Smearing of propagators by gravitational fluctuations on the Planck scale

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 60, Issue 10, 1999, Pages

  Ohanian, Hans C ^a  

^a RENSSELAER POLYTECHNIC INSTITUTE   (United States)

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EID: 18244384734     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.60.104051     Document Type: Article

References (21)

1. H. C. Ohanian, Phys. Rev. D 55, 5140 (1997).
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5. B. DeWitt, in Relativity, Groups, and Topology, edited by B. DeWitt and C. DeWitt (Gordon and Breach, New York, 1964), pp. 739–745.
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11. Details of the application of this procedure to gravitons are given in C. J. Isham, A. Salam, and J. Strathdee, Phys. Rev. D 3, 1805 (1971).
12. If we treat a (weak) gravitational-wave field as a perturbation of a fixed classical background, we obtain a linear differential equation in that background. But this does not help us here, since any attempt at rearranging the path integral (Formula presented) into a double path integral (Formula presented) over the background and the perturbation would mean that we are adopting two independent gravitational fields, with a doubling of the degrees of freedom.
13. B. DeWitt, in Relativity, Groups, and Topology, edited by B. DeWitt and C. DeWitt (Gordon and Breach, New York, 1964), p. 817. In Eq. (39), a total divergence (Formula presented) has been omitted, and a term (Formula presented)has been eliminated by means of the Gauss-Bonnet identity for Euclidean topology.
14. A negative value of B is required for the cancellation of the divergences in B and (Formula presented) The sign of B depends on the signs of (Formula presented) and of (Formula presented) since (Formula presented) Hence (Formula presented) must be taken spacelike if (Formula presented) and timelike if (Formula presented) The conformal case (Formula presented) does not lend itself to the proposed renormalization.
15. K. S. Stelle, Phys. Rev. D 16, 953 (1977).
16. K. S. Stelle, in Quantum Theory of Gravity, edited by S. M. Christensen (Hilger, Bristol, 1984).
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18. L. D. Landau and I. Pomeranchuk, Dokl. Akad. Nauk SSSR 102, 489 (1955).
19. H. C. Ohanian, Nuovo Cimento A 110, 751 (1997).
20. See, e.g., K. Johnson, in Lectures on Particles and Field Theory, edited by K. Johnson, D. B. Lichtenberg, J. Schwinger, and S. Weinberg (Prentice-Hall, Englewood Cliffs, NJ, 1964), Sec. 2.2,or S. L. Adler, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. Deser, M. Grisaru, and H. Pendleton (MIT Press, Cambridge, MA, 1970), Sec. 6.1.
21. For the gauge-invariant part of the vacuum polarization, we elect to perform the integrations over the spacetime variables last, so that we can garner the benefits of the smeared propagator.