0  δG (x, x ) x, x > a ∂t∂t0 G (x, t; x , t ) − G (x, t; x , t ) ω,3 i x=x0,t=t0 √ The different components δG(2δ)(x, x0) are computed us- α Z ∞ dω ω2e2i ω2−m2|x| ω,j = √ √ . (57) 2 2 2 2 ing formula (28): i −∞ 2π 2i ω − m (2i ω − m − α) 8

To ensure that all operator products are well-defined and D. Two-delta stress tensor and Casimir energy therefore that all integrals are well-defined it is necessary to perform an euclidean rotation ω = iξ (see references [17] and [24]) The calculation of the quantum vacuum interaction energy between two delta plates requires to subtract not n ˆ1δ ˆ0 o h0| T00 (x) − T00(x) |0i = only the vacuum energy of the constant background but √ the vacuum energies of each single delta plate as well. α Z ∞ dξ ξ2e−2 ξ2+m2 |x| Accordingly the renormalised two-delta Green’s function = − . (58) p 2 2 p 2 2 that we must use to compute the quantum vacuum in- 2 −∞ 2π ξ + m (2 ξ + m + α) teraction energy is: Using the euclidean rotated energy density the single delta Casimir energy is Z ∞ G¯(2δ)(x, x0; α, β, a) = 1δ n ˆ1δ ˆ(0) o ω EC (α) = dx h0| T00 (x) − T00 (x) |0i = (2δ) 0 (0) 0 −∞ = Gω (x, x ; α, β, a) − Gω (x − x ) − α Z ∞ dξ ξ2 −G(1δ)(x + a, x0 + a; α) − G(1δ)(x − a, x0 − a; β). = − . (59) ω ω 2 2 p 2 2 2 −∞ 2π (ξ + m )(2 ξ + m + α) 2 2 2 Performing the change of variables to ξ = z − m as The vacuum expectation value of the renormalised two- done in reference [24] we obtain the stress tensor Casimir delta stress tensor Tˆ2δR(x) = Tˆ2δ(x)−Tˆ1δ(x)−Tˆ1δ(x)− 1δ 00 00 00 00 energy (EC (α)) ˆvac ST T00 (x) is given in terms of the renormalised reduced √ Z ∞ 2 2 Green function written above as 1δ α z − m EC (α) = − dz . (60) ST 2π m z(2z + α) Z ∞ dw h0|Tˆ2δR(x)|0i = ω2G¯(2δ)(x, x; α, β, a) . (63) This result was also obtained in references [4, 13] by fol- 00 2πi ω lowing the same stress tensor procedure. Later on in this −∞ paper we will compare the Casimir energy achieved in this calculation with the outcomes for the same magni- Note that the integrand in this last expression for the tude obtained through the use of the T operator and the quantum vacuum interaction energy must coincide with heat trace methods. the spectral density: The integration in (60) is ultraviolet divergent, thus we 1δ choose to regularize EC by using an ultraviolet cutoff ST Z ∞ (2δ) Λ = 1 : 1 ¯(2δ) % (k, α, β, a) ε dx Gω (x, x; α, β, a) = , (64) i −∞ k 1δ −ip 2 2 EC (α, µ, ε) = 4m − α × ST 4π ! √ −α + ip(4m2 − α2)(1 − m22) − 2m2 where k = ω2 − m2. To pass from vacuum energy den- × log m(α + 2) sities to Casimir energies we need the following integra- tions of the x-dependent functions over the three scatter- r ! 1 p α 1 1 ing zones in equation (63): + m + 4m2 − α2 − log − 1 + 4 4π m22 m

−1/2! Z −a 2iak Z ∞ m 1 −2ikx e 2ikx − arctan 2 2 − 1 (61) dx e = − = dx e , 2π m −∞ 2ik a Z a sin2ak Z a To set the scale in the regulator we select the infinite mass dx e2ikx = = dx e−2ikx , renormalization criterion: the finite part of the regular- −a k −a ized Casimir energy must be zero because when m → ∞ Z a there are no massive quantum fluctuations at all !. To 2 dx = 4a . −a comply with this freezing condition of very heavy quantum fluctuations it is necessary to re-scale the regulator to be (see [34]): The integral in the third row arises from the cosine term 2 in the Green function where both arguments lie in zone ε = · ε˜ (62) 0 me 1. x tends to x either from the left or from the right. Therefore in the modulus of the argument of the cosine 1δ such that EC (m → ∞) = 0. The divergence of both possibilities must be accounted for separately, hence ST,fin E1δ(α, µ, ε˜) is thus regularized: E1δ(α, µ, ε˜) = the factor 2 must be included. C ST C ST, div α log(˜ε)/4π. Using the equations (54)-(56) we obtain the contribu- 9 tion to the spectral density from each zone: plate and the quantum vacuum interaction energy between two semitransparent delta plates. We start by pro- 1 i(1 − e4iak) α(2k + iβ) α %(2δ)(k) = − + viding the general formula for the T operator generated k 1 4k2 ∆(k) 2k + iα by a potential concentrated in one point. β(2k + iα) β 4iaαβ + − − e4iak ∆(k) 2k + iβ k∆(k) A. The transfer matrix for potentials concentrated 1 i α(2k + iβ) α %(2δ)(k) = − + on points. k 2 4k2 ∆(k) 2k + iα β(2k − iα) β The general form of the scattering states for a potential + − e4iak ∆(k) 2k + iβ concentrated at x = 0 is given by 1 (2δ) i α(2k − iβ) α 4iak ikx −ikx % (k) = − e + (R) e + rR (k)e x < 0 k 3 4k2 ∆(k) 2k + iα ψk (x) = ikx tR (k)e x > 0 β(2k + iα) β −ikx + − . (L) tL (k)e x < 0 ∆(k) 2k + iβ ψk (x) = −ikx ikx , e + rL (k)e x > 0 √ The total spectral density per wave number, provided where again k = ω2 − m2. by the fluctuations in the three zones, is the sum of the Notice that from general scattering theory we can be quantities above, and is equal to: sure that tR (k) = tL (k) = t(k) [25, 27]. By using formula 1 1 1 1 (28) the general form of the Green function for a point %(2δ)(k, ) = %(2δ)(k) + %(2δ)(k) + %(2δ)(k) k k 1 k 2 k 3 potential is:

2αβ[1 + 2a(α − 2ik)]  (0) r (k) 0 = − .(65) G − R e−ik(x+x ), x, x0 < 0 k(2k + iα) [(2k + iα)(2k + iβ)e−4iak + αβ]  ω 2ik 0 (0) r (k) ik(x+x0) 0 Gω(x, x ) = L . Gω − 2ik e , x, x > 0 (2δ)  (0) t(k)−1 ik|x+x0| 0 The Casimir energy is the integral of % (k)/k weighted  Gω − e , sgn(xx ) = −1 with ω2 over all the range of frequencies: 2ik From the definition of the transfer matrix Z ∞ (2δ) dω 2 (2δ) p 2 2 EC (α, β, a) = √ ω % ( ω − m ). (0) 2π ω2 − m2 Gω(x, y) = Gω (x, y) −∞ Z Z 0 (0) 0 (0) 0 The integral over frequencies has convergence problems − dz dz Gξ (x, z)Tω(z, z )Gξ (z , y) due to oscillatory functions in the integrand and unitarity problems posed by bound states in Minkowski space. To and using the free Green function differential equation avoid these problems we perform the Euclidean rotation 2 ω = iξ and change the variable to the imaginary momen- 2 d (0) −ω − Gω (x, y) = δ(x − y) (67) tum κ = −ik = pξ2 + m2 (see references [17, 24]). The dx2 Casimir energy is finally written as the integral: we get an alternative general formula for the transfer

(2δ) matrix EC (α, β, a) = (66) ST √ d2 d2 Z ∞ 2 2 2 2 1 αβ(1 + 2a(α + 2κ)) κ − m Tω(x, y) = − ω + 2 ω + 2 δGω(x, y), (68) = − dκ 4aκ . dy dx π m (2κ + α) [(2κ + α)(2κ + β)e − αβ] where we deﬁned In the case of the double delta the integral obtained is not ultraviolet divergent because all the divergences have δG (x, y) = G (x, y) − G(0)(x, y) (69) been subtracted by taking into account the renormalised ω ω ω ¯(2δ) 0 reduced Green function Gω (x, x ; α, β, a). The vacuum 2 2 2 2 2 2 interaction energy between both deltas is the part of the Acting with (ω +d /dx ) (equivalently for (ω +d /dy )) total energy that depends on the distance between them over δGω(x, y) we always get 0 when sgn(x) = sgn(y) (see [4, 16, 24, 33, 35]). because in these zones the exponentials do not contain absolute values. Hence the only non trivial contribution to the transfer matrix comes when we act with (ω2 + d2/dx2) and (ω2 + d2/dy2) over δG in the case IV. CASIMIR ENERGIES FROM THE ω TRANSFER MATRIX where sgn(x) 6= sgn(y). The derivatives of functions that depend on absolute values are

In this section we will use the T operator to compute d f(|x|) = f 0(|x|)sgn(x) (70) the vacuum energy for a single semitransparent delta dx 10

d2 f(|x|) = 2f 0(|x|)δ(x) + f 00(|x|). (71) The analytic integration of (75) gives, up to the leading dx2 log approximation, the following result: Taking into account that 1 m22 E1δ(α, µ, ) = − α ln − 2 + 2πm C 8π 4 eik(|x|+y), x < 0, y > 0 eik|x−y| = iµ (−2µ + iα)2 eik(x+|y|), x > 0, y < 0 − ln − iπ + O() 4π 4m2 immediately we obtain for sgn(x) 6= sgn(y) Because log(x + iy) = log(x2 + y2)/2 + i arctan(y/x), we d2 d2 check that the regularized vacuum energy is indeed real: ω2 + ω2 + eik|x−y| = −4k2δ(x)δ(y). dy2 dx2 1 m22 E1δ(α, µ, ) = − α ln − 2 + 2πm C 8π 4 With this result the expression of the transfer matrix for µ α arbitrary point-like potentials is given by − 2 arctan + π + O() . (76) 4π 2µ Tω(x, y) = 2ik(t(k) − 1)δ(x)δ(y). (72) 1δ In the physical limit → 0 EC (α, µ, ) is logarithmic Note that when the potential is concentrated in another divergent. To set the scale in the regulator we select the point other than zero this result is valid by simply trans- infinite mass renormalization criterion. The µ → ∞ limit forming x 7→ x − x0, y 7→ y − x0. of (76) is 1 α 2 1 E1δ(α, µ, ) = ln − 2µ + O( ) (77) B. One-delta transfer matrix and Casimir energy C 4 π µ µ We thus re-scale the regulator in the form Using the general formulas (72) and (39) the euclidean rotated T operator for the delta potential is − 2πµ 2e α = e (78) (1δ) 0 2κα 0 µ Tiξ (x, x ) = δ(x)δ(x ) , (73) 2κ + α in order to fit in the infinite mass renormalization criterion: where κ = pξ2 + m2. The TG formula for the Casimir energy of the 1δ con- α m2 4πµ E1δ(α, µ, ) = − ln − + ln(2) − 2 figuration reads (see refs. [16, 17, 24]) C e 8π µ2 α e (0) (1δ) (0) 1 α Z ∞ G 1 − T · G − 2πm + 2µ 2 arctan + π + O()(79) 1δ 1 dξ iξ iξ iξ 8π 2µ e E (α) = − Tr 2 ln C 2 π L (0) 0 Giξ Neglecting the logarithmic divergence, we obtain the uni- ! Z ∞ dξ α versal finite part = ln 1 − p 2 2 2 0 2π 2 ξ + m + α 1δ α m 4πµ EC |FP (α, µ) = − ln − − 2 1 Z ∞ κdκ α 8π µ2 α = √ ln 1 − . (74) 2π 2 2 2κ + α 1 α m κ − m − 2πm + 2µ 2 arctan + π (80) 8π 2µ In the last part of this section we compare this result with the vacuum energy obtained from the energy momentum that goes to zero in the µ → ∞ limit. Thus, a fine tuning tensor. of the finite renormalizations is necessary to take into account the fact that the massive quantum fluctuations are frozen in the infinite mass limit. C. One-delta TG Casimir energy ultraviolet Note that with this re-scaling of the regulator, different regularization from the re scaling used in the stress tensor version of 1− 2πµ the Casimir energy by the factor e α , the logarithmic We regularize the vacuum energy of massive fluctu- divergences of the TG and ST one-delta Casimir energies ations in one-delta configuration backgrounds by cut- are the same. ting the TG formula at a finite ultraviolet momentum 1 κuv = ε : D. Two-delta transfer matrix and Casimir energy 1 Z 1/ κdκ α E1δ(α, µ, ) = − √ ln 1 − . C 2 2 2π m κ − m α + 2κ The T-matrices for the displaced delta’s are immedi- (75) ately obtained from the T operator for a single delta 11 placed at the origin given in formula (73): to estimate the kernel of the operator Mξ for the double- delta system: (α) 2κα T (x1, x2) = δ(x1 + a)δ(x2 + a) · (81) 2κ + α αβe−2aκ M 2δ(x, x0) = e−κ|x+a|δ(x0 − a). (83) ξ (2κ + α)(2κ + β)

(β) 2κβ T (x1, x2) = δ(x1 − a)δ(x2 − a) · (82) 2δ 0 2κ + β From the kernel Mξ (x, x ) we obtain after the corresponding integration over the whole real line the trace of where the M-matrix in terms of κ = pξ2 + m2: p κ = ξ2 + m2. Z ∞ −4κa 2δ 2δ αβe TrL2 Mξ = Mξ (x, x) = . (84) The general expression (18) and the explicit expression −∞ (2κ + α)(2κ + β) for the kernels of the T operator for each delta allow us

2δ FIG. 2. Level curves of the real part of the non dimensional Casimir energy 2πaEc as a function of the αa and βa non dimensional parameters

The formal series expansion of log(1 − x) shows that formula 2δ Tr 2 ln 1 − M L ξ Z ∞ ∞ 2δ 1 dξ 2δ n+1 E (α, β, a) = Tr 2 ln 1 − M (85) X (−1) 2δn 2δ C L ξ = Tr 2 M = ln 1 − Tr 2 M . 2 0 2π n L ξ L ξ n=1 Applying these results in the integrand of the T GT G we obtain in the case of the two-δ potential the following 12

T GT G Casimir quantum vacuum energy: Therefore, the Casimir force

2δ 2δ 1 dEC (α, β, a) FC (α, β, a) = − = Z ∞ dξ αβe−4κa 2 da 2δ ∞ 2 EC (α, β, a) = · ln 1 − , 2αβ Z κ dκ 1 0 4π (2κ + α)(2κ + β) = − √ 2 2 4aκ π m κ − m e (2κ + α)(2κ + β) − αβ may be attractive or repulsive. Generically, we find at- or, more explicitly: traction if the signs of the two δ-potential plates are equal and repulsion when sgnα 6= sgnβ. Null Casimir energies and forces are found when one moves throughout the E2δ(α, β, a) = α : β-plane of couplings. Identical qualitative behav- C √ ior has been found in the Casimir interaction between 2 2 ! Z ∞ dξ αβe−4a ξ +m two magneto-electric δ-plates with dual electro-magnetic = ln 1 − p 2 2 p 2 2 0 2π (2 ξ + m + α)(2 ξ + m + β) properties, see reference [12]. Relative minus signs in the reflection coefficients of both transverse electric and mag- 1 Z ∞ κdκ αβe−4κa = √ ln 1 − , (86) netic modes coming respectively from the electric permit- 2 2 2π m κ − m (2κ + α)(2κ + β tivity and the magnetic permeabilty of the δ-plate pro- duce this interesting situation. Our scalar model will give rise to repulsive Casimir force when one of the two cou- where a change of integration variable from the Euclidean plings of the δ-potentials is negative. As a consequence, energy to the Euclidean momentum has been performed an imaginary part in the Casimir energy arises, unless a in the last step. These integrals can not be carried out an- mass on the scalar fluctuations is introduced to avoid this alytically in general. Alternatively Figure 2 shows Math- dissipative phenomenon. In the case of electromagnetic ematica plots of these integrals numerically estimated as fluctuations such as those considered in [12] no such cut- functions of α and β. off is needed because there are no poles in the reflection coefficients coming from bound states. The results are assembled in the Figure 2 showing The choice µ = 0 in the definition of m2 leaves room selected graphs of level curves of the real part of the for an small imaginary part of the Casimir energy in a Casimir energy over the αa : βa plane. We observe that little region of the α : β-plane shown in Figure 3.This the Casimir energy is negative when the two δ-potential happens because for these very weak negative values of plates have the same sign, they are either repulsive or αa and βa the eigenvalue of the unique bound state of attractive. If the signs are different, in the second and the double δ-well is more negative than the sum of the four quadrants, the Casimir energy, however, is positive. eigenvalues of the two individuals δ-wells: −α/2 − β/2.

FIG. 3. Zone in the (αa)(βa)-plane where the imaginary part of the Casimir energy is non zero and level curves of the imaginary 2δ part of the non dimensional Casimir energy 2πaEc in this region

2 2 α2 β up enough the negative eigenvalue to reverse its sign and Thus, the m = 4 + 4 massive fluctuations do not push 13 the quantum field theory is non-unitary [36]. When the quantum field theory is unitary the norm of the operator Mξ is lower than one. Hence the Taylor series expansion of the logarithm that appears in the integrand of the T GT G formula makes sense. Nevertheless when de Schrödingerproblem that gives the one particle states has negative energy states the norm of Mξ becomes bigger than one and the vacuum interaction energy has a complex value. The physical meaning of a complex vacuum energy is the surge of particle creation and annihilation in the vacuum. This effect is a bosonic cousin of the Schwinger effect where electron/positron pairs are cre- ated from the vacuum in the background of strong electric fields (see refs. [37–39]). Here, absorption and/or emission of the scalar field fluctuations by the plates is the physical phenomenon responsible of the imaginary part of the energy. FIG. 4. Level curves of the non dimensional Casimir energy 2δ 2πa EC (α, β, a) in the region α, β > 0 m=0 The massless Dirichlet limit: perfectly conducting plates.

When we impose µ = ipα2/ + β2/4 the quantum ﬂuc- (Dirichlet boundary conditions; see reference [33]): tuations become massless, i. e. m = 0. Hence from formula (86) we obtain the quantum vacuum interaction 2δ π lim EC (α, β, a) = − . (88) energy for massless quantum ﬂuctuations α,β→∞ m=0 24 · (2a)

Z ∞ −4κa 2δ dκ αβe This is the scalar one-dimensional version of the original EC (α, β, a) = ln 1 − . m=0 0 2π (2κ + α)(2κ + β result obtained by H. B. G. Casimir for the electromagnetic field in the three-dimensional case in reference [3]. Integrating by parts in this last expression we obtain an alternative formula for E2δ(α, β, a) : C m=0 Z ∞ 2δ κdκ E. Comparison between the TG and stress-tensor EC (α, β, a) = − m=0 0 2π Casimir energies d αβe−4κa × ln 1 − dκ (2κ + α)(2κ + β It is worthwhile to compare the results achieved within the TG formalism with the answer obtained from the that can be directly related to the trace of operator stress tensor calculation. In the case of one single δ a 2δ 1/2 K m=0 by using the Cauchy’s residue theorem to simple partial integration makes the link between these compute the sum over zeroes of a holomorphic function two alternative methods: f(z), 1 Z ∞ zdz α I zpdz d E1δ | (α) = √ log 1 − = X p C TG 2 2 kn = ln (f(z)) , (87) 2π m z − m 2z + α C 2πi dz ∞ kn∈zeros(f) 1 p α = z2 − m2 log 1 − − 2π 2z + α where C is a contour in the complex z plane that encloses √ m 2δ Z ∞ 2 2 all the zeroes of f(z). We stress that EC (α, β, a) is α z − m m=0 − dz only well-defined for α, β > 0 because the single delta 2π m z(2z + α) vacuum energy subtraction induces an imaginary con- α ⇒ E1δ | (α) = − + E1δ | (α) . (89) tribution when any of the delta couplings is negative. C TG 4π C ST For any α, β > 0 the vacuum energy E2δ(α, β, a) C m=0 must be computed numerically. The level curves of We see that the Casimir energies computed by these two 2δ EC (α, β, a) as a function of αa, and βa can be seen different methods differ in a finite constant term that is m=0 1 in Figure 4. When α, β → ∞ the integral arising in precisely equal to minus 2π times the bound/anti-bound E2δ(α, β, a) can be carried out exactly, giving rise state energy. C m=0 to the very well known result of the quantum vacuum in- One can also compare the Casimir energies of two δ- teraction energy between two perfectly conducting plates function plates calculated within these two different pro- 14 cedures through partial integration in (86): in this last section is to attack the calculation of Casimir 2δ energies in δ backgrounds using the theoretical machin- EC (α, β, a) = ery developed in the study of one-loop kink fluctuations. ∞ 1 p αβe−4aκ From this point of view the quantum vacuum interac- = κ2 − m2 ln 1 − − tion between two Dirac deltas can be understood as the 2π (2κ + α)(2κ + β) m √ quantum vacuum interaction between two sine-Gordon αβ Z ∞ κ2 − m2(1 + 2a(α + 2κ)) kinks when both kinks can be considered infinitely thin − dκ 4aκ . π m (2κ + α)[(2κ + α)(2κ + β)e − αβ] compared with the distance between them (see reference In contrast to what happens with a single δ the two meth- [24]). ods lead to exactly the same result for the double delta system because the boundary term is zero: A. The one-δ Dashen-Hasslacher-Neveu formula 2δ 2δ EC (α, β, a) |TGTG = EC (α, β, a) |ST . (90) We apply the Dashen-Hasslacher-Neveu formula to calculate in a third approach the Casimir energy of a δ- V. CASIMIR ENERGIES FROM SPECTRAL function plate. The ingredients are spectral data of the FUNCTIONS operators

d2 d2 The (1 + 1)-dimensional sine-Gordon model of one K1δ = − + m2 + αδ(x) , K0 = − + m2 , scalar field is characterised by the action functional: dx2 dx2 √ Z ( 4 !) considered in preceeding sections (as before, we denote 2 1 µ m λ 2 S[φ] = d x ∂µφ∂ φ − 1 − cos φ . m2 = µ2 + α ). The necessary data entering in the DHN 2 λ m 4 formula are collected from the discrete and continuous (91) spectra of K1δ and K0: The first-order or BPS equations in this system √ 1. Discrete spectrum. dφ m2 λ = ∓2√ sin φ (92) • There exists a (singlet) half-bound state of K0, dx λ 2m the constant function. Half-bound states are char- are solved by the kinks or solitary waves: acterised by energies that lie in the threshold of the continuous spectrum. Thus, the K0 half-bound 2 m a ±m(x−b) state eigenvalue is precisely: ω2 = m2 = µ2 + α . φK (x) = √ (−1) 4arctan e + 2πn , (93) m 4 λ • There is a bound state in the spectrum of K1δ if where a = 0, 1, b ∈ R, and n ∈ Z. Small deformations of and only if α < 0. The corresponding eigenvalue is these non-linear waves that are also solutions of the BPS 2 2 α2 2 known to be: ωµ = m − 4 = µ . equations (92) belong to the kernel of the first-order dif- d 2. Continuous spectrum. ferential operator ∂ = dx +m·tanh(mx) (we have chosen a = b = n = 0). The second-order sine-Gordon kink fluc- The information needed from the continuous spec- tuations are governed by the Schrödingeroperator: trum is the spectral density. Choosing a very long normalisation interval of length L we impose pe- d2 2m2 KsG = ∂†∂ = − + m2 − . (94) riodic boundary conditions on both the eigenfunc- dx2 cosh2mx tions of K0 and K1δ:

The spectrum of this operator is thus the basic infor- iqL i(kL+δ1δ (k)) mation needed to calculate the quantum energy induced e = e = 1 1δ by one-loop fluctuations in the kink background. In ⇒ qL = kL + δ (k) = 2πn , n ∈ Z . (96) the regime of strongly coupled and infinitely thin√ kink 2 2π (kink string limit), m, λ → +∞ such that m / λ = Here q = L n denotes the plane wave momenta of 2 the K0 eigenfunctions compatible with the periodic const. ≡ α/√2 < ∞, we have the approximation m · α boundary conditions. k, in turn, are the momenta tanh(mx)/ λ ' 2 sgn(x). In this limit the second-order operator (94) becomes of the scattering eigen-waves of K1δ determined from the equations on the right in (96) in terms d2 α2 of the total phase shifts induced by the one − δ po- K1δ = − + − αδ(x) , α > 0 , (95) dx2 4 tential. For very large L one defines the spectral densities characterising the continuous spectra to i.e., the Hamiltonian of the one-δ-well shifted in such a be: way that the bound state energy is precisely zero. This is required by soliton physics and the reason for introduc- dn L dn L dδ1δ %0 = = ,%1δ = = + (k) . ing the one-half factor in the kink string limit. Our goal dq 2π dk 2π dk 15

• The eigenvalues in the continuous spectrum of in solitonic as in constant backgrounds, is determined K0 are thus ω2(k) = k2 + m2 whereas the spectral from the phase shift at infinity and at the origin. Ap- 0 L density is constant: % = 2π . plied to the one-δ background the equation (99) leads to • From the phase shifts in the even and odd chan- the precise DHN formula for the Casimir energy: nels √ Z ∞ 2 2 1δ θ(−α) α α k + m 1δ α E | = (µ − m) + + dk , 1δ 2ik ± α δ (k) = −arctan C DHN 2 2 2iδ± (k) + 2k 2 4π π 0 4k + m e = ≡ 1δ 2ik − α δ− (k) = 0 (100) i.e., only finite terms modify the result previously shown we identify the total phase shift: δ1δ(k) = in (97) and there is no need to repeat the regularization α −arctan 2k . The eigenvalues in the continuous spec- already achieved in (98). trum of K1δ are identical to those of K0: ω2(k) = We mention, however, that in [40] a φ4 self- k2 + m2. The spectral densities, however, differ: interactionterm has been added to the Lagrangian giving rise to a four-order vertex that induces a mass renormal- 1 dδ1δ 1 2α ization counter-term. Taking into account the contribu- %1δ(k) = L + = L + . 2π dk 2π 4k2 + α2 tion of this term to the Casimir energy the autor shows that the ultraviolet divergence disappears, just like in The Casimir energy of the δ-function is now provided soliton physics. by “almost ”the DHN formula: Z ∞ 1δ µ m dk 1δ 0 p 2 2 B. One-delta heat trace and spectral zeta function EC = θ(−α) − + % − % · k + m 2 4 −∞ 2 µ m = θ(−α) − We now implement the spectral zeta function regular- 2 4 isation method to control the ultraviolet divergences in Z ∞ dk α p 2 2 the one-δ Casimir energy. We need an intermediate tool: + 2 2 · k + m , (97) −∞ 2π 4k + α the associated heat trace. where θ(z) is the Heaviside step function and the addi- tional 1/2 factor entering in the subtraction of the half- 1. The K1δ-heat trace bound state of K0 is due to the one-dimensional Levinson theorem. The integration in (97) is ultraviolet divergent. The one-δ heat trace is defined from the spectrum of The easiest regularisation is to use of a cutoff in the en- K1δ in the form: ergy: −τK1δ √ h 1δ [τ] = Tr 2 e (101) Z 1/ε 2 2 K L 1δ µ m α k + m ∞ 2 Z 2 2 EC (ε) = θ(−α) − + dk 2 2 −τµ 1 2α −τ(k +m ) 2 4 π 0 4k + α = θ(−α)e + dk (L + 2 2 )e 2π −∞ 4k + α µ m µ 2µ 1 2 2 = θ(−α) − + arctan √ + e−τm e−τµ α√ 2 4 2π α m2ε2 + 1 = L √ + 1 − Erf[ τ] . " √ # 4πτ 2 2 α (1 + 1 + m2ε2)2 + log . (98) 8π m2ε2 From the power series representation of the complemen- tary error function The “true ”DHN formula, unveiled in quantum theory of 2 ∞ solitons, requires a subtraction mode-by-mode to perform e−z X 2n Erfc[z] = 1 − Erf[z] = 1 − √ z2n−1 the vacuum zero point renormalisation because the soli- π (2n − 1)!! tonic backgrounds, in contrast to constant backgrounds, n=1 have bound states. In practical terms this procedure re- we infer the K1δ-heat trace high-temperature expansion quires a partial integration in the contribution to the −τµ2 Casimir energy of the continuous part of the spectrum: L −τm2 e hK1δ [τ] = √ e + − Z ∞ Z ∞ 4πτ 2 1 kδ(k) 1 δ p 2 2 dk √ = dk (k) k + m − 2 ∞ 2π k2 + m2 2π dk e−τm X 2n α√ 0 0 − √ ( τ)2n−1 , (102) ∞ 1 p 4π (2n − 1)!! 2 − · δ(k) k2 + m2 , (99) n=1 2π 0 In this case the series (102) is truly convergent rather such that the integral on the left-hand member of this than asymptotic series because the integrability of the equation (99) must be used in the Casimir energy for- one-δ spectral problem. Previous calculations of the heat mula. Thus, the difference between taking into account kernels coefficients for non smooth backgrounds has been all the modes up to a given energy, not equal in number performed in reference [41]. From the least equality of 16 equation 101 and taking into account the series expansion 3. One-delta Casimir energy from the spectral zeta function for the error function and the exponential function the heat kernel coefficients for the delta function potential The standard zeta function regularisation procedure can be easily computed: prescribes a finite value for the divergent one-δ Casimir (1δ) √ energy by assigning the result obtained from the spectral a = L/ 4π, (103) −1/2 zeta function at a regular point s ∈ C:

r 2r 1 (1δ) 1δ 2s+1 (−1) µ EC (s, µ, α, M) = M (ζK1δ [s] − ζK0 [s]) (109) ar = , (104) 2 2 · r! 1 2s 1 mL Γ(s − 2 ) ζK0 [s] = 2 + √ . r+1 2(r+1) 2s 2s (1δ) (−1) Lm m 4π m Γ(s) ar+1/2 = √ 4π (r + 1)! Here M is a parameter of dimensions of inverse length  r introduced to keep the dimensions of energy L−1 at ev- X (α/2µ)2j + µ2rα , (105) ery point s ∈ . Note that we subtracted the vacuum j!(r − j)! · (2j + 1) C j=0 zero point energy also regularised by means of the corresponding spectral zeta function. where r = 0, 1, 2, ... is a natural number. The heat trace 1 The limit s → − 2 where the physical Casimir energy for the single delta system is written in terms of the heat 1δ 1δ arise, E (α, µ) = lim 1 E (s, µ, α, M), is very deli- kernel coeﬃcients as the series: C s→− 2 C cate because it is a pole of the one-δ spectral zeta func- ∞ X (1δ) (1δ) −1/2 r tion. Nevertheless, analysis of the Casimir energy near h 1δ [τ] = (a + a τ )τ (106) K r r−1/2 the pole allows us to isolate the divergent part: r=0

1δ 1δ 1 EC (α, µ) = lim EC (− + ε, µ, α, M) = ε→0 2 2. The spectral K1δ-zeta function 1 α 1 1 3 α2 = (µ − m) − lim · 2F1[ , 0, ; − ] + 4 8π ε→0 ε 2 2 4µ2 The one-δ spectral zeta function ζK1δ [s] is the Mellin’s 2 transform of the heat trace. Therefore, we obtain α 0 1 3 α + · 2F1[ , 0, ; − 2 ] , Z ∞ 8π 2 2 4µ 1 s−1 ζK1δ [s] = dτ τ hK1δ [τ] (107) Γ(s) 0 the singularity arising at the pole of the Γ-function at 1 1 L 1 Γ(s − 1 ) the origin. To derive this formula we used: √ 2 = 2s + 2s−1 − 2 µ 4π m Γ(s) 1 1 3 α2 1 2 Γ(ε) = Γ(1 + ε) , 2F1[ , ε, ; − 2 ] = α 1 Γ(s + 2 ) 1 1 3 α ε 2 2 4µ −√ 2F1[ , + s, ; − ] 4π m2s+1 Γ(s) 2 2 2 4µ2 1 3 α2 1 3 α2 = F [ , 0, ; − ] + ε · F 0[ , 0, ; − ] with the spectral information encoded in the Gauss hy- 2 1 2 2 4µ2 2 1 2 2 4µ2 pergeometric function 2F1[a, b, c; z]. The meromorphic structure of the spectral zeta func- and the prime means derivative of the hypergeometric tion is deciphered from the Mellin’s transform of the heat function with respect to the second argument. trace expansion: The one-δ Casimir energy calculated from the heat kernel expansion ζK1δ [s] = ∞ −τm2 ! 1δ 1 1 Z e L 1 2 E (α, µ) = (µ − m) − (110) = dτ τ s−1 √ + )e−τµ − C 4 Γ(s) 4πτ 2 ∞ 0 1 X α2n Γ(−1 + ε + n) ∞ 2 ∞ − √ lim 1 Z e−τm X 2n α√ α 4π ε→0 2n(2n − 1)!! Γ(− 1 ) − dτ τ s−1 √ ( τ)2n−1 n=1 2 Γ(s) 0 4π (2n − 1)!! 2 n=1 shows that the singularity only arises in the n = 1 term. L 1 Γ(s − 1 ) 1 1 = √ 2 + − 4π m2s−1 Γ(s) 2 µ2s ∞ 1 X α2n Γ(s + n − 1 ) C. Two-delta Casimir energy from the spectral −√ · 2 (108) 2n(2n − 1)!! m2s+2n−1Γ(s) heat and zeta functions 4πα n=1 and we check that the poles of ζK1δ arise at the points: In order to compute the two-δ Casimir energies from 1 s + n − 2 = 0, −1, −2, ··· , or, equivalently, s = the spectral functions we rewrite the phase shifts of the 1 1 3 5 2δ 2 , − 2 , − 2 , − 2 , ··· . Schr¨odingeroperator K for two delta’s of the same 17 strength in the form: DHN formula applies to single-body objects. To general- ize this expression to two-body structures our criterion is 2iδ±(k) 2δ e = t(k) ± r(k) , δ (k) = δ+(k) + δ−(k) to reproduce the rigorous result obtained from the T GT G 2ik + α(1 ± e−2iak) formula. Thus we must work with a renormalized spec- e2iδ±(k) = 2ik − α(1 ± e2iak) tral density 2δ 2δ 1δ 1δ 0 The choice of δ’s of identical weights is aimed to simplify % (k) = % (k) − % (k, a) − % (k, −a) − % = the formulas. The spectral density on an interval of very 1 d α2e−4iak − (2ik + α)2 = · ln − large length, L → ∞, reads: 4πi dk α2e4iak − (2ik − α)2 " −1#) %2δ(k) = (2ik + α)2 − α2e−4ika α2e4ika − ln · 1 − L 1 d α2e−4iak − (2ik + α)2 (2ik − α)2 − α2e4ika (2ik − α)2 = + ln .(111) 2π 4πi dk α2e4iak − (2ik − α)2 1 d α2e4ika = · ln 1 − (113) 4πi dk (2ik − α)2

1. The K2δ-spectral heat trace and zeta function where the subtraction of the individual δ’s displaced with respect to each other in 2a exactly gives the T GT G den- The knowledge of the scattering data of the K2δ sity. From the renormalized spectral density we write the Schr¨odingeroperator allows us to write the spectral heat renormalized DHN two-δ Casimir energy by means of the trace and zeta function. The heat trace is two body DHN formula: 2δ ω1 ω2 1 m 2 1 2 −τω1 −τω2 EC |DHN (α, a) = θ(−α) + θ(−α − ) − − hK2δ (τ) = θ(−α)e + θ(−α − )e + 2 2 a 4 a ∞ Z ∞ 1 Z kdk α2e4ika 2δ −τ(k2+m2) −µθ(−α) − √ ln 1 − . + dk % (k) · e , 2 2 2 4πi 0 k + m (2ik − α) −∞ Note that we have subtracted also the square roots of the formula that encodes the contribution of the (condi- eigenvalues of the half-bound state of the free Hamilto- tional) bound states (ﬁrst row) and the scattering states nian and the two bound state eigenvalues of the individ- (second row). After a partial integration we obtain: ual δ’s. hK2δ (τ) = (112) −τω2 1 −τω2 L −τm2 = θ(−α)e 1 + θ(−α − )e 2 + √ · e + VI. PROSPECTS AND FURTHER COMMENTS a 4πτ Z ∞ 2 −4iak 2 kdk α e − (2ik + α) −τ(k2+m2) A. Vacuum energy of many δ-interactions +τ ln 2 4iak 2 · e . −∞ 2πi α e − (2ik − α) The concepts explored and the techniques developed The Mellin transform of these expressions leads respec- in the core of this paper can be extended to analyse an tively to the two-δ spectral zeta function array of 2N + 1 Dirac δ-potentials. Here N ∈ N∗ or Z N ∈ ∗, i.e., N is a positive integer or half-integer in 1 s−1 N2 ζ 2δ (s) = dτ τ h 2δ (τ) = K Γ(s) K such a way that 2N + 1 is either an odd or even integer. If n = −N, −N+1, ··· ,N−1,N span the integers or half- L Γ(s − 1 ) 1 1 1 1 √ 2 integers between −N and N we consider the background = 2s−1 + 2s θ(−α) + 2s θ(−α − ) + 4π Γ(s) m ω1 ω2 a N Z ∞ kdk α2e−4iak − (2ik + α)2 X U(x) = αn δ(x − 2na) , (114) +s 2 2 s+1 · ln 2 4iak 2 −∞ 2πi(k + m ) α e − (2ik − α) n=−N

1 e.g., for N = 0 it gives the one-δ-potential, the N = 2 2. The two body DHN formula case corresponds to two-δ’s, N = 1 to three δ’s, etc. The quantum vacuum energy induced by the fluctuations of Strict translation of the DHN formula would give the an scalar field on this background (114) is computable by 1 plugging the kernel of the M-operator two-δ vacuum energy in the s = − 2 value of the spectral zeta function as: (α−N ,α−N+1··· ,αN−1,αN ) Mω (x1, x2) Z h 2δ 1 (0) (α−N ) E | = lim (ζ 2δ (s) − ζK0 (s)) . = dz1dz2 ··· dz4N+1 Gω (x1, z1)Tω (z1, z2) · C DHN 1 K 2 s→− 2 (0) (α−N+1) ·Gω (z2, z3)Tω (z3, z4) ··· By doing this we fail to subtract the contributions of the i (0) (αN ) self-energies of the individual δ’s. In fact the original ··· Gω (z4N , z4N+1)Tω (z4N+1, x2) 18 in formula (16). Recall that in the Euclidean version we 1. One starts from the “supercharges ” have d 0 0 0 dx + W (x) † (0) 1 Q = , Q = d G (x , x ) = · e−κ|x1−x2| 0 0 − + W (x) 0 iξ 1 2 2κ dx where W (x) is a real function called the superpo- 2κα tential. T (αn)(x , x ) = δ(x − 2na)δ(x − 2na) · n iξ 1 2 1 2 2κ + α n 2. The supersymmetric Hamiltonian is: formulas from which we derive 1 † † (α−N ,··· ,αN ) K = QQ + Q Q = Miξ (x1, x2) = 2 2 2 ! N α 1 − d + dW dW + d W 0 −κ|x1+2Na| Y n −2|n|κa dx2 dx dx dx2 = e δ(x2 − 2Na) · · e = d2 dW dW d2W . 2κ + αn 2 0 − 2 + − 2 n=−N dx dx dx dx and 3. The supersymmetry algebra Z ∞ (α ,··· ,α ) (α ,··· ,α ) −N N −N N † † TrL2 Miξ = dxMiξ (x, x) = {Q, Q } = 2H , [H, Q] = [H, Q ] = 0 −∞ N shows that the the supercharges are “super ”sym- Y αn = e−4Nκa · · e−2|n|κa . (115) metry operators. 2κ + αn n=−N The two kinds of scalar fluctuations give rise to a su- The Lipmann-Schwinger equation determining the trans- persymmetric quantum mechanical problem if and only fer matrix provides us with a result for the vacuum en- if: ergy of an array of 2N + 1-δ’s where the vacuum ener- 2 gies of any array with a lower number of δ’s (including dW dW d W U±(x) = ± . the constant background) are subtracted. Thus, the TG dx dx dx2 procedure applied to 2N +1 δ’s not only subtracts the di- In the case of two δ-function plates supporting the fluctu- vergent vacuum zero point energy and the 2N +1 Casimir ations of two scalar fields one spectral problem in super- “self-energies ”of the individual δ’s but also the finite two- symmetric quantum mechanics arises if the backgrounds body, three-body, ··· , 2N-body Casimir energies. In the are of the form: recent reference [42], however, the interactions between 2 2 a lower number of δ’s than N have been also considered. U±(x) = ±αδ(x+a)±βδ(x−a)+α θ(−x−a)+β θ(x−a) . The hidden reason for supersymmetry is the existence of a superpotential B. Supersymmetric δ-interactions α β β − α W (x) = · |x + a| + · |x − a| + · x The fluctuations of two real scalar fields on two pos- 2 2 2 sibly different static bacgrounds are governed by the ac- that defines the supercharges of this problem. Standard tion: lore in supersymmetric quantum mechanics, see e.g. [43]

S[Φ+, Φ−] = and references quoted therein, classifies the characteris- Z tics of the spectrum of a supersymmetric Hamiltonian as 1 2 µ 2 2 = d x ∂µΦ+∂ Φ+ + (m + U+(x))Φ (t, x)+ follows: 2 + + µ 2 2 • The ground state of K may be unique or doubly + ∂µΦ−∂ Φ− + (m− + U−(x))Φ−(t, x) . (116) degenerate. If it is unique the ground state belongs The one-particle states of the quantum field theories are to the spectrum of either K+ or K−. In this case obtained through a Fourier transform from time to fre- there exists spectral asymmetry, and supersymme- quency of the eigenfunctions of the Schrödingeropera- try is unbroken. Degenerate ground states form tors: a doublet: one member belongs to the spectrum 2 d of K+ and the other is an eigenstate of K−, and K = − + U (x). ± dx2 ± supersymmetry is spontaneously broken. The two- δ supersymmetric Hamiltonian exhibits a unique Generically, this system is no more than the system con- ground state and supersymmetry is unbroken. sidered in the main core of this paper counted twice. 2 There is, however, an interesting situation: if m+ = • Higher excited levels in the discrete spectrum of K 2 2 m− = m it may be possible that K+ and K− are super- come in degenerate pairs living respectively in the symmetric partners and their spectra have the symmetry spectrum of K+ and K−. Their positive eigenval- properties arising in supersymmetric quantum mechan- ues are identical. Hence, operators K+ and K− are ics. Briefly, the structure is the following: almost isospectral. 19

• The continuous spectrum eigenvalues of K+ and In this situation (no time reversal invariance) the trans- K− are also positive and identical. The spectral mission amplitudes are also diﬀerent: densities of the SUSY pair of operators, however, 2q 2k are diﬀerent but are related by the supercharges. tR(ω) = , tL(ω) = ; (118) ∆sδ ∆sδ q − k − iα k − q − iα rR(ω) = , rL(ω) = (119) ∆sδ ∆sδ

2 2 2 2 In summary, once we know the Casimir energy due to one ∆sδ(ω, α) = k + q + iα , ω = k = q + s , (120) kind of ﬂuctuations the contribution of the other kind follows from supersymmetry. A warning: in order to apply the T GT G formula in this supersymmetric context (R) (L) 4ikq W (ψω , ψω ) = (121) we must rely on the Green’s function and the T -matrix ∆sδ in one step/delta background, e.g., located at the origin: where k and q are the momenta of the plane waves on the x > 0 and x < 0 half-lines respectively. From the Wronskian and the scattering waves incoming from the left or the right we obtain the reduced Green’s function 2 Usδ(x; α, s) = αδ(x) + s (1 − θ(x)) . (117) and the kernel of the T-operator in four pieces:

 1 2k  − eik|x−y| − ( − 1)eik(x+y) , x > 0 , y > 0  2ik ∆sδ    i 1 ei(kx−qy) , x > 0 , y < 0  ∆sδ (0)  Gω (x, y) = . (122)  i 1 ei(ky−qx) , x < 0 , y > 0  ∆sδ     − 1 eiq|x−y| − ( 2q − 1)eiq(x+y) , x < 0 , y < 0 2iq ∆sδ

 iα2  α + ∆ δ(x)δ(y) , x > 0 , y > 0  sδ   2 2  iα δ(x)δ(y) + iαs e−iqyδ(x) , x > 0 , y < 0  ∆sδ ∆sδ (α,s)  Tω (x, y) = . (123)  iα2 iαs2 −iqx  ∆ δ(x)δ(y) + ∆ e δ(y) , x < 0 , y > 0  sδ sδ   2 2  α + iα δ(x)δ(y) + iαs e−iqyδ(x) + e−iqxδ(y) , x < 0 , y < 0 ∆sδ ∆sδ

From these formulas for the Green’s function and the points x = ±a: kernel of the T-matrix one derives the kernel of the two 0 step/δ’s M-matrix by shifting the origin to the x = ±a U(x) = α1δ(x + a) + λ1δ (x + a) points and bearing in mind that at x = a one must ex- + α δ(x − a) + λ δ0(x − a) . (124) change k and p in the solutions at the left and the right 2 2 of the s − δ-potential. We understand the δ0 interaction in (124) as the self- adjoint extension of the free particle Hamiltonian proposed in reference [44]. Because this potential responds to interactions concentrated in isolated points the general expression (72) provides us with the kernel of the C. Miscellanea: some exotic backgrounds T-matrix for a δ/δ0 interaction at the origin:

2κ(α + κλ2) Other more exotic backgrounds include two linear com- T (α,λ)(x , x ) = δ(x )δ(x ) . 0 ω 1 2 λ2 1 2 binations of δ and δ interactions concentrated at the 2κ(1 + 4 ) + α 20

2δ 2 The reduced Green’s function is, of course, the free parti- of the spectral problem K ψω(~x) = ω ψω(~x): cle Green’s function and simply application of the T GT G Z 2 −i~p~x formula, after the appropriate shiftings to x = ±a and i~k~x d p e ψω(~x) = e − × use of these ingredients, leads to the Casimir energy. We (2π)2 |~p|2 − |~k|2 − iε skip writing the quantum vacuum energy of the dou- × α e−i~p~aψ (−~a) + α ei~p~aψ (~a) , ble δ/δ0 system but we mention that re-interpretation of − ω + ω where |~k|2 = ω2 − m2 and the scattering amplitudes are: these point interactions as boundary conditions encom- pass for some special values of the parameters not only ~ ~ 1 + α∓ I1[|k|] − I2[|k|, |~a|] Dirichlet but also Neumann and even Robin boundary ψ (±~a) = ω ~ conditions. ∆(|k|, |~a|, α±) A promising avenue is the study of scalar ﬁeld ﬂuctua- Z 2 ~ d p 1 tions on solitonic or curved back grounds. For instance, I1[|k|] = (2π)2 |~p|2 − |~k|2 − iε the background

Z 2 ±2i~p~a ~ d p e U(x) = αδ(x + a) + βδ(x − a) + I2[|k|, |~a|] = (2π)2 2 ~ 2 2 |~p| − |k| − iε + m2 1 − θ(a − x)θ(a + x) · , cosh2 mx ~ ~ ~ ∆(|k|, |~a|, α±) = 1 + α+I1[|k|] 1 + α−I1[|k|] − see [45], can be interpreted in comparison with the Hamil- −α α I2[|~k|, |~a|] . tonian KsG (94) as a sine-Gordon kink background con- + − 2 strained to the finite interval (−a, a) with two δ-function The integrals I1 and I2 are ultraviolet divergent but can interactions at the endpoints. In this paper the Green’s be regularised by using, e.g., a cutoff in the momentum. function and the energy-momentum tensor of the double The identification of the scattering data from these so- delta/Pösch-Teller configuration were computed paving lutions is extremely delicate from an analytical point of the way to the calculation of the quantum vacuum en- view because the lack of rotational invariance of the po- ergy. In reference [33] two of us analysed the scattering tential. problem for this potential. In particular, the scattering Fortunately, the TG formalism only requires address- amplitudes, as well as the bound state structure, were ing the problem of one centre (and the knowledge of the fully unveiled. The limit of Dirichlet boundary condi- free particle Green’s function), an issue solved by Jackiw tions revealed many subtleties, particularly relevant to in [46] where he found two striking results: (1) the scale the ground state (a zero mode) of the system. By switch- invariance of the δ(2)-interaction is broken by the qua- ing on the δ0-interactions on the endpoints more general tization process; (2) The scattering waves only emerge boundary conditions can be achieved. Nevertheless, the in s-waves, the only non null phase shifts corresponding full analysis of the quantum vacuum energy remains to be to zero angular momentum. To work Jackiw formulas (0) (PT) performed. More interestingly, replacing Gω by Gω , in a point away from the origin is the remaining task to the particle Green’s function in the Pösch-Teller back- perform in order to compute the two δ(2)-lines Casimir ground, we should be able to generalise the TG procedure energy from the T GT G formula. Physically this mathe- in this situation. matical structure arises as quantized vortex lines in su- The last background that we consider is formed by perfuid Helium 4 or other Bose condensates. If the phase two δ-interactions placed on antipodal points on a circle of the order parameter (the complex scalar field) is of the l x2 2 form χv(~x,t) = , arctan , l ∈ , the gradient of the around the origin in the R -plane. If ~ea, a = 1, 2, ~ea ·~eb = π x1 Z δab, is an orthonormal basis in this plane, ~x = x1~e1 +x2~e2 ~ l x1 x2 fluid velocity ∇χv(~x,t) = 2 ~e1 + 2 ~e2 is precisely: denotes the particle position and ~a = a ~e +a ~e is a fixed π |~x| |~x| 1 1 2 2 2 l (2) vector, we write the background and the 2D Schrödinger ∇ χv(~x,t) = π δ (~x). Therefore, the T GT G formalism operator that governs the fluctuations of the scalar field allows the calculation of the quantum vacuum energies in two spatial dimensions in the form: between two vortex lines in superfluid liquids (see ref. [47]).

(2) (2) U(~x) = α−δ (~x + ~a) + α+δ (~x − ~a) ACKNOWLEDGEMENTS

∂2 ∂2 The authors would like to thank Michael Bordag, I. K2δ = −∇2 + m2 + U(~x) = − − + m2 + U(~x) , ∂x2 ∂x2 Cavero-Pelaez, M. Asorey and G. Marmo for illuminating 1 2 discussions, and S. Ratcliﬀe for language support. This work has been supported by the German DFG grant BO (2) where δ (~v) = δ(v1)δ(v2). 1112/18-1, and the European Union ESF Research Net- Fourier analysis unveils the scattering wave solutions work CASIMIR. 21

[1] To cope with the problems of the continuous spectrum [19] J. M. Munoz-Castaneda. Boundary effects in quantum in many QFT textbooks the system is enclosed in a field theory (in spanish). Ph. D. thesis. Zaragoza Univer- “normalization”box of finite but very large volume. Pe- sity, 2009. riodic boundary conditions are imposed on the fields in [20] M. Asorey and J.M. Munoz-Castaneda. Vacuum Bound- such a way that the wave vectors take values on a 3D- ary Effects. J.Phys., A41:304004, 2008. 3 lattice: ~k ∈ Z . One deals with QFT on a 3D torus [21] Roger F. Dashen, Brosl Hasslacher, and Andre Neveu. 3 1 1 1 T = S × S × S and at the end of the calculations The Particle Spectrum in Model Field Theories from sends the radius of the circles to +∞. Semiclassical Functional Integral Techniques. Phys.Rev., [2] K. Symanzik. Schrodinger Representation and Casimir D11:3424, 1975. Effect in Renormalizable Quantum Field Theory. [22] Alberto Alonso-Izquierdo and Juan Mateos Guilarte. Nucl.Phys., B190:1, 1981. One-loop kink mass shifts: A Computational approach. [3] H.B.G. Casimir. On the Attraction Between Two Per- Nucl.Phys., B852:696–735, 2011. fectly Conducting Plates. Indag.Math., 10:261–263, 1948. [23] S. Mandelstam. Soliton Operators for the Quantized [4] Michael Bordag, D. Hennig, and D. Robaschik. Vacuum Sine-Gordon Equation. Phys.Rev., D11:3026, 1975. energy in quantum field theory with external potentials [24] M. Bordag and J.M. Munoz-Castaneda. Quantum vac- concentrated on planes. J.Phys., A25:4483–4498, 1992. uum interaction between two sine-Gordon kinks. J.Phys., [5] Michael Bordag and D.V. Vassilevich. Heat kernel expan- A45:374012, 2012. sion for semitransparent boundaries. J.Phys., A32:8247– [25] A. Galindo and P. Pascual. Quantum mechanics I. Quan- 8259, 1999. tum Mechanics. Springer-Verlag, 1990. [6] C.D. Fosco, F.C. Lombardo, and F.D. Mazzitelli. Deriva- [26] Note that the transfer matrix (name commonly used in tive expansion for the boundary interaction terms in the scattering theory) is nothing else but the so called T op- Casimir effect: Generalized delta-potentials. Phys.Rev., erator. D80:085004, 2009. [27] L. J. Boya. Quantum mechanical scattering in one di- [7] Prachi Parashar, Kimball A. Milton, K.V. Shajesh, and mension. Nuovo Cimento Rivista Serie, 31(2):020000– M. Schaden. Electromagnetic semitransparent δ-function 139, February 2008. (i) plate: Casimir interaction energy between parallel in- [28] When operators Kω , i = 1, 2 have discrete and contin- finitesimally thin plates. Phys.Rev., D86:085021, 2012. uum spectrum the Hilbert space spanned by eigenstates (i) [8] G Barton. Casimir energies of spherical plasma of Kω , i = 1, 2 is not isomorphic to the one spanned by shells. Journal of Physics A: Mathematical and General, eigenstates of operator K0. 2 37(3):1011, 2004. [29] We assume that there are no zero modes, ωj 6= 0, ∀j, [9] Michael Bordag. The Casimir effect for thin plasma 2 2 and there are no half-bound states, ωl < m , in the sheets and the role of the surface plasmons. J.Phys., spectrum of K. Physically, the τ = 1 parameter is mkB T A39:6173–6186, 2006. understood as an auxiliary fictitious inverse temperature [10] Michael Bordag. Reconsidering the quantization of elec- of dimensions of area: [τ] = L2. trodynamics with boundary conditions and some mea- [30] G. Barton. Levinson’s theorem in one-dimension: heuris- surable consequences. Phys.Rev., D70:085010, 2004. tics. J.Phys., A18:479–494, 1985. [11] Cesar D. Fosco, Fernando C. Lombardo, and Fran- [31] R. Rajaraman. Solitons and Instantons: An Introduction cisco D. Mazzitelli. Derivative expansion of the electro- to Solitons and Instantons in Quantum Field Theory. magnetic Casimir energy for two thin mirrors. Phys.Rev., North-Holland Personal Library. North-Holland, 1987. D85:125037, 2012. [32] Michael Bordag. Interaction of a charge with a thin [12] Kimball A. Milton, Prachi Parashar, Martin Schaden, plasma sheet. Phys.Rev., D76:065011, 2007. and K.V. Shajesh. Casimir interaction energies for [33] J. Mateos Guilarte and Jose M. Munoz-Castaneda. magneto-electric δ-function plates. arXiv:1302.0313, Ac- Double-delta potentials: one dimensional scattering. The cepted for publication in Il Nuovo Cimento C, 2013. Casimir effect and kink fluctuations. Int.J.Theor.Phys., [13] Kimball A. Milton. The Casimir effect: Recent contro- 50:2227–2241, 2011. versies and progress. J.Phys., A37:R209, 2004. [34] Note that the new regulatorε ˜ is dimensionless. [14] N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, [35] C.D. Fosco and E.L. Losada. Functional approach to the O. Schroeder, et al. The Dirichlet Casimir problem. fermionic Casimir effect. Phys.Rev., D78:025017, 2008. Nucl.Phys., B677:379–404, 2004. [36] Of course, there is a minimum value of µ > 0 such that [15] N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, all the eigenvalues become positive and the QFT of the M. Scandurra, et al. Calculating vacuum energies in system becomes unitary. renormalizable quantum field theories: A New approach [37] W. Heisenberg and H. Euler. Consequences of Dirac’s to the Casimir problem. Nucl.Phys., B645:49–84, 2002. theory of positrons. Z.Phys., 98:714–732, 1936. [16] Oded Kenneth and Israel Klich. Casimir forces in a T [38] Julian S. Schwinger. On gauge invariance and vacuum operator approach. Phys.Rev., B78:014103, 2008. polarization. Phys.Rev., 82:664–679, 1951. [17] M. Bordag, G.L. Klimchitskaya, U. Mohideen, and V.M. [39] Gerald V. Dunne. New Strong-Field QED Effects at ELI: Mostepanenko. Advances in the Casimir effect. Oxford Nonperturbative Vacuum Pair Production. Eur.Phys.J., University press, 2009. D55:327–340, 2009. [18] J. M. Munoz-Castaneda and M. Asorey. Attractive and [40] David J. Toms. Renormalization and vacuum energy repulsive Casimir vacuum energy with general boundary for an interacting scalar field in a δ-function potential. conditions. In preparation. 22

J.Phys., A45:374026, 2012. [45] Ines Cavero-Pelaez and Juan Mateos Guilarte. Local [41] Michael Bordag and D.V. Vassilevich. Nonsmooth back- analysis of the sine-Gordon kink quantum fluctuations. grounds in quantum field theory. Phys.Rev., D70:045003, 2009. 2004. [46] R.W. Jackiw. Diverse Topics in Theoretical and Math- [42] K.V. Shajesh and M. Schaden. Many-Body Contri- ematical Physics, section I.3: Delta function potentials butions to Green’s Functions and Casimir Energies. in two-dimensional and three-dimensional quantum me- Phys.Rev., D83:125032, 2011. chanics. Advanced Series in Mathematical Physics. [43] Andreas Wipf. Non-perturbative methods in supersym- World Scientific, 1995. metric theories. 2005. [47] R.J. Donnelly. Quantized Vortices in Helium II. Cam- [44] M. Gadella, J. Negro, and L.M. Nieto. Bound states and bridge Studies in Low Temperature Physics. Cambridge scattering coefficients of the −aδ(x) + bδ0(x) potential. University Press, 1991. Physics Letters A, 373(15):1310 – 1313, 2009.