Green's Distributions and the Cauchy Problem for the Multi‐Mass Klein‐Gordon Operator
dc.contributor.author | Bowman, John Judson | en_US |
dc.contributor.author | Harris, Joseph David | en_US |
dc.date.accessioned | 2010-05-06T22:23:39Z | |
dc.date.available | 2010-05-06T22:23:39Z | |
dc.date.issued | 1962-11 | en_US |
dc.identifier.citation | Bowman, John Judson; Harris, Joseph David (1962). "Green's Distributions and the Cauchy Problem for the Multi‐Mass Klein‐Gordon Operator." Journal of Mathematical Physics 3(6): 1281-1290. <http://hdl.handle.net/2027.42/70586> | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/70586 | |
dc.description.abstract | Explicit forms of the Green's functions (which are to be regarded as distributions in the sense of Schwartz) for the multi‐mass Klein‐Gordon operator in n‐dimensional spaces are presented. The homogeneous Green's functions GN(x) and GN1(x), defined in the usual way by independent paths of integration in the k0 plane, are investigated in the neighborhood of the light cone. The parameter N indicates the total number of masses involved. The singularities on the light cone reflect the well‐known difference between even‐ and odd‐dimensional wave propagation. It is found that GN(x; odd n) contains a finite jump on the light cone as well as a linear combination of derivatives up to order ☒(n − 2N − 1) of δ(x2); the singular part of GN1(x; odd n) consists of a logarithmic singularity ln (∣x2∣) along with a polynomial in (x2)−1 of degree ☒(n − 2N − 1). For even‐dimensional spaces, the singular part of both Green's functions consists of a polynomial in (x2)−1∕2 of degree n − 2N + 1 vanishing outside the light cone for GN and vanishing inside the light cone for GN1. In all cases no singularities or finite jumps occur when the order 2N of the operator is greater than the number n + 1 of space‐time dimensions. The general solution of the Cauchy problem is given both for the data carrying surface t = 0 and for arbitrary spacelike data surfaces. | en_US |
dc.format.extent | 3102 bytes | |
dc.format.extent | 634872 bytes | |
dc.format.mimetype | text/plain | |
dc.format.mimetype | application/pdf | |
dc.publisher | The American Institute of Physics | en_US |
dc.rights | © The American Institute of Physics | en_US |
dc.title | Green's Distributions and the Cauchy Problem for the Multi‐Mass Klein‐Gordon Operator | en_US |
dc.type | Article | en_US |
dc.subject.hlbsecondlevel | Physics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | Department of Physics, University of Michigan, Ann Arbor, Michigan | en_US |
dc.contributor.affiliationother | Departments of Physics and Biochemistry, Dartmouth University, Hanover, New Hampshire | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/70586/2/JMAPAQ-3-6-1281-1.pdf | |
dc.identifier.doi | 10.1063/1.1703872 | en_US |
dc.identifier.source | Journal of Mathematical Physics | en_US |
dc.identifier.citedreference | Multi‐mass equations also arise naturally when one considers particles of higher spin; see, e.g., J. D. Harris, Phys. Rev. 112, 2124 (1958). | en_US |
dc.identifier.citedreference | W. Pauli and F. Villars, Revs. Modern Phys. 21, 434 (1949). | en_US |
dc.identifier.citedreference | A. Pais and G. E. Uhlenbeck, Phys. Rev. 79, 145 (1950). | en_US |
dc.identifier.citedreference | J. Rzewuski, Acta Phys. Polon. 12, 100 (1953). | en_US |
dc.identifier.citedreference | J. J. Bowman and J. D. Harris, J. Math. Phys. 3, 396 (1962), hereafter called (I). | en_US |
dc.identifier.citedreference | L. Schwartz, Théorie des distributions I, II (Hermann et Cie, Paris, 1950–51). | en_US |
dc.identifier.citedreference | Green’s functions for multi‐mass operators like [□m−(−μ2)m]l[□m−(−μ2)m]l may be calculated directly without recourse to a partial fraction expansion. Such operators have been investigated by J. J. Bowman and J. D. Harris, J. Math. Phys. 3, 1291 (1962). | en_US |
dc.identifier.citedreference | G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1944), 2nd ed. | en_US |
dc.identifier.citedreference | A rigorous proof showing that (61) satisfies Eq. (59) directly is not hard to construct, but seems longer than this demonstration. | en_US |
dc.identifier.citedreference | See, e.g., J. Rzewuski, Field Theory (Hafner Publishing Company, New York, 1958). | en_US |
dc.identifier.citedreference | Compare with reference 4. | en_US |
dc.identifier.citedreference | A complete treatment of partial fractions is given by J. A. Serret, Cours d’algàbre supréieure (Gauthier‐Villars, Paris, 1885), Tome I. | en_US |
dc.identifier.citedreference | H. Bateman, Higher Transcendental Functions (McGraw‐Hill Book Company, Inc., New York, 1953), Vols. I, II, III. | en_US |
dc.owningcollname | Physics, Department of |
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