Analytical regularisation for confined quantum fields between parallel surfaces

a r X i v :q u a n t -p h /0511230v 1 24 N o v 2005Analytical regularisation for confined quantum fields betweenparallel surfacesF C Santos ∗,A C Tort †Departamento de F´ısica Te´o rica -Instituto de F´ısicaUniversidade Federal do Rio de JaneiroCaixa Postal 68.528;CEP 21941-972Rio de Janeiro,Braziland E Elizalde ‡Instituto de Ciencias del Espacio (CSIC)&Institut d’Estudis Espacials de Catalunya (IEEC/CSIC)Campus UAB,Facultat de Ci`e ncies,Torre C5-Parell-2a plantaE-08193Bellaterra (Barcelona)SpainFebruary 9,2008AbstractWe review a simple technique for evaluating the vacuum energy stemming from non-trivial boundary con-ditions and present results for the Casimir energy of a massive fermionic field confined in a d +1-dimensional slab-bag and the effect of a uniform magnetic field on the vacuum energy of confined massive bosonic and fermionic fields.New results concerning the Casimir energy and the evaluation of the rate of creation of quanta in κ-deformed theories are presented.PACS:11.10.-z;12.20.-m;12.39.Ba1IntroductionIn quantum field theory under external conditions,the macroscopically observable zero-point energy shift is de-fined as the regularised difference between the vacuum expectation value of the hamiltonian with and without the presence of the external conditions.Under certain assumptions,external conditions can be approximated by boundary conditions and one-loop calculations of the energy shift lead to the Casimir effect [1]–see Ref.[2]for the most recent review on this subject.In the evaluation of the zero point energies some configurations which depend on the nature of the particular quantum field (scalar,spinorial,etc.),the type of spacetime manifold and its dimensionality,and the specific boundary condition imposed on the quantum field on certain surfaces,may lead to rather complex spectra.The heart of the matter in these calculations is the evaluation of the spectral sum that results at the one-loop level from the shift of the zero-point energy.This evaluation requires regularisation andrenormalisation,and recipes for accomplishing this task range from the relatively simple cutoff method,employed by Casimir himself [1],to a number of powerful and generalised zeta function techniques [3].Contour integral rep-resentations of spectral sums are a great improvement in the techniques of evaluating zero-point energies and have been employed before [4].A simple technique for evaluating vacuum energy stemming from non-trivial boundary conditions based on Cauchy’s integral formula and the Mittag-Leffler expansion theorem was proposed in [5,6].The method was employed in a number of cases,in particular it was applied to calculate the Casimir energy of a massive fermionic field confined in a d +1-dimensional slab-bag [6]and to calculate the effect of a uniform magnetic field on the vacuum energy of confined massive bosonic and fermionic fields [7].Recently this technique was applied to evaluate the Casimir energy of κ-deformed theories under Dirichlet boundary onditions [8].Here we review these cases and extend the method to the evaluation of the rate of creation of quanta in κ-deformed theories.2A general formula for the regularised vacuum energy shiftFor a field theory in d +1flat spacetime under boundary conditions imposed on two parallel planes of area equal to L d −1which are kept at a fixed distance ℓfrom each other,and with L ≫ℓ,the non-regularised vacuum energy is given byE 0(d )=α(d )L d −1(2π)d −1Ωn(1)where α(d )is a dimensionless factor that takes into account the internal degrees of freedom andΩn =ℓ+m 2(2)with p 2⊥=p 21+p 22+p 23+···+p 2d −1and m is the mass of an elementary excitation of the field and the λn are the real roots of the equation determined by the boundary conditions.Consider nownΩn =− Γdqq 2+Ω2n (3)which follows from Cauchy’s integral formula.The curve Γis a Jordan curve on the complex q -plane with Im q >0that we choose to be a semicircle with very large radius the diameter of which is the entire real axis of the auxiliary variable q .Defining z =ℓ2πℓ2q 2z 2+λ2n(5)This summation can be performed in the following way:Let G (z )be a complex function of the complex variable z symmetrical on the real axis and suppose its roots λn are simple,nonzero and symmetrical with respect to the origen.Notice that if z =0is a root of G we can divide G by some suitable power eliminating this root from the set of roots without introducing new singularities.Since the roots are symmetrical we can order them in such a way that λn =−λ−n with n =±1,±2,....Defining the auxiliary function J (z )byJ (z )=′n1where the prime indicates the the term n=0is to be omitted.This function is meromorphic with simple poles at iλn and residua equal to one.Defining the function K(z)=G(iz)we see that the function K′(z)/K(z)has simple poles at z=iλn with residua also equal to one.Due to the symmetry of the roots we can writeJ(z)=K′(z)K(z)= 2z2 d d−1p⊥2πq d ln[K(z)]2 d d−1p⊥2πln[K(z)](9)At this point it is possible to simplify this last expression by decomposing the function K(z)into two parts, namely K(z)=K1(z)+K2(z)where K1(z)contains all terms whose integrals diverge when Re>0and K2(z)contains all terms whose integrals diverge when Re<0;also K1(z)=K2(−z).In this way the integral in equation(9)can be written asI= Γ1dq2πln 1+K2(z)2πln[K1(z)]+ Γ2dq K2(z) (10) The regularised vacuum energy shift is thenE0(z)=α(d)L d−1(2π)dln 1+K1(z)p2+m2with p:=x/ℓand integrating out the angular part we obtainE0(ℓ,µ,d)=α(d)L d−1K2(z) (12)All along the real axis the function z does not changes its sign and is a function of q and p⊥.This approach to the evaluation of the Casimir energy can be also applied when the integrand is a more complex function ofΩn.It is convienient to make the replacement z→ω=2dπd/2Γ d2−1ln1+K1(ω)3Quantumfields confined between parallel surfacesEquation(12)is a general formula that can be applied to all cases of quantumfields confined between two parallel surfaces on which we impose boundary conditions.A non-trivial example is the one of the Casimir energy for a massive fermionicfield confined in a d+1-dimensional slab-bag under MIT boundary conditions[9],[6].If the surface of the bag is perpendicular to the d-direction MIT boundary cconditions lead to the functionF(p dℓ)=µsin(p dℓ)+p dℓcos(p dℓ)(14) Hence we can choose G(ω)asG(ω)=µsin(ω)ω+cosh(ω)(16)It follows thatK1(ω)=1ω e−ω(17)andK2(ω)=1ω eω(18)The regularised Casimir energy of the fermionfield isE0(ℓ,µ,d)=−C(d)L d−12 ℓd ∞µdωω ω2−µ2 dω+µe−2ω (19)Setting d=3we obtain the the result given in[10].We can explore this result and lengthy calculation shows that upon expanding the log function we end up with a convergent sum of integrals that can be expressed each one of them as a derivative of the Whittaker function Wνµwith respect to an auxiliary variableλatλ=1.This is as far as we can go analytically.But it is possible to consider and obtain analytical results in the massless limit and the very massive limit,see[6]for details.4Confined bosonic and fermionicfields in a uniform magneticfieldThe analytical regularisation scheme can be also applied to massive bosonic and fermionicfields in a uniform magneticfield perpendicular to the confining surfaces.Let us considerfirst the case of a charged scalarfield under Dirichlet boundary conditions.In this case we will haveE0(ℓ,µ,eB)=α eB2∞ n=0∞ 0dp3ℓ2p23+eBℓ2(2n+1)+µ2(22)4whereµ:=mℓ.The integrand can be expanded and the Casimir energy can be expressed as a convergent infinite series of integrals that can be evaluated in terms of the Whittaker functions.Thefinal result readsE0(ℓ,µ,eB)=−eBL2eBℓ2+µ2∞k=11eBℓ2+µ2−eBL2(2n+1)eBℓ2+µ2∞k=11(2n+1)eBℓ2+µ2 (23)Notice that this is an exact result.If eB≫ℓ2the behavior of the Bessel function of the third kind leads us to writeE0(ℓ,eB)π1/2ℓ3e−2√2 eBL22πln 1−z−µℓ2p23+(2n+1−α)eBℓ2+µ2n=0,1,2,3, (27)after expanding the log and summing overαwe arrive atE0(ℓ,µ,eB)=−2eBL2kI pk(M p)(28)where the prime means that the term corresponding to p=−1must be multiplied by1/2andI pk(M p):=∞dx x2+M2p 1/2+µ −k x2+M2p 1/2−µ k e−2k(x2+M2p)1/2(29)with x=p3ℓand M2p:=2(p+1)eBℓ2+µ2for p=−1,0,1,2,3,···.These integrals are non-trivial ones but a numerical evaluation of equation(28)is feasible[7].Two limiting cases,however can be solved analytically.The first one is limitµ→0and the other one is the limitµ≫1.For the massless limit we obtainE0(ℓ,µ→0,eB)≈−eBL232π3/2ℓe−2µ5Vacuum energy shift inκ-deformed theoryA quantumfield theory isκ-deformed when its spacetime symmetries are described by theκ-deformed Poincar´e algebra.These theories which may be important in some models of the early universe lead in general to highly nontrivial dispersion relations.See[12]and references therein.Consider for example scalarκ-deformed electro-dynamics.The frequency spectrum is given byω(p)=sinh−1 1p2+m2 (32)The parameterκis a measure of the departure from the Poincar´e algebra.For Dirichlet boundary coditions imposed on a certain direction,part of the frequency sepectrum is discretised and the vacuum energy shift is given byE0(ℓ,η,m)=L2 n d2p⊥ηsinh−1(ηΩn)(33) where asΩn=2π n2q2n d2p⊥(2π)31/ηdq1−η2q2log 1+K1(z)(2π)21/ηdq1−η2q2I q2 (37)where4ℓ2 q2+m2 ℓq2−Li2 e−2ℓ√q2+m2 ℓq2− q2+m2 (38) Forη→0and m→0we obtain the standard Casimir energy of a massless scalarfield under Dirichlet boundary conditions,Forη=0and m=0we obtainE0(ℓ,η,m=0)=−L2n2 1/η0dy1+1n2y2ℓ2(39)in agreement with[12].66Photon creation rate for κ-deformed theoryIn a κ-deformed theory there is the possibility of quanta creation.The rate at which photons are created is related to the quantityS =iη2πd 2p ⊥ηsinh−1ηp 2⊥+p 2n=−ndqq 2+Ω2n (42)then the photon creation rate can be rewritten asS =−L 2q 2+Ω2nsin −1(ηq )24π3d 2p ⊥∞1/ηdqdq 1−η2q 2log1+K 1(z )4π3d 2p ⊥∞1/ηdq η2q 2−1log1−e−2ℓ√4π2ℓ3∞ n =11ℓ2−11+1References[1]Casimir H B G1948Proc.K.Ned.Akad.Wet.51793[2]Bordag M,Mohideen U and Mostepanenko V M2001Phys.Rept.3531;see 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