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Prog. Theor. Phys. Vol. 79 No. 1 (1988) pp. 217-226
Geometry, Heat Equation and Path Integrals on the Poincaré Upper Half-Plane
Reijiro Kubo
Research Institute for Theoretical Physics,
Hiroshima University, Takehara, Hiroshima 725
(Received August 13, 1987)
Abstract:
Geometry, heat equation and Feynman's path integrals are studied on the Poincaré upper half-plane. The fundamental solution to the heat equation ∂f/∂t = ΔH f is expressed in terms of a path integral defined on the upper half-plane. It is shown that Kac's statement that Feynman's path integral satisfies the Schrödinger equation is also valid for our case.
URL :
http://ptp.ipap.jp/link?PTP/79/217/
DOI : 10.1143/PTP.79.217
References:
- For reviews of string theories see J. H. Schwarz, Superstrings, The First Fifteen Years of Superstring Theory, two volumes (World Scientific, Singapore, 1985).
Unified String Theories ed. M. Green and D. Gross (World Scientific, Singapore, 1986).
M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, two volumes (Cambridge University Press, 1987).
- A. M. Polyakov, Phys. Lett. 103B (1981), 207; ibid. 103B (1981), 211.
O. Alvarez, Nucl. Phys. B216 (1983), 125.
D. Friedan, “Introduction to Polyakov's String Theory” in Recent Advances in Field Theory and Statistical Mechanics, Les Houches 1982, ed. J. B. Zuber and R. Stora (Elsevier, Amsterdam, 1984).
-
J. Polchinski, Commun. Math. Phys. 104 (1986), 37[CrossRef].
- E. D'Hoker and D. H. Phong, Nucl. Phys. B269 (1986), 205;
Phys. Rev Lett. 56 (1986), 912[APS];
Commun. Math. Phys. 104 (1986), 537[CrossRef].
G. Gilbert, Nucl. Phys. B277 (1986), 102.
M. A. Namazie and S. Rajeev, Nucl. Phys. B277 (1986), 332.
- A. Selberg, J. Indian Math. Soc. 20 (1956), 47.
D. A. Hejhal, The Selberg Trace Formula for PSL ( 2, R), I, Lecture Notes in Mathematics 548 (Springer, Berlin, 1976); Duke Math. J. 43 (1976), 441.
F. Steiner, Phys. Lett. B188 (1987), 447.
- M. Kac, Probability and Related Topics in Physical Sciences (Interscience, N. Y., 1959).
- A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I (Springer, Berlin, 1985).
- S. Helgason, Groups and Geometric Analysis (Academic, N. Y., 1984).
- H. P. Mckean, Comm. Pure and Appl. Math. 25 (1972), 225.
- R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, N. Y., 1961), vol. I.
-
N. L. Balazs and A. Voros, Phys. Rep. 143 (1986), 109[CrossRef].
- C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1983).
-
B. S. DeWitt, Rev. Mod. Phys. 29 (1957), 377[APS]; Dynamical Theory of Groups and Fields (Gordon and Breach, N. Y., 1965);
Phys. Rep. 19 (1975), 295[CrossRef].
- K. Ito, Mem. Am. Math. Soc. 4 (1951), 51.
- L. S. Schulman, Techniques and Applications of Path Integrals (Wiley, N. Y., 1981).
