We apply quantum field theory in quantum space-time techniques to study the
Casimir effect for large spherical shells. As background we use the recently
constructed exact quantum solution for spherically symmetric vacuum space-time
in loop quantum gravity. All calculations are finite and one recovers the usual
results without the need of regularization or renormalization. This is an
example of how loop quantum gravity provides a natural resolution to the
infinities of quantum field theories.Comment: 4 page

Gambini, Rodolfo

Olmedo, Javier

Pullin, Jorge

Classical and Quantum Gravity

English

arXiv

We apply quantum field theory in quantum space-time techniques to study the Casimir effect for large spherical shells. As background we use the recently constructed exact quantum solution for spherically symmetric vacuum space-time in loop quantum gravity. All calculations are finite and one recovers the usual results without the need for regularization or renormalization. This is an example of how loop quantum gravity provides a natural resolution to the infinities of quantum field theories

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 6-11-2015 Casimir effect in a quantum space-time Rodolfo Gambini Universidad de Montevideo Javier Olmedo Universidad de Montevideo Jorge Pullin Louisiana State University Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Gambini, R., Olmedo, J., & Pullin, J. (2015). Casimir effect in a quantum space-time. Classical and Quantum Gravity, 32 (11) https://doi.org/10.1088/0264-9381/32/11/115002 This Article is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. arXiv:1410.4479v1  [gr-qc]  16 Oct 2014Casimir effect in a quantum space-timeRodolfo Gambini1, Javier Olmedo1,2, Jorge Pullin21. Instituto de F́ısica, Facultad de Ciencias, Iguá 4225, esq. Mataojo, 11400 Montevideo, Uruguay.2. Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001We apply quantum field theory in quantum space-time techniques to study the Casimir effectfor large spherical shells. As background we use the recently constructed exact quantum solutionfor spherically symmetric vacuum space-time in loop quantum gravity. All calculations are finiteand one recovers the usual results without the need of regularization or renormalization. This isan example of how loop quantum gravity provides a natural resolution to the infinities of quantumfield theories.Quantum field theory has been developed over the years with a series of techniques for dealing with the infinitiesthat arise, namely regularization and renormalization. On curved space-times, however, certain problems remainwhich have hampered efforts, for instance, to compute the back reaction of black hole evaporation due to Hawkingradiation [1]. It has been a long held belief that when a theory of quantum space-time is developed, the point of viewon these issues may shift and some of the problems solved. Loop quantum gravity has been steadily developed andin recent years has been used to treat vacuum spherically symmetric space-times. In particular the space of physicalstates of the theory has been found in closed form [2, 3] for that case. One can therefore study quantum field theoriesliving on such a quantum space-time. A first step has already been taken in computing Hawking radiation in suchan approach [4]. A remarkable characteristic is that the discreteness of the quantum space-time acts naturally toregulate the infinities of the quantum field theory. It is therefore of interest to study how such new finite quantumfield theories address potentially measurable effects like the Casimir effect [5, 6]. The latter is a striking phenomenonstemming from the energy of the vacuum in quantum field theory. Generically speaking it implies that when onecreates a bounded region in space-time, forces will arise on the boundaries due to the suppression of certain quantummodes that the boundary conditions impose on the quantum fields. From a technical point of view continuumquantum field theory requires additional ingredients in order to deal with divergences, like regularization by pointsplitting and renormalization, exponential cutoffs or even the use of functional analysis techniques like the Riemannzeta function for the computation of the infinite (usually divergent) mode sums. Though these techniques give finiteresults independent of the scheme of regularization (and renormalization) employed, we still lack a description freeof these drawbacks. The intention of this paper is to address some of these questions and their consequences fora quantum field theory in a quantum space-time. Since loop quantum gravity puts at our disposal [2, 3] the exactquantum states corresponding to an effective background that is spherically symmetric (generically containing a blackhole) we will study the Casimir effect between two spherical shells. They will be embedded inside a bounded region(provided, for instance, by two auxiliary spherical shells) such that it covers an extensive portion of the entire space-time. Although we can carry out our study throughout the whole space-time, to simplify calculations we will restrictthe study to the interior of the shells and assume they are far away from the black hole horizon that generically ispresent in spherically symmetric vacuum space-times, so one can use a planar approximation in the computation ofthe Casimir effect. We will consider a quantum scalar field living in a quantum spacetime as discussed in [4], but thistime interacting with two neighboring shells where the field vanishes. The quantization of the field is done withinthe Fock representation whereas the background is quantized via loop quantum gravity in spherical symmetry. Thebackground quantum space-time [2, 3] is characterized by a vector ~k that corresponds to the valences of the links ofthe spin network it is based on and the value of the mass at spatial infinity M . The components of ~k are proportionalto the values of the areas of the spheres of symmetry intersected by the spin network Ai = 4πℓ2Planckki. If one considersstates that are eigenstates with precise values of ~k,M , as opposed to superpositions, the main effect of considering aquantum field theory on quantum space-time treatment is to discretize the equations for the scalar field (since M isa continuous parameter we strictly speaking consider a narrow superposition of values of M). The field takes valuesat the vertices of the spin networks only, as it is the usual treatment of scalar fields in loop quantum gravity. If oneconsiders quantum states that are in a superposition, there exist additional effects to the ones discussed here, butit does not change the main conclusions, i.e. that all quantities are well defined without infinities. We will assumethat the quantum space-time is such that it approximates well a continuum smooth geometry. This places someconstraints on the values of ~k: i) one would like them to grow monotonically to avoid coordinate singularities, andii) to have small differences between successive components to avoid large “jumps” in the values of the areas of thespheres of symmetry. We will also for simplicity take the changes in adjacent values of ~k along the spin network toyield a uniform spacing in the radial coordinate. Since r2i = ℓ2Planckki, the difference between two successive values ofthe radial coordinate has to be at least ℓ2Planck/(2r). For instance, in the exterior of a black hole one can choose auniform spacing with the lowest possible separation ∆ = ℓ2Planck/(2M). Notice that the bound allows lattice spacings2that are considerably smaller than the Planck scale, even for small black holes. This in turn implies that the discreteequations for the quantum fields are extremely well approximated by equations in the continuum, except in theextreme trans-Planckian ultraviolet region of the modes of the fields.We will proceed with the computation of the Casimir effect. On the one hand, we consider a the scalar fieldinteracting with two spherical shells of radii r0 and r0 + L. We demand that the scalar field vanish at the shells asis usual for calculations of the Casimir effect. Besides, since we assume that we are in the asymptotic region wherethe background quantum metric is flat, we require r0 ≫ 2M . Otherwise the contributions due to the Schwarszchildgeometry must be considered. For convenience, we will carry out all the calculations with respect to an observer atrest in r = r0. On the other hand, we must consider contributions of the field outside the shells as well. To do this,we place two auxiliary shells at r0 + L0 and r0 − L1, such that L0 ≫ L. The value of L1 can be selected arbitrarily,covering a large portion of space-time, but still in the asymptotic region so we can still make the approximation thatthe gravitational potential in the wave equation vanishes.We will adopt a mode decomposition for the scalar field of the form un,ℓ,m =exp(−iωt)Rℓ(ω, rj)Yℓ,m(θ, ϕ)/(√2πωrj), where Yℓ,m(θ, ϕ) are the standard spherical harmonics. Besides, rj = r0+j∆is the discrete radial coordinate, with j a suitable integer and where ∆ is the separation of the vertices of the spinnetwork of the quantum space-time. Therefore, the number of vertices on each region is given by the quotient of itslength over the step. For instance, inside the slab of length L we have NL = L/∆ vertices.The radial modes fulfill the difference equation,Rj+1 − 2Rj +Rj−1∆2+ ω2Rj −ℓ (ℓ+ 1)r2jRj = 0, (1)with Dirichlet boundary conditions, where we have neglected the potential due to the curvature of the Schwarzschildspace-time since we assume the shells are far away from the horizon. In the continuum, the corresponding solutionsare linear combinations of Bessel functions, and the dispersion relation is not known in closed form but involves(unbounded) discrete frequencies. On the lattice, the radial functions will approximate very well the continuum limit.Then, the field decomposition is well approximated by these modes asφ (t, rj) =√2πNL∆NL−1∑n=12r0∆∑ℓ=0ℓ∑m=−ℓan,ℓ,me−iωn,ℓt√2πωn,ℓsin(πnj∆NL∆)rjYℓ,m (θ, ϕ) +a†n,ℓ,meiωn,ℓt√2πωn,ℓsin(πnj∆NL∆)rjY ∗ℓ,m (θ, ϕ) , (2)where the dispersion relation is in very good agreement withω2n,ℓ ≃4∆2sin2(∆kn2)+ℓ(ℓ+ 1)r20, (3)and kn = (nπ)/(NL∆). The prefactor√2πNL∆implies the modes are normalized. Moreover, and for the sake ofsimplicity, we have replaced Bessel functions by sinusoidal ones. For modes with small ℓ this is a good approximation,but a more elaborate calculation would be needed for modes of high angular momentum. Preliminary numericalstudies justify these approximations. Let us notice that the sum in n is finite due to the finiteness of the slab. Thesum in ℓ is truncated since the dominant contribution in our calculations will be due to modes of angular momentumℓ lower than 2r0/∆ (the maximum mode frequency on the lattice times the radius r0). This bound is due to theasymptotic behavior of the Bessel functions for high angular momentum. It is not oscillatory anymore but exponential.Therefore no eigenfunctions can be found compatible with the Dirichlet boundary conditions. In consequence, we willdisregard them in our calculations.Now, to compute the force due to the Casimir effect we will need to evaluate the integral of the expectation value ofthe T00 component of the stress-energy tensor of the field in the region between the shells and compute its derivativewith respect to the separation L between them. The relevant component of the energy-momentum tensor in thecontinuum [8] is given byT00 =12φ̇2 +12∇φ · ∇φ = 12φ̇2 − 12φ∇2φ+ 12∇ (φ∇φ) , (4)where the dot indicates derivation with respect time, ∇φ = (∂rφ) êr + 1r (∂θφ) êθ + 1r sin θ (∂ϕφ) êϕ is the standardgradient and ∇2φ = ∂2rφ + L̂2r2φ is the Laplace operator in terms of the square of the standard angular momentumoperator L̂2. Explicitly, on the basis of spherical harmonics it fulfills L̂2Yℓ,m(θ, ϕ) = −ℓ(ℓ+1)Yℓ,m(θ, ϕ). If we integratethis energy density inside the shells, the total divergence contributes at the boundary, and it can be neglected since the3field vanishes on the shells (in the discrete theory this might not be true but the nonvanishing contributions are of theorder of the step of the lattice and can be disregarded). Besides, the equations of motion can be employed to replaceφ∇2φ = φφ̈. At the end of the day we just need to compute the expectation value of T00 = 12 φ̇2− 12φφ̈ with respect toa vacuum state compatible with the Dirichlet boundary conditions imposed by the shells. The corresponding creationand annihilation operators satisfy [an,ℓ,m, a†n′,ℓ′,m′ ] = δn,n′δℓ,ℓ′δm,m′ . In particular, the vacuum state |0〉L for the slabof width L is defined such that an,ℓ,m|0〉L = 0.To compute the expectation value of this component of the stress-energy tensor in this region we would need to con-centrate on the derivatives of the field in the Green’s function associated with the slab GL+(x, x′) = 〈0L|φ(x)φ(x′)|0L〉.On the one hand, after some calculations,GL+ (x;x′) ≃ 1LNL−1∑n=12r0∆∑ℓ=0ℓ∑m=−ℓe−iωn,ℓ(t−t′)ωn,ℓsin (knz)rjsin (knz′)rj′Yℓ,m (θ, ϕ) Y∗ℓ,m (θ′, ϕ′) (5)with z = r − r0, and similarly for z′, and ωn,ℓ is given in (3). It gives a good approximation for the discrete Green’sfunction.We can calculate the first contribution to the T00 component of the stress-energy tensor. For it, we only needto evaluate the Green’s function for coincident radial and angular coordinates (but different times), and employ theidentityℓ∑m=−ℓYℓ,m (θ, ϕ)Y∗ℓ,m (θ, ϕ) =2ℓ+ 14π. (6)We also replace the sum in ℓ by an integral as well as we only keep the dominant terms. On then obtains the density〈0L|φ̇2|0L〉 =∂2∂t ∂t′GL+∣∣∣∣x=x′=14πLr20NL−1∑n=12r203sin2 (knz)[8∆3+12∆3sin2(∆kn2)− 8∆3sin3(∆kn2)+3∆3sin4(∆kn2)+ . . .]. (7)In this expression we have taken rj = r0 for simplicity. Let us emphasize that these sums can be explicitly computedand give finite results at any z ∈ [0, L], contrary to the usual situation in continuum quantum field theory (due to thedivergences at the boundary). We have also replaced sin(2πz/∆) = 0 and cos(2πz/∆) = 1 anywhere, since z = ∆j.For the remaining contributions in T00, recalling that we have employed the (discrete) equations of motion of thefield, it is straightforward to see that〈0L| − φφ̈|0L〉∣∣∣= − ∂2∂2t′GL+∣∣x=x′= 〈0L|φ̇2|0L〉. (8)With this in mind, we can compute the T00-component of the stress-energy tensor in the region between the smallestshells,〈0L|T00|0L〉 = 〈0L|φ̇2|0L〉, (9)by means of (7) and (8), though its explicit expression is rather lengthy.For the scalar field inside the region with separation L̃0 = (L0 − L) the same construction can be adopted withoutadditional considerations, just replacing in the densities (7) and (8) the width L → (L0−L) and afterwards z → (z−L).About the slab of width L1, since we are interested in variations of the energy of the system with respect to L, thiscontribution will be a constant and we will not give it explicitly. The corresponding expectation values of the T00component of the stress-energy tensor will be then 〈0L̃0 |T00|0L̃0〉 and 〈0L1 |T00|0L1〉, respectively.To obtain the Casimir force, we simply calculate (minus) the derivative of the energy of the system with respect toL. In particular, in the limit in which L0 ≫ L and for small ∆, we get for the force per unit area,P = − ddL(∫ 0−L1dz〈0L1 |T00|0L1〉+∫ L0dz〈0L|T00|0L〉+∫ L0Ldz〈0L̃0 |T00|0L̃0〉)= − π2480L4+O(L−10 ) +O(∆). (10)4We then obtain the exact result [8, 9]. Notice that since we are working in the asymptotic region, it coincides withthe planar case as well. Let us emphasize that for the improved stress-energy tensor [7, 8], since it differs from thecanonical one (in the continuum) by a divergence, the very same result is obtained (up to corrections of the orderof the step of the lattice, precisely the contributions we are neglecting). For the sake of completeness, we have alsocarried out the same calculation for the s-mode of the scalar field (ℓ = 0), obtaining the exact result found in thecontinuum for the Casimir effect in 1+1 dimensions. It is remarkable that without the cutoff that arise naturally fromthe quantum discreteness of the space-time this sum of energies would be ill defined.It is worth commenting that the configuration we have adopted here to deal with the problem of the Casimir effectmight not have been employed before. We obtain the force as the variation of the energy of a closed system, withoutany (possibly artificial) subtraction of an hypothetical external vacuum energy. In fact, we need to add, nor subtract,appropriately the energy of both slabs, considering the system as a whole, and take variations of the total energy inorder to achieve a meaningful physical result. Therefore, our example sheds light in the understanding of one of themost prominent and confirmed predictions of quantum field theory.It is known that the Casimir force for spheres and planes is finite, but it diverges for more general situations. Itwould be interesting to see if quantum field theory in quantum space-times yields finite results in those cases andcompare them with (future) experiment. Moreover, of great interest will be to extend these results to address thetrace anomaly, which seems eminently feasible with the techniques we used. Also our study has application in blackhole evaporation [10], where the expectation value of the stress-energy tensor with respect to a suitable vacuum stateplays an essential role for the computation of the backreation. Our preliminary calculations yield a finite polarizationof the vacuum, avoiding as well the divergences of the continuum theory.We have shown that one can compute the Casimir effect using quantum field theory in quantum space-time tech-niques on the exact quantum spacetime of spherically symmetric vacuum gravity of loop quantum gravity. Allcalculations result finite and no regularization nor renormalization is needed. Remarkably, we reproduce the result ofquantum field theory in curved classical space-time which require regularization and renormalization. Some authorshave speculated [8] that gravitational effects might address those issues and this paper suggests that they may. Ourcalculation is an example in which quantum gravity successfully deals with the singularities that arise in ordinaryquantum field theory without additional hypothesis about them and yield sensible physical results.This work was supported in part by grant NSF-PHY-1305000, funds of the Hearne Institute for Theoretical Physics,CCT-LSU and Pedeciba.[1] R. Wald, “Quantum field theory in curved space-time and black hole thermodynamics”, University of Chicago Press,Chicago, (1994).[2] R. Gambini and J. Pullin, “Loop quantization of the Schwarzschild black hole,” Phys. Rev. Lett. 110, no. 21, 211301(2013) [arXiv:1302.5265 [gr-qc]].[3] R. Gambini, J. Olmedo and J. Pullin, “Quantum black holes in Loop Quantum Gravity,” Class. Quant. Grav. 31, 095009(2014) [arXiv:1310.5996 [gr-qc]].[4] R. Gambini and J. Pullin, “Hawking radiation from a spherical loop quantum gravity black hole,” Class. Quant. Grav. 31,115003 (2014) [arXiv:1312.3595 [gr-qc]].[5] B. DeWitt, Phys. Rep. 19, 295 (1975).[6] K. Milton, “The Casimir effect”, World Scientific, Singapore (2001).[7] C. Callan, S. Coleman, R. Jackiw, Ann. Phys. 59, 4273 (1970).[8] S. Fulling, “Aspects of quantum field theory in curved space-time”, Cambridge University Press, Cambridge (1989/1996).[9] M. Ozcan, “Scalar Casimir effect between two concentric spheres,” Int. J. Mod. Phys. A 27, 1250082 (2012) [arXiv:1207.4183[quant-ph]].[10] P. Candelas, Phys. Rev. D 21, 2125 (1980).

Casimir effect in a quantum space-time

Rodolfo Gambini

Javier Olmedo

Jorge Pullin

Crossref

Casimir effect in a quantum space–time

https://digitalcommons.lsu.edu/cgi/viewcontent.cgi?article=5404&context=physics_astronomy_pubs

Casimir effect in a quantum space-time

Abstract

Similar works

Full text

Available Versions

LSU Scholarly Repository (Louisiana State Univ.)

Crossref