Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 094, 23 pages      arXiv:0706.2831      https://doi.org/10.3842/SIGMA.2007.094
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Vacuum Energy as Spectral Geometry

Stephen A. Fulling
Department of Mathematics, Texas A&M University, College Station, Texas, 77843-3368, USA

Received June 21, 2007, in final form September 14, 2007; Published online September 26, 2007

Abstract
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and total energies are demonstrated. This material provides background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approximation and other methods.

Key words: Casimir; periodic orbit; energy density; cylinder kernel.

pdf (338 kb)   ps (234 kb)   tex (66 kb)

References

  1. Balian R., Bloch C., Distribution of eigenfrequencies for the wave equation in a finite domain. III. Eigenfrequency density oscillations, Ann. Physics 69 (1972), 76-160.
  2. Balian R., Bloch C., Solution of the Schrödinger equation in terms of classical paths, Ann. Physics 85 (1974), 514-545.
  3. Balian R., Duplantier B., Electromagnetic waves near perfect conductors. I. Multiple scattering expansions. Distribution of modes, Ann. Physics 104 (1977), 300-335.
    Balian R., Duplantier B., Electromagnetic waves near perfect conductors. II. Casimir effect, Ann. Physics 112 (1978), 165-208.
  4. Bär C., Moroianu S., Heat kernel asymptotics for roots of generalized Laplacians, Internat. J. Math. 14 (2003), 397-412.
  5. Bender C.M., Hays P., Zero-point energy of fields in a finite volume, Phys. Rev. D 14 (1976), 2622-2632.
  6. Bernasconi F., Graf G.M., Hasler D., The heat kernel expansion for the electromagnetic field in a cavity, Ann. Henri Poincaré 4 (2003), 1001-1013, math-ph/0302035.
  7. Bordag M., Mohideen U., Mostepanenko V.M., New developments in the Casimir effect, Phys. Rep. 353 (2001), 1-205, quant-ph/0106045.
  8. Boyer T.H., Van der Waals forces and zero-point energy for dielectric and permeable materials, Phys. Rev. A 9 (1974), 2078-2084.
  9. Brack M., Bhaduri R.K., Semiclassical physics, Addison-Wesley, Reading, 1997.
  10. Branson T.P., Gilkey P.B., The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), 245-272.
  11. Brown L.S., Maclay G.J., Vacuum stress between conducting plates: an image solution, Phys. Rev. 184 (1969), 1272-1279.
  12. Brownell F.H., Extended asymptotic eigenvalue distributions for bounded domains in n-space, J. Math. Mech. 6 (1957), 119-166.
  13. Casimir H.B.G., On the attraction between two perfectly conducting plates, Konink. Nederl. Akad. Weten., Proc. Sec. Sci. 51 (1948), 793-795.
  14. Cognola G., Vanzo L., Zerbini S., Regularization dependence of vacuum energy in arbitrarily shapted cavities, J. Math. Phys. 33 (1992), 222-228.
  15. Colin de Verdière Y., Spectre du laplacien et longuers des géodésiques périodiques. II, Compos. Math. 27 (1973), 159-184.
  16. Combescure M., Ralston J., Robert D., A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Comm. Math. Phys. 202 (1999), 463-480, math-ph/9807005.
  17. DeWitt B.S., Quantum field theory in curved spacetime, Phys. Rep. 19 (1975), 295-357.
  18. Dowker J.S., 1. The counting function. 2. Hybrid boundary conditions, Nuclear Phys. B Proc. Suppl. 104 (2002), 153-156.
  19. Duistermaat J.J., Guillemin V.W., The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39-79.
  20. Estrada R., The Cesáro behavior of distributions, Proc. Roy. Soc. London A 454 (1998), 2425-2443.
  21. Estrada R., Fulling S.A., Distributional asymptotic expansions of spectral functions and of the associated Green kernels, Electron. J. Differential Equations 1999 (1999), 07, 37 pages, funct-an/9710003.
  22. Estrada R., Fulling S.A., Functions and distributions in spaces with thick points, in Special Issue for the Tricentennial of Leonard Euler, Internat. J. Appl. Math. Stat., to appear, available at this URL http://www.math.tamu.edu/~fulling/fatpts4.pdf.
  23. Estrada R., Fulling S.A., Kaplan L., Kirsten K., Liu Z.H., Milton K.A., Classical paths and quantum vacuum stress in rectangular cavities, in preparation.
  24. Ford L.H., Svaiter N.F., Vacuum energy density near boundaries, Phys. Rev. D 58 (1998), 065007, 8 pages, quant-ph/9804056.
  25. Fulling S.A., The local geometric asymptotics of continuum eigenfunction expansions I, SIAM J. Math. Anal. 13 (1982), 891-912.
  26. Fulling S.A., Systematics of the relationship between vacuum energy calculations and heat kernel coefficients, J. Phys. A: Math. Gen. 36 (2003), 6857-6873, quant-ph/0302117.
  27. Fulling S.A., Global and local vacuum energy and closed orbit theory, in Quantum Field Theory under the Influence of External Conditions (6th Workshop ..., Norman, 2003). Editor K.A. Milton, Rinton Press, Princeton, 2004, 166-174, available at this URL http://www.math.tamu.edu/~fulling/funorman.pdf.
  28. Fulling S.A., Gorbar E.V., Romero C.T., Spectral Riesz-Cesàro means: How the square root function helps us to see around the world, in Mathematical Physics and Quantum Field Theory (E.H. Wichmann Symposium, Berkeley, 1999), Electron. J. Differ. Equ. Conf. 04 (2000), 87-101, available at this URL http://ejde.math.unt.edu/conf-proc/04/f3/abstr.html.
  29. Fulling S.A., Gustafson R.A., Some properties of Riesz means and spectral expansions, Electron. J. Differential Equations 1999 (1999), 06, 39 pages, physics/9710006.
  30. Fulling S.A., Wilson J.H., Vacuum energy and closed orbits in quantum graphs, in Program on Analysis on Graphs and Its Applications (Isaac Newton Institute, Cambridge, 2007), Editor P. Kuchment, to appear.
  31. Gilkey P.B., The spectral geometry of the higher order Laplacian, Duke Math. J. 47 (1980), 511-528.
  32. Gilkey P.B., Invariance theory, the heat equation and the Atiyah-Singer index theorem, CRC Press, Boca Raton, 1995.
  33. Gilkey P.B., Grubb G., Logarithmic terms in asymptotic expansions of heat operator traces, Comm. Partial Differential Equations 23 (1998), 777-792.
  34. Gradshteyn I.S., Ryzhik I.M., Table of Integrals, series, and products, 5th ed., Academic Press, New York, 1980.
  35. Greiner P., An asymptotic expansion for the heat equation, Arch. Ration. Mech. Anal. 41 (1971), 163-218.
  36. Gutzwiller M.C., Periodic orbits and classical quantization conditions, J. Math. Phys. 12 (1971), 343-358 (and related papers).
  37. Hays P., Vacuum fluctuations of a confined massive scalar field in two dimensions, Ann. Physics 121 (1979), 32-46.
  38. Helson H., Harmonic analysis, Addison-Wesley, Reading, 1983.
  39. Hertzberg M.P., Jaffe R.L., Kardar M., Scardicchio A., Casimir forces in a piston geometry at zero and finite temperatures, arXiv:0705.0139.
  40. Hörmander L., The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218.
  41. Iannuzzi D., Lisanti M., Munday J.N., Capasso F., The design of long-range quantum electrodynamical forces and torques between macroscopic bodies, Solid State Commun. 135 (2005), 618-626.
  42. Isham C.J., Twisted quantum fields in a curved space-time, Proc. Roy. Soc. London A 362 (1978), 383-404.
  43. Jaekel M.T., Reynaud S., Casimir force between partially transmitting mirrors, J. Physique I 1 (1991), 1395-1409.
  44. Kac M., Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), Part II, 1-23.
  45. Kirsten K., Spectral functions in mathematics and physics, Chapman & Hall/CRC, Boca Raton, 2002.
  46. Lebedev S.L., Casimir effect in the presence of an elastic boundary, Zh. Eksp. Teor. Fiz. 110 (1996), 769-792 (English transl.: JETP 83 (1996), 423-434).
  47. Littlejohn R.G., The Van Vleck formula, Maslov theory, and phase space geometry, J. Stat. Phys. 68 (1992), 7-50.
  48. Lukosz W., Electromagnetic zero-point energy shift induced by conducting closed surfaces, Z. Physik 258 (1973), 99-107.
  49. Mazzitelli F.D., Sánchez M.J., Scoccola N.N., von Stecher J., Casimir interaction between two concentric cylinders: exact versus semiclassical results, Phys. Rev. A 67 (2003), 013807, 11 pages, quant-ph/0209097.
  50. Milton K.A., The Casimir effect: physical manifestations of zero-point energy, World Scientific, Singapore, 2001.
  51. Minakshisundaram S., Eigenfunctions on Riemannian manifolds, J. Indian Math. Soc. 17 (1953), 159-165.
  52. Plunien G., Müller B., Greiner W., The Casimir effect, Phys. Rep. 143 (1986), 87-193.
  53. Romeo A., Saharian A.A., Casimir effect for scalar fields under Robin boundary conditions on plates, J. Phys. A: Math. Gen. 35 (2002), 1297-1320.
  54. Scardicchio A., Jaffe R.L., Casimir effects: an optical approach. I. Foundations and examples, Nuclear Phys. B 704 (2005), 552-582, quant-ph/0406041.
    Scardicchio A., Jaffe R.L., Casimir effects: an optical approach. II. Local observables and thermal corrections, Nuclear Phys. B 743 (2006), 249-275, quant-ph/0507042.
  55. Schaden M., Sign and other aspects of semiclassical Casimir energies, Phys. Rev. A 73 (2006), 042102, 16 pages, hep-th/0509124.
  56. Schaden M., Semiclassical electromagnetic Casimir self-energies, hep-th/0604119.
  57. Schaden M., Spruch L., Infinity-free semiclassical evaluation of Casimir effects, Phys. Rev. A 58 (1998), 935-953.
  58. Stewart J., Calculus, 3rd ed., Brooks/Cole, Pacific Grove, 1995.
  59. Wilson J.H., Vacuum energy in quantum graphs, Undergraduate Research Fellow Thesis, Texas A&M University, 2007, available at this URL http://handle.tamu.edu/1969.1/5682.
  60. Wolfram S., The Mathematica book, 4th ed., Cambridge University Press, Cambridge, 1999.

Previous article   Next article   Contents of Volume 3 (2007)