Abstract
The most familiar connection between quantum mechanics and functional integration is provided by Feynman’s path integral formalism. Path integrals are special linear functionals defined on an appropriate space of paths. They are close to the concept of an ordinary integral but cannot be represented as an integral w.r. to some measure (cf. 1). On the other hand — in classical physics — we are acquainted with the theory of integration in function spaces by means of integrals which are defined by some measure. In non-equilibrium statistical mechanics, e.g., the Wiener integral is widely used (cf. 2).
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References
“Feynman Path Integrals”, S. Albeverio, Ph. Combe, R. Hoegh-Krohn et al. eds., Springer, Berlin (1979).
D. Dürr and A. Bach, The Onsager-Machlup Function as the Most Probable Path of a Diffusion Process, Comm. math. Phys. 60:153 (1978).
A. Bach, Quantum Mechanics and Integration in Hilbert Space, Phys. Lett. 73A:287 (1979).
A. V. Skorohod, “Integration in Hilbert Space”, Springer, Berlin (1974).
H.-H. Kuo, “Gaussian Measures in Banach Spaces”, Springer, Berlin (1975).
M. Jammer, “The Philosophy of Quantum Mechanics”, Wiley, New York (1974)
J. v. Neumann, “Die mathematischen Grundlagen der Quantenmechanik”, Springer, Berlin (1968).
A. M. Gleason, Measures on the Closed Subspaces of a Hilbert Space, J. Math. and Mech. 6:885 (1957).
S- Kochen and E. P. Specker, The Problem of Hidden Variables in Quantum Mechanics, J. Math. and Mech., 17:59 (1967).
J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1:195 (1964).
J. Schwartz, The Wiener-Siegel Causal Theory of Quantum Mechanics, in: “Integration of Functionals”, K. O. Friedrichs and H. N. Shapiro, eds., Courant Institute, New York (1957).
J. R. Klauder and E. C. G. Sudarshan, “Fundamentals of Quantum Optics”, Benjamin, New York (1968).
Y. Takahashi and F. Shibata, Spin Coherent State Representation in Non-Equilibrium Statistical Mechanics, J. Phys. Soc. Japan 38:656 (1975).
J. M. Radcliffe, Some Properties of Coherent Spin States, J. Phys. A4:313 (1971).
I. M. Gelfand und N. J. Wilenkin, “Verallgemeinerte Funktionen (Distributionen)”, Band 4, VEB Deutscher Verlag der Wissenschaften, Berlin (1964).
D. Kölzow, A Survey of Abstract Wiener Spaces, in: “Proceedings of the Summer Research Institute on Statistical Interference for Stochastic Processes”, Vol. 1, Madan Lal Puri, ed., Academic Press, Bloomington (1975).
A. Einstein, Physik und Realität, in: A. Einstein, “Aus meinen späteren Jahren”, Deutsche Verlags-Anstalt, Stuttgart (1952).
K. Popper, Quantum Mechanics without Observer, in: “Quantum Theory and Reality”, M. Bunge, ed., Springer, Berlin (1967).
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© 1980 Plenum Press, New York
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Bach, A. (1980). Integration in Hilbert Space and Quantum Theory. In: Antoine, JP., Tirapegui, E. (eds) Functional Integration. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-7035-6_6
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DOI: https://doi.org/10.1007/978-1-4615-7035-6_6
Publisher Name: Springer, Boston, MA
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