Abstract
We study the options for boundary conditions at the conical singularity for quantum mechanics on a two-dimensional cone with deficit angle ≦ 2π and for classical and quantum scalar fields propagating with a translationally invariant dynamics in the 1+3 dimensional spacetime around an idealized straight infinitely long, infinitesimally thin cosmic string. The key to our analysis is the observation that minus-the-Laplacian on a cone possesses a one-parameter family of selfadjoint extensions. These may be labeled by a parameterR with the dimensions of length—taking values in [0, ∞). ForR=0, the extension is positive. WhenR≠0 there is a bound state. Each of our problems has a range of possible dynamical evolutions corresponding to a range of allowedR-values. They correspond to either finite, forR=0, or logarithmically divergent, forR≠0, boundary conditions at zero radius. Non-zeroR-values are a satisfactory replacement for the (mathematically ill-defined) notion of δ-function potentials at the cone's apex.
We discuss the relevance of the various idealized dynamics to quantum mechanics on a cone with a rounded-off centre and field theory around a “true” string of finite thickness. Provided one is interested in effects at sufficiently large length scales, the “true” dynamics will depend on the details of the interaction of the wave function with the cone's centre (/field with the string etc.) only through a single parameterR (its “scattering length”) and will be well-approximated by the dynamics for the corresponding idealized problem with the sameR-value. This turns out to be zero if the interaction with the centre is purely gravitational and minimally coupled, but non-zero values can be important to model nongravitational (or non-minimally coupled) interactions. Especially, we point out the relevance of non-zeroR-values to electromagnetic waves around superconducting strings. We also briefly speculate on the relevance of theR-parameter in the application of quantum mechanics on cones to 1+2 dimensional quantum gravity with massive scalars.
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References
Vilenkin, A.: In: 300 Years of Gravity. Hawking, S.W., Israel, W. (eds.). Cambridge: Cambridge University Press 1987
Aryal, M., Ford, L.H., Vilenkin, A.: Phys. Rev. D34, 2263 (1986)
Linet, B.: J. Gen. Rel. Grav.17, 1109 (1985)
Smith, A.G.: Gravitational effects of an infinite straight cosmic string on classical and quantum fields: self-forces and vacuum fluctuations. Tufts University Preprint, June 1986
Helliwell, T.M., Konkowski, D.A.: Phys. Rev. D34, 1918 (1986)
Linet, B.: Phys. Rev. D35, 536 (1987)
Frolov, V.P., Serebrinary, E.M.: Phys. Rev. D35, 3779 (1987)
Dowker, J.S.: Phys. Rev. D36, 3095 (1987)
Dowker, J.S.: Class. Quantum Grav.4, L157 (1987)
Davies, P.C.W., Sahni, V.: Class. Quantum Grav.5, 1 (1988)
De Witt, B.S.: Phys. Rep.19, 295 (1975)
Kay, B.S.: Phys. Rev. D20, 3052 (1979)
't Hooft, G.: Commun. Math. Phys.117, 685 (1988)
Deser, S., Jackiw, R.: Commun. Math. Phys.118, 495 (1988)
de Barros Cobra Damgaard, B., Roemer, H.: Lett. Math. Phys.13, 189 (1987)
Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. San Francisco: Freeman 1973
Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. I. Functional analysis. New York, London: Academic Press 1972
Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. II. Fourier analysis, self-adjointness. New York, London: Academic Press 1975
Stakgold, I.: Green's functions and boundary value problems. New York: Wiley 1979
Wald, R.M.: J. Math. Phys.20, 1056 (1979); Erratum, J. Math. Phys.21, 218 (1980)
Kay, B.S., Wald, R.M.: Class. Quantum Grav.4, 893 (1987)
Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasi-free states on spacetime with a bifurcate Killing horizon. Phys. Rep., in press, 1991
Segal, I.E.: Mathematical problems of relativistic physics. Providence: American Mathematical Society 1963
Kay, B.S.: Commun. Math. Phys.62, 55 (1978)
Studer, U.M.: Quantum mechanics and quantum field theory on spacetimes with conical singularities. Diploma Thesis, ETH Zürich 1988
Chernoff, P.R.: J. Funct. Anal.12, 401 (1973)
Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955
Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. Berlin, Heidelberg, New York: Springer 1988
Linet, B.: Phys. Rev. D33, 1833 (1986)
Kay, B.S.: Springer Lect. Notes in Math., Vol. 905, p. 272. Berlin, Heidelberg, New York: Springer 1982
Dowker, J.S.: J. Phys. A10, 115 (1977)
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. New York, London: Academic Press 1980
de Sousa Gerbert, P., Jackiw, R.: Commun. Math. Phys.124, 229 (1989)
Dimock, J., Kay, B.S.: Ann. Phys. (NY)175, 366 (1987)
Gott, J.R.: Ap. J.288, 422 (1985)
Hiscock, W.A.: Phys. Rev. D31, 3288 (1985)
Frolov, V.P., Israel, W., Unruh, W.G.: Phys. Rev. D39, 1084 (1989)
Hiscock, W.A.: Phys. Lett. B188, 317 (1987)
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. III. Scattering Theory. New York, London: Academic Press 1979
Newton, R.G.: Scattering theory of waves and particles. New York: McGraw-Hill 1966
Gibbons, G.W., Ruiz Ruiz, F., Vachaspati, T.: Commun. Math. Phys.127, 295–312 (1990)
Kay, B.S.: When can a small object have a big effect at large scales?, in preparation
Kay, B.S.: Theorems on sequences of Schrödinger potentials with supports which decrease to zero, in preparation
Alford, M.G., March-Russell, J., Wilczek, F.: Nucl. Phys. B328, 140 (1989)
Blatt, J.M., Weisskopf, V.F.: Theoretical nuclear physics. New York: Wiley (London: Chapman and Hall) 1952
Bollé, D., Gesztesy, F.: Phys. Rev. Lett.52, 1469 (1984)
Bollé, D., Gesztesy, F.: Phys. Rev. A30, 1279 (1984)
Gregory, R.: Phys. Rev. Lett.59, 740 (1987)
Landsman, K.: Quantization and superselection sectors II: Dirac monopole and Aharonov-Bohm effect, Rev. Math. Phys.2, 73 (1990)
Aharonov, Y., Bohm, D.: Phys. Rev.119, 485 (1959)
Everett, A.E.: Phys. Rev. D24, 858 (1981)
Witten, E.: Nucl. Phys. B249, 557 (1985)
Kay, B.S.: On the electromagnetic self-force due to a superconducting cosmic string, in preparation
Vilenkin, A.: Phys. Rev. D23, 852 (1981)
Garfinkle, D.: Phys. Rev. D32, 1323 (1985)
Everett, A.E.: Phys. Rev. D61, 1807 (1988)
Preskill, J.: Vortices and Monopoles. In: Architecture of the fundamental interactions at short distances. Ramond, R., Stora, R. (eds.). Amsterdam: North Holland 1987
Allen, B., Ottewill, A.: Effects of curvature couplings for quantum fields on cosmic string space-times. Phys. Rev. D42, 2669 (1990)
Exner, P., Šeba, P. (eds.): Application of self-adjoint extensions in quantum physics. (Proceedings, Dubna.) Lect. Notes in Phys., Vol. 324. Berlin, Heidelberg, New York: Springer 1989
Emmrich, C., Römer, H.: Commun. Math. Phys.129, 69 (1990)
de Sousa Gerbert, Ph.: Phys. Rev. D40, 1346 (1989)
Perkins, W.B., Davis, A.-C.: Nucl. Phys. B349, 207 (1991)
Gregory, R., Perkins, W.B., Davis, A.-C., Brandenberger, R.H.: Cosmic Strings and Baryon Decay Catalysis. In: The Structure and Evolution of Cosmic Strings. Gibbons, G.W., Hawking, S.W., Vachaspati, T. (eds.) Cambridge, Cambridge University Press 1990
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Kay, B.S., Studer, U.M. Boundary conditions for quantum mechanics on cones and fields around cosmic strings. Commun.Math. Phys. 139, 103–139 (1991). https://doi.org/10.1007/BF02102731
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DOI: https://doi.org/10.1007/BF02102731