Time-ordered operators. I. Foundations for an alternative view of reality
HTML articles powered by AMS MathViewer
- by Tepper L. Gill
- Trans. Amer. Math. Soc. 266 (1981), 161-181
- DOI: https://doi.org/10.1090/S0002-9947-1981-0613790-2
- PDF | Request permission
Abstract:
The purpose of this paper is to present the proper framework for the mathematical foundations of time-ordered operators. We introduce a new mathematical process which we call the chronological process. This process generalizes the notion of a limit and allows us to recapture many properties lost in time-ordering. We then construct time-ordered integrals and evolution operators. We show that under reasonable assumptions, the time-ordered sum of two generators of contraction semigroups is a generator. This result resolves a question that has been debated in physics for forty years.References
- R. P. Feynman, Theory of positrons, Phys. Rev. 76 (1949), 749-759.
- R. P. Feynman, Space-time approach to quantum electrodynamics, Phys. Rev. (2) 76 (1949), 769–789. MR 35687, DOI 10.1103/PhysRev.76.769 —, Mathematical formulation of the quantum theory of electromagnetic radiation, Phys. Rev. 80 (1950), 440.
- Richard P. Feynman, An operator calculus having applications in quantum electrodynamics, Phys. Rev. (2) 84 (1951), 108–128. MR 44379, DOI 10.1103/PhysRev.84.108
- 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
- Tepper L. Gill, New perspectives in time-ordered operators and divergencies. I, II, Hadronic J. 3 (1979/80), no. 6, 1575–1596, 1597–1621. MR 602052 E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957; reprinted 1974.
- Takashi Ichinose, On the spectra of tensor products of linear operators in Banach spaces, J. Reine Angew. Math. 244 (1970), 119–153. MR 278096, DOI 10.1515/crll.1970.244.119
- Takashi Ichinose, Operators on tensor products of Banach spaces, Trans. Amer. Math. Soc. 170 (1972), 197–219. MR 322553, DOI 10.1090/S0002-9947-1972-0322553-1
- Takashi Ichinose, Spectral properties of tensor products of linear operators. I, Trans. Amer. Math. Soc. 235 (1978), 75–113. MR 472915, DOI 10.1090/S0002-9947-1978-0472915-2
- Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
- C. T. Taam (ed.), Lectures in modern analysis and applications. II, Lecture Notes in Mathematics, Vol. 140, Springer-Verlag, Berlin-New York, 1970. MR 0263578
- J. von Neumann, On infinite direct products, Compositio Math. 6 (1939), 1–77. MR 1557013
- Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022, DOI 10.1007/978-3-642-46066-1
- Ralph Henstock, Theory of integration, Butterworths, London, 1963. MR 0158047
- Jaroslav Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7(82) (1957), 418–449 (Russian). MR 111875, DOI 10.21136/CMJ.1957.100258
- Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824
- Amnon Pazy, Semi-groups of linear operators and applications to partial differential equations, University of Maryland, Department of Mathematics, College Park, Md., 1974. Department of Mathematics, University of Maryland, Lecture Note, No. 10. MR 0512912
- James D. Bjorken and Sidney D. Drell, Relativistic quantum fields, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0187642 A. Beck, Flows in the plane, Springer-Verlag, New York, 1975.
Bibliographic Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 266 (1981), 161-181
- MSC: Primary 47D05; Secondary 22E65, 81C99
- DOI: https://doi.org/10.1090/S0002-9947-1981-0613790-2
- MathSciNet review: 613790