Prog. Theor. Phys. Vol. 58 No. 6 (1977) pp. 1964-1972
On the Path Integral in the Curved Space
Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima 725
(Received July 9, 1977)
An attempt is made to determine the coefficient of the scalar curvature appeared in the quantal Lagrangian and Hamiltonian for a system of harmonic oscillators in the curved space with non-vanishing Riemann-Christoffel curvature tensor. The c-number Hamiltonian appearing in the path integral in the phase space is derived from the quantal one of Kawai and Kamo by the correspondence rule of Weyl ordering. In connection with the consideration of point transformations in the normal coordinate system, the arbitrary scalar curvature in the Hamiltonian is determined so that additive vacuum energy due to two-loop bubble Feynman diagrams is eliminated. The transformation function characterized by the Hamiltonian with the determined scalar curvature satisfies the equation obtained by Feynman's theory due to Pauli and DeWitt, which is based on Van Vleck's work.
DOI : 10.1143/PTP.58.1964
J.-L. Gervais and A. Jevicki, Nucl. Phys. B 110 (1976), 93[CrossRef].
M. M. Mizrahi, J. Math. Phys. 16 (1975), 2201[CrossRef].
- W. Pauli, Feldquantisierung, lecture note, Zürich (1950-1951).
B. S. DeWitt, Rev. Mod. Phys. 29 (1957), 377[APS].
- J. H. Van Vleck, Proc. Natl. Acad. Sci. 14 (1928), 178.
C. Garrod, Rev. Mod. Phys. 38 (1966), 483[APS].
- T. Kawai, Found. of Phys. 5 (1975), 143.
H. Kamo and T. Kawai, Prog. Theor. Phys. 50 (1973), 680[PTP].
Within the framework of q-number variation, a similar quantal Hamiltonian was obtained by T. Ohtani and R. Sugano, Prog. Theor. Phys. 50 (1973), 1715[PTP].
L. Cohen, J. Math. Phys. 7 (1966), 781[CrossRef].
J. S. Dowker, J. Math. Phys. 17 (1976), 1873[CrossRef].
- M. Sato, Prog. Theor. Phys. 58 (1977), 1262[PTP].
- L. P. Eisenhart, Riemannian Geometry (Princeton Univ. Press, 1949) §18 and Appendix 3.
For instance, E. S. Abers and B. W. Lee, Phys. Rep. 9 (1973), 1[CrossRef].