Abstract
We consider problems related to integration with respect to the Bogoliubov measure in the space of continuous functions and calculate some functional integrals with respect to this measure. Approximate formulas that are exact for functional polynomials of a given degree and also some formulas that are exact for integrable functionals belonging to a broader class are constructed. An inequality for traces is proved, and an upper estimate is derived for the Gibbs equilibrium mean square of the coordinate operator in the case of a one-dimensional nonlinear oscillator with a positive symmetric interaction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. Fr´echet, Bull. Soc. Math. France, 43, 249 (1915).
P. J. Daniell, Ann. Math., 19, 279 (1917–1918).
P. J. Daniell, Ann. Math., 20, 281 (1918–1919).
L. H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand, Toronto (1953).
F. Riesz and B. Sz.-Nagy, Leçons d'analyse fonctionnelle (6th ed.), Akadèmiai Kiadó, Budapest (1972); B. Sz.-Nagy, “Unitary dilations of Hilbert space operators and related topic, ” in: Expository Lectures from the CBMS Regional Conference Held at the University of New Hampshire, June 7–11, 1971, Am. Math. Soc., Providence (1974).
N. Wiener, Ann. Math., 22, 66 (1920).
R. Feynman, Rev. Mod. Phys., 20, 367 (1948).
R. Cameron, J. Anal. Math., 10, 287 (1962–1963).
S. F. Edwards and R. E. Pierls, Proc. Roy. Soc., A224, 24 (1954).
K. Symanzik, Z. Naturforsch, 9a, 809 (1954).
I. M. Gelfand and R. A. Minlos, Dokl. Akad. Nauk SSSR, 97, 209 (1954).
E. S. Fradkin, Dokl. Akad. Nauk SSSR, 98, 47 (1954).
N. N. Bogoliubov, Dokl. Akad. Nauk SSSR, 99, 225 (1954).
A. V. Svidzinskii, Zh. Eksp. Teor. Fiz., 31, 324 (1956).
N. N. Bogoliubov and N. N. Bogolyubov Jr., Aspects of Polaron Theory [in Russian] (Comm. No. P17–81–65), Joint Inst. Nucl. Res., Dubna (1981).
D. P. Sankovich, Theor. Math. Phys., 119, 670 (1999).
Hui-Hsiung Kuo, Gaussian Measures in the Banach Spaces, Springer, Berlin (1975).
G. E. Shilov and Fan Dyk Tin', Integral, Measure, and Derivative on Linear Spaces [in Russian], Nauka, Moscow (1967).
A. D. Egorov and L. A. Yanovich, Dokl. Akad. Nauk Belorus. SSR, 14, 873 (1970).
L. A. Yanovich, Approximate Calculation of Functional Integrals with Respect to Gaussian Measures [in Russian], Nauka i Tekhnika, Minsk (1976).
J. Kuelbs, J. Funct. Anal., 5, 354 (1970).
B. S. Rajput, J. Multivariate Anal., 2, 282 (1972).
L. Gross, “Abstract Wiener spaces,” in: Proc. 5th Berkeley Symp. Math. Stat. and Prob. (L. M. LeCam and J. Neyman, eds.), Vol. 2, Univ. of California Press, Berkeley (1967), p. 31.
M. Kac, “On some connections between probability theory and differential and integral equations,” in: Proc. 2nd Berkeley Symp. Math. Stat. and Prob. (J. Neyman, ed.), Univ. of California Press, Berkeley (1951), p. 189.
P. I. Sobolevskii and L. A. Yanovich, Dokl. Akad. Nauk Belorus. SSR, 18, 965 (1974).
H. Falk and L. W. Bruch, Phys. Rev., 180, 442 (1969).
J. Fröhlich, Bull. Am. Math. Soc., 84, 165 (1978).
D. P. Sankovich, Theor. Math. Phys., 79, 656 (1989).
N. N. Bogoliubov Jr. and D. P. Sankovich, Phys. Lett. A, 137, 179 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sankovich, D.P. Some Properties of Functional Integrals with Respect to the Bogoliubov Measure. Theoretical and Mathematical Physics 126, 121–135 (2001). https://doi.org/10.1023/A:1005262400667
Issue Date:
DOI: https://doi.org/10.1023/A:1005262400667