Time-ordered operators. II
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- by Tepper L. Gill
- Trans. Amer. Math. Soc. 279 (1983), 617-634
- DOI: https://doi.org/10.1090/S0002-9947-1983-0709572-5
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Abstract:
In this paper, we substantially improve on the work of [G1]. After constructing the general mathematical foundations for linear time-ordered evolution equations, we apply our results to show that both the perturbation expansion and the Feynman diagram method are mathematically sound. We provide a remainder term so that the expansion may be considered exact at all orders. We then show that time-ordered operators naturally induce an operator-valued path integral whenever a transition kernel is given.References
- Paul R. Chernoff, Product formulas, nonlinear semigroups, and addition of unbounded operators, Memoirs of the American Mathematical Society, No. 140, American Mathematical Society, Providence, R.I., 1974. MR 0417851
- J. R. Dorroh, A linear evolution equation without a common dense core for the generators, J. Differential Equations 31 (1979), no. 1, 109–116. MR 524819, DOI 10.1016/0022-0396(79)90154-2
- R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367–387. MR 0026940, DOI 10.1103/revmodphys.20.367
- Richard P. Feynman, An operator calculus having applications in quantum electrodynamics, Phys. Rev. (2) 84 (1951), 108–128. MR 44379
- Tepper L. Gill, Time-ordered operators. I. Foundations for an alternative view of reality, Trans. Amer. Math. Soc. 266 (1981), no. 1, 161–181. MR 613790, DOI 10.1090/S0002-9947-1981-0613790-2
- Tepper L. Gill, Infinite tensor products of Banach spaces. I, J. Functional Analysis 30 (1978), no. 1, 17–35. MR 513475, DOI 10.1016/0022-1236(78)90052-6 —, New perspectives in time-ordered operators and divergencies. I, Hadronic J. 3 (1980), 1597-1621. —, New perspectives in time-ordered operators and divergencies. II, Hadronic J. 3 (1980), 1597-1621.
- Jerome A. Goldstein, On the absence of necessary conditions for linear evolution operators, Proc. Amer. Math. Soc. 64 (1977), no. 1, 77–80. MR 500284, DOI 10.1090/S0002-9939-1977-0500284-2
- A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Séminaire Bourbaki, Vol. 2, Soc. Math. France, Paris, 1995, pp. Exp. No. 69, 193–200 (French). MR 1609222
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- J. R. Holub, Hilbertian operators and reflexive tensor products, Pacific J. Math. 36 (1971), 185–194. MR 301487
- Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
- Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
Bibliographic Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 279 (1983), 617-634
- MSC: Primary 47D05; Secondary 35K22, 81C35
- DOI: https://doi.org/10.1090/S0002-9947-1983-0709572-5
- MathSciNet review: 709572