Abstract
We consider a region of Minkowski spacetime bounded either by one or by two parallel, infinitely extended plates orthogonal to a spatial direction and a real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize these two systems within the algebraic approach to quantum field theory using the so-called functional formalism. As a first step we construct a suitable unital ∗-algebra of observables whose generating functionals are characterized by a labelling space which is at the same time optimal and separating and fulfils the F-locality property. Subsequently we give a definition for these systems of Hadamard states and we investigate explicit examples. In the case of a single plate, it turns out that one can build algebraic states via a pull-back of those on the whole Minkowski spacetime, moreover inheriting from them the Hadamard property. When we consider instead two plates, algebraic states can be put in correspondence with those on flat spacetime via the so-called method of images, which we translate to the algebraic setting. For a massless scalar field we show that this procedure works perfectly for a large class of quasi-free states including the Poincaré vacuum and KMS states. Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the extended algebras, built in globally hyperbolic subregions can be collected in a global counterpart only after a suitable deformation which is expressed locally in terms of a *-isomorphism. As a last step, we construct explicitly the two-point function and the regularized energy density, showing, moreover, that the outcome is consistent with the standard results of the Casimir effect.
Similar content being viewed by others
References
Bär, C.: Green-hyperbolic operators on globally hyperbolic spacetimes. Comm. Math. Phys. 333(3), 1585 (2015). ar**v:1310.0738 [math-ph]
Brunetti, R., Duetsch, M., Fredenhagen, K.: Perturbative Algebraic Quantum Field Theory and the Renormalization Groups. Adv. Theor. Math. Phys. 13, 1541 (2009). ar**v:0901.2038 [math-ph]
Brunetti, R., Fredenhagen, K., Ribeiro, P.L.: Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields. ar**v:1209.2148 [math-ph]
Benini, M., Dappiaggi, C., Hack, T.P.: Quantum Field Theory on Curved Backgrounds – A Primer. Int. J. Mod. Phys. A 28, 1330023 (2013). ar**v:1306.0527 [gr-qc]
Benini, M., Dappiaggi, C., Murro, S.: Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states. J. Math. Phys. 55, 082301 (2014). ar**v:1404.4551 [gr-qc]
Benini, M., Dappiaggi, C., Schenkel, A.: Quantum field theory on affine bundles. Annales Henri Poincare 15, 171 (2014). ar**v:1210.3457 [math-ph]
Benini, M., Dappiaggi, C., Schenkel, A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123 (2014). ar**v:1303.2515 [math-ph]
Benini, M.: Optimal space of linear classical observables for Maxwell k-forms via spacelike and timelike compact de Rham cohomologies. J. Math. Phys. 57, 053502 (2016). ar**v:1401.7563 [math-ph]
Brunetti, R., Fredenhagen, K.: Quantum Field Theory on Curved Backgrounds. ar**v:0901.2063 [gr-qc]
Brunetti, R., Fredenhagen, K., Verch, R.: The Generally covariant locality principle: A New paradigm for local quantum field theory. Commun. Math. Phys. 237, 31 (2003). ar**v:math-ph/0112041
Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization, 1St Edn. (Eur Math. Soc., Zürich, 2007)
Brown, L.S., Maclay, G.J.: Vacuum stress between conducting plates: An Image solution. Phys. Rev. 184, 1272 (1969)
Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Indag. Math 10, 261 (1948)
Callan, C.G. Jr., Coleman, S.R., jackiw, R.: A New improved energy - momentum tensor. Annals Phys. 59, 42 (1970)
Casimir, H.B.G., Polder, D.: The Influence of retardation on the London-van der Waals forces. Phys. Rev. 73, 360 (1948)
Deutsch, D., Candelas, P.: Boundary effects in quantum field theory. Phys. Rev. D 20, 3063 (1979)
Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219 (1980)
Fewster, C.J., Higuchi, A.: Quantum field theory on certain nonglobally hyperbolic space-times. Class. Quant. Grav. 51 (1996). ar**v:gr-qc/9508051
Fewster, C.J., Hunt, D.S.: Quantization of linearized gravity in cosmological vacuum spacetimes. Rev. Math. Phys. 25, 1330003 (2013). ar**v:1203.0261 [math-ph]
Fewster, C.J., Pfenning, M.J.: A Quantum weak energy inequality for spin one fields in curved space-time. J. Math. Phys. 44, 4480 (2003). ar**v:gr-qc/0303106
Fredenhagen, K., Rejzner, K.: Perturbative algebraic quantum field theory. ar**v:1208.1428 [math-ph]
Fulling, S.A., Reijsenaars, S.N.M.: Temperature, periodicity and horizons. Phys. Rep. 152, 135 (1987)
Fulling, S.A.: Aspects of Quantum Field Theory in Curved Space-time. London Math. Soc. Student Texts 17, 1 (1989)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals Series and Products, 7th. Academic, New York (2007)
Haag, R., Kastler, D.: An Algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)
Hack, T. -P.: On the Backreaction of Scalar and Spinor Quantum Fields in Curved Spacetimes. ar**v:1008.1776 [Gr-Qc], PhD thesis, Universität Hamburg (2010)
Herdegen, A.: Quantum backreaction (Casimir) effect. I. What are admissible idealizations?. Annales Henri Poincare 6, 657 (2005). ar**v:hep-th/0412132
Herdegen, A.: Quantum backreaction (Casimir) effect. II. Scalar and electromagnetic fields. Annales Henri Poincare 7, 253 (2006). ar**v:hep-th/0507023
Herdegen, A., Stopa, M.: Global versus local Casimir effect. Annales Henri Poincare 11, 1171 (2010). ar**v:1007.2139 [hep-th]
Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved space-time. Commun. Math. Phys. 223, 289 (2001). ar**v:gr-qc/0103074
Hollands, S., Wald, R.M.: Quantum fields in curved spacetime. ar**v:1401.2026 [gr-qc]
Hörmander, L.: The Analysis of Linear Partial Differential Operators I, 2nd. Springer, Berlin (1990)
Kay, B.S.: Casimir effect in quantum field theory. Phys. Rev. D 20, 3052 (1979)
Kay, B.S.: The Principle of locality and quantum field theory on (nonglobally hyperbolic) curved space-times. Rev. Math. Phys. SI 1, 167 (1992)
Kennedy, G., Critchley, R., Dowker, J.S.: Finite Temperature Field Theory with Boundaries: Stress Tensor and Surface Action renorMalization. Annals Phys. 125, 346 (1980)
ühn, H.K: Thermische Observablen gekoppelter Felder in Casimir-Effekt, Diploma Thesis in German, Universität Hamburg. available at www.desy.de/uni-th/lqp/psfiles/dipl-kuehn.ps.gz (2005)
Lee, J.M.: Introduction to smooth manifolds. Springer, Berlin (2000)
Milton, K.A.: The Casimir Effect: Physical Manifestations of Zero-point Energy. World Scientific (2001)
Moretti, V.: Comments on the stress energy tensor operator in curved space-time. Commun. Math. Phys. 232, 189 (2003). ar**v:gr-qc/0109048
Niekerken, O.: Quantum and Classical Vacuum Forces at Zero and Finite Temperature, Diploma Thesis in German. Universität Hamburg, available at http://www-library.desy.de/preparch/desy/thesis/desy-thesis-09-019.pdf (2009)
Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529 (1996)
Radzikowski, M.J.: A Local to global singularity theorem for quantum field theory on curved space-time. Commun. Math. Phys. 180, 1 (1996)
Sahlmann, H., Verch, R.: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214, 705 (2000). ar**v:math-ph/0002021
Sanders, K., Dappiaggi, C., Hack, T.-P.: Electromagnetism, Local Covariance, the Aharonov-Bohm Effect and Gauss Law. Commun. Math. Phys. 328, 625 (2014). ar**v:1211.6420 [math-ph]
Sommer, C.: Algebraische Charakterisierung von Randbedingungen in der Quantenfeldtheorie , Diploma Thesis in German, Universität Hamburg. available at http://www.desy.de/uni-th/lqp/psfiles/dipl-sommer.ps.gz (2006)
Sopova, V., Ford, L.H.: Energy density in the Casimir effect. Phys. Rev. D 66, 045026 (2002)
Wald, R.M.: General Relativity, 1st. The University of Chicago Press, Chicago (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dappiaggi, C., Nosari, G. & Pinamonti, N. The Casimir Effect from the Point of View of Algebraic Quantum Field Theory. Math Phys Anal Geom 19, 12 (2016). https://doi.org/10.1007/s11040-016-9216-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-016-9216-y