Producción CientíficaWe obtain new expressions for the Casimir energy between plates that are mimicked
by the most general possible boundary conditions allowed by the principles of quantum field theory. This result enables to provide the quantum vacuum energy for scalar fields propagating under the influence of a one-dimensional crystal represented by a periodic potential formed by an infinite array of identical potentials with compact support.MINECO (MTM2014-57129-C2-1-P) and Junta de Castilla y Leon (BU229P18 and VA137G18)

Bordag, Michael

Muñoz Castañeda, José María

Santamaría Sanz, Lucía

English

Repositorio Documental de la Universidad de Valladolid

arXiv:1910.08142v1  [math-ph]  17 Oct 2019REVISITING THE CASIMIR ENERGY WITHGENERAL BOUNDARY CONDITIONS ANDAPPLICATIONS IN 1D CRYSTALSJ. M. Mun˜oz-Castan˜eda∗†, M. Bordag⋆, and L. Santamar´ıa-Sanz††Departamento de F´ısica Teo´rica, Ato´mica y O´ptica, Universidad de Valladolid, 47011, Spain⋆Institut fu¨r Theoretische Physik, Universita¨t Leipzig, 04103, GermanyAbstractWe obtain new expressions for the Casimir energy between plates that are mimickedby the most general possible boundary conditions allowed by the principles of quantumfield theory. This result enables to provide the quantum vacuum energy for scalar fieldspropagating under the influence of a one-dimensional crystal represented by a periodicpotential formed by an infinite array of identical potentials with compact support.Keywords: Quantum field theory under the influence of classical backgrounds;Casimir effect; boundary conditions.1 IntroductionAs it is well known, two plane parallel homogeneous isotropic D − 1-dimensional plates ina D-dimensional space interacting with a scalar field living between them can be mimickedwith boundary conditions over the scalar field [1]. The Schro¨dinger eigenvalue problem thatcharacterises the modes of a quantum scalar field living in the space1 M = [0, L]× RD−1 is−∆Mψω(x,y) = −(∆RD−1 +d2dx2)ψω(x,y) = ω2ψω(x,y). (1)Unitarity of the scalar quantum field theory requires that −∆M has to be selfadjoint andnon-negative [1]. After separation of variables we arrive to the eigenvalue problem in theorthogonal direction to the plates−d2fk(x)dx2= k2fk(x), x ∈ [0, L], (2)∗jose.munoz.castaneda@uva.es1We will denote coordinates in R×M as y = (t, x,y) ∈ R× [0, L]× RD−1. Making this election for thephysical space M implies that we assume the plates orthogonal to the OX-axis and placed at x = 0 andx = L respectively.1meanwhile the eigenvalue problem for the spatial directions parallel to the plates becomes−∆RD−1gk(y) = k2gk(y), with y,k ∈ RD−1. Since ∆RD−1 is selfadjoint, the obstructionfor ∆M to be selfadjoint is enclosed in d2/dx2 for x ∈ [0, L]. The most general boundaryconditions, including topology changes [1], for the normal modes of the quantum field are inthis case given by(f(0) + iλf ′(0)f(L)− iλf ′(L))= U(f(0)− iλf ′(0)f(L) + iλf ′(L))with U ∈MF ⊂ U(2), (3)where λ is parameter with units of length that characterises the plates (from now onwardswe will assume λ = 1), and MF is the set of U(2) matrices that ensures a non-negativespectrum for the eigenvalue problem (2) (see Ref. [1]). Taking for the group U(2) thestandard parametrisationU(α, β,n) = eiα [cos(β)I+ i sin(β)(n · σ)] , (α, β,n) ∈ [0, 2pi]× [−pi2,pi2]× S2, (4)being S2 the 2-sphere, and σ = (σ1, σ2, σ3) the Pauli matrices. The spectrum of discretemomenta k orthogonal to the plates is given by the zeroes of the function [1, 2]hU (k, L) = 2i eiα[((k2 − 1) cosβ + (k2 + 1) cosα) sin(kL)− 2k sinα cos(kL)− 2kn1 sin β] = −2k(det(U)− 1) cos(kL)− 2k(U12 + U21)+ i((k2 + 1)(det(U) + 1) + (k2 − 1)tr(U))sin(kL). (5)2 Casimir Energy between Plates in arbitrary Dimen-sionThe general formula for the finite Casimir energy per unit area between two isotropic homo-geneous plates mimicked by the boundary conditions (3) is given by,[1]E(D)(U)S= w(D)LD0LD − LD0∫∞0dk kD(L− L0 −ddklog(hU(ik, L)hU(ik, L0))), (6)where w(D) is a global factor just depending on the dimension D of the physical spacewhere the quantum field lives, and L0 a regularisation length that enables to subtract thesubdominant divergent terms [1]. The Eq.(6) as it appears in Ref. [1] gives the correct answerfor the quantum vacuum energy between plates in arbitrary dimension whenever there is nobackground potential and is valid as well when L0 > L. Nevertheless it is necessary toremove the length L0 to provide a formula that can be generalised whenever the operatorthat characterises the modes in the normal directions has the formK = −d2dx2+ V (x), x ∈ [0, L], (7)with V (x) a potential with compact support in [0, L] and no bound states. In order to removethe regularising length L0 we take the limit L0 →∞ in Eq. (6). After some straightforwardcalculations we obtainlimL0→∞−L0 +ddklog (hU(ik, L0)) =ddklog(12(k − i)2cU(−k + ik − i)), (8)2where cU(z) ≡ det(U)− tr(U)z + z2 = det(z − U). It is of note that12(k − i)2cU(−k + ik − i)= limL0→∞hU (ik, L0)ekL0≡ h∞U (ik). (9)Hence, introducing (8) into (6) we obtain a new formula for the Casimir interaction energyper unit area that is independent of the regularisation length L0:E(D)(U)S= −w(D)∫∞0dk kD[L−ddklog (hU(ik, L)) +ddklog (h∞U (ik))]. (10)This last equation is valid for the case in which there is a potential of compact supportbetween plates that depends only in x as shown in (7).3 Applications to the Quantum Vacuum Energy perUnit Cell for 1D CrystalsRecently the quantum vacuum energy per unit cell for a comb formed by potentials of com-pact support with no bound states was obtained in Ref. [3]. In particular it was computedthe quantum vacuum energy per unit cell of a scalar quantum field interacting with thebackground potentialV (x) =∑n∈Z(w0δ(x− na) + 2w1δ′(x− na)), (11)where the δ-δ′potential was defined as in Ref. [4]. In Ref. [3] it is demonstrated thata quantum scalar field propagating along a one-dimensional crystal formed as an array ofpotentials of compact support smaller than the lattice spacing is equivalent to a scalarquantum field theory in an interval of length given by the lattice spacing, interacting withthe potential of compact support that defines the comb centred in the middle point of theinterval. In addition, the boundary conditions that the modes of the quantum field mustsatisfy are given by Bloch semi-periodicity condition, which are quasi-periodic boundaryconditions,[3]Uqp =(0 eiθe−iθ 0), (12)where in this situation the parameter θ is to be interpreted as θ = −qL with q the quasi-momentum and L the lattice spacing, just after the limit L0 → ∞ is done as in Ref. [3].The secular equation that determines the spectrum for the normal modes in this case canbe written in terms of the scattering data {t(k), rR(k), rL(k)} for the potential of compactsupport when placed over the real line:fθ(k) ≡ cos(θ)−12t[e−ikL + eikL(t2 − rRrL)]. (13)In this case the limit in (8) replacing hU(ik) by fθ(ik) gives −∂k log (t(ik)) which enablesusing straightforward Eq. (10) to obtain a much more physically meaningful formula than3the one obtained in Ref. [3]Efin0 (θ) = −12pi∫∞0dk k[−L+ddklog (fθ(ik)) +ddklog (t(ik))]. (14)From this last formula, following the previous results from [3] we can obtain the energyper unit cell of the comb as a summation over the quasi-momentum of the discrete spectracharacterised by fθ = 0 :Efincomb =∫ pi−pidθ2piEfin0 (θ). (15)4 Concluding RemarksWe have provided a much more general formula for the Casimir energy with general boundaryconditions than the one originally obtained in Ref. [1], that is independent of the regular-isation length L0. This result enables to extend (10) to much more general situations inwhich there is a potential of compact support between plates as shown in Eq. (7). As anapplication we provided a formula for the quantum vacuum energy per unit cell of a scalarfield interacting with a one-dimensional crystal lattice built as an infinite array of identicalpotentials with compact support.AcknowledgmentsJMMC and LSS acknowledge MINECO (MTM2014-57129-C2-1-P) and Junta de Castilla yLeo´n (BU229P18 and VA137G18).References[1] M. Asorey and J. M. Mun˜oz-Castan˜eda, Nucl. Phys. B 874, 852 (2013).[2] J. M. Mun˜oz-Castan˜eda, K. Kirsten and M. Bordag Lett. Math. Phys 105, 523 (2015).[3] M. Bordag, J. M. Mun˜oz-Castan˜eda and L. Santamar´ıa-Sanz, Front. Phys., 7 38 (2019).[4] M. Gadella, J. Negro and L. M. Nieto, Phys. Lett. A 373, 1310 (2009).4

Revisiting the Casimir Energy with General Boundary Conditions, and applications in 1D Crystals

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Revisiting the Casimir Energy with General Boundary Conditions, and applications in 1D Crystals

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