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On quantum theory in terms of white noise *

Published online by Cambridge University Press:  22 January 2016


T. Hida
Affiliation:
Department of Mathematics Nagoya University
L. Streit
Affiliation:
Fakultät für Physik, Universität Bielefeld

Extract

It has often been pointed out that a much more manageable structure is obtained from quantum theory if the time parameter t is chosen imaginary instead of real. Under a replacement of t by i·t the Schrödinger equation turns into a generalized heat equation, time ordered correlation functions transform into the moments of a probability measure, etc. More recently this observation has become extremely important for the construction of quantum dynamical models, where criteria were developed by E. Nelson, by K. Osterwalder and R. Schrader and others [8] which would permit the reverse transition to real time after one has constructed an imaginary time (“Euclidean”) model.


Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1977

Footnotes

*

This work was done at Zentrum für interdisziplinäre Forschung of Bielefeld University under the support of DFG.


References

[1] Albeverio, S. and Hoegh-Krohn, R., Drichlet Forms and Diffusion Processes on Rigged Hilbert Spaces, Oslo Univ. Preprint, December 1975.Google Scholar
[2] Balakrishnan, A. V., Stochastic Optimization Theory in Hilbert Spaces—1. Applied Math, and Optimization 1 (1974), 97120.Google Scholar
[3] Ezawa, H., Klauder, J. R. and Shepp, L. A., A Path Space Picture for Feynman-Kac Averages. Ann. of Phys. 88 (1974), 588620.Google Scholar
[4] Hegerfeldt, G. C., From Euclidean to Relativistic Fields and on the Notion of Markoff Fields. Commun. Math. Phys. 35 (1974), 155171.Google Scholar
[5] Hida, T., Canonical Representations of Gaussian Processes and their Applications, Mem. Coll. Sci. Univ. Kyoto, Ser. A33 (1960), 109155. Hida, T., “Functional of Brownian Motion II”, ZiF Lectures 1976.Google Scholar
[6] Hida, T., Stationary Stochastic Processes. Princeton Univ. Press 1970.Google Scholar
[7] Okabe, Y., Stationary Gaussian Processes with Markovian Property and M. Sato’s Hyperfunctions. Japanese Journ. of Math. 41 (1973), 69112.Google Scholar
[8] for a review of B. Simon: The P(φ)2 Euclidean (Quantum) Field Theory, Princeton Univ. Press 1974.Google Scholar
[9] e.g. Widder, D. V.: The Laplace Transform. Princeton Univ. Press 1946.Google Scholar

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