de Sousa Gerbert, P.; Jackiw, R. Classical and quantum scattering on a spinning cone. (English) Zbl 0685.35093 Commun. Math. Phys. 124, No. 2, 229-260 (1989). Summary: Solutions are presented for the Klein-Gordon and Dirac equations in the \(2+1\) dimensional space-time created by a massive point particle, with arbitrary angular momentum. A universal formula for the scattering amplitude holds when a required self-adjoint extension of the Dirac operator is specified uniquely. Various obstacles to a consistent quantum mechanical interpretation of these results are noted. Cited in 44 Documents MSC: 35Q99 Partial differential equations of mathematical physics and other areas of application 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35P25 Scattering theory for PDEs Keywords:Klein-Gordon and Dirac equations; scattering amplitude PDFBibTeX XMLCite \textit{P. de Sousa Gerbert} and \textit{R. Jackiw}, Commun. Math. Phys. 124, No. 2, 229--260 (1989; Zbl 0685.35093) Full Text: DOI References: [1] ’t Hooft, G.: Commun. Math. Phys.117, 685 (1988) · Zbl 0663.47007 · doi:10.1007/BF01218392 [2] Deser, S., Jackiw, R.: Commun. Math. Phys.118, 495 (1988) · Zbl 0655.58038 · doi:10.1007/BF01466729 [3] Gott, J.: Ap. J.288, 422 (1985); for a review see Vilenkin, A.: Phys. Rep.121, 263 (1985). The spinning cosmic string was considered by Mazur, P.: Phys. Rev.D34, 1925 (1986), and simultaneously to our work by Harari, D. and Polychronakos, A.: Phys. Rev.D38, 3320 (1988) · doi:10.1086/162808 [4] Deser, S., Jackiw, R., ’t Hooft, G.: Ann Phys. (NY)152, 220 (1984) · doi:10.1016/0003-4916(84)90085-X [5] The generalization to arbitrary numbers of spinning point particles is given by Clément, G.: Int. J. Theor. Phys.24, 281 (1985), who also studies classical motion [6] Sommerfeld, A.: Math. Ann,47, 317 (1896) and Optics. New York, NY: Academic Press 1954 · JFM 27.0706.03 · doi:10.1007/BF01447273 [7] We see no reason for energy to be quantized or timet to be periodic as has been claimed by Mazur, P.: Phys. Rev. Lett.57, 929 (1986). Our objections to the reasoning in that paper coincide with those raised by Samuel, J., Iyer, B.: Phys. Rev. Lett.59, 2379 (1987) and are apparently accepted by Mazur, P.: Phys. Rev. Lett.59, 2380 (1987) · doi:10.1103/PhysRevLett.57.929 [8] Wigner, E.: Phys. Rev.98, 145 (1955) · Zbl 0064.21804 · doi:10.1103/PhysRev.98.145 [9] A mathematical investigation of Dirac operators on spaces with conical singularities is by Chou, A.: Trans. Am. Math. Soc.289 1 (1985) · doi:10.1090/S0002-9947-1985-0779050-8 [10] Deser, S., Jackiw, R., Templeton, S.: Ann Phys. (NY)140, 372 (1982) The normalization in (5.4) is chosen so that a well-defined zero mass limit may be taken. But recall that the interpretation of the massless theory is quite different, as might be expected since spin is discontinuous whenm passes through zero. A redefinition of the fields in the massless Dirac theory \(\phi (k)e^{ikr} = \left( {\begin{array}{*{20}c} {\sqrt {k_x - ik_y } } & 0 0 & {\sqrt {k_x + ik_y } } \end{array} } \right)\exp \left\{ {\frac{\pi }{4}\hat k^i \sigma ^i } \right\}\psi _k^0 (r)\) exhibits their almost scalar nature, with discontinuous behaviour under rotationR ? , \(\phi (k)\xrightarrow[{R_\theta }]{}\left\{ {\begin{array}{*{20}c} { \phi (R_\theta ^{ - 1} k) if 0 \leqq \theta \leqq 2\pi } { - \phi (R_\theta ^{ - 1} k) if 2\pi< \theta< 4\pi } \end{array} } \right.\) because the functions \(\sqrt {k_x \mp ik_y } \) are double-valued; see Binegar, B.: J. Math. Phys.23, 1511 (1982). Parity is not violated in this case · doi:10.1016/0003-4916(82)90164-6 [11] One has to choose thedreibein that yields the proper flat space limit [S=0, ?=1], with solutions to the Dirac equation that are single-valued and periodic in ?. This would not be the case if in (5.13) ?=0 [12] Similar problems have been studied in monopole physics by Goldhaber, A.: Phys. Rev.D16, 1815 (1977) and Callias, C.: Phys. Rev.D16, 3068 (1977), who found that thes-wave radial Dirac Hamiltonian is not self-adjoint. Chou in ref. 9 recognized the possible loss of self-adjointness of the Dirac operator on conical spaces, but did not study extensions [13] Alvarez-Gaumé, L., Della Pietra, S., Moore, G.: Ann. Phys. (NY)163, 288 (1985); Goni, M., Valle, M.: Phys. Rev.D34, 648 (1986); Vuorio, I.: Phys. Lett.B175, 176 (1986) · Zbl 0584.58049 · doi:10.1016/0003-4916(85)90383-5 [14] Deser, S., Jackiw, R., Templeton, S.: Phys. Rev. Lett.48, 975 (1982) and Ann. Phys. (NY)140, 372 (1982 · doi:10.1103/PhysRevLett.48.975 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.