Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T21:44:39.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Foundations of the Theory of Dynamical Systems of Infinitely Many Degrees of Freedom, II

Published online by Cambridge University Press:  20 November 2018

I. E. Segal
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The notion of quantum field remains at this time still rather elusive from a rigorous standpoint. In conventional physical theory such a field is defined in essentially the same way as in the original work of Heisenberg and Pauli (1) by a function ϕ(x, y, z, t) on space-time whose values are operators. It was recognized very early, however, by Bohr and Rosenfeld (2) that, even in the case of a free field, no physical meaning could be attached to the values of the field at a particular point—only the suitably smoothed averages over finite space-time regions had such a meaning. This physical result has a mathematical counterpart in the impossibility of formulating ϕ(x, y, z, t) as a bona fide operator for even the simplest fields (in any fashion satisfying the most elementary non-trivial theoretical desiderata), while on the other hand for suitable functions f, the integral ∫ϕ(x, y, zy t)f(x, y, z, t)dxdydzdt could be so formulated.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 1 - 18
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Heisenberg, W. and Pauli, W., Quantum mechanics of wave fields, Zeits. f. Physik, 56 (1929), 1–61.Google Scholar
2. Bohr, N. and Rosenfeld, L., Zur frage der Messbarkeit der Elektromagnetischen Feldgrossen, Kgl. Danske Vidensk. Selsk., mat.-fys. Medd. XII, 8 (1953), 3–65.Google Scholar
3. Fock, V., Konfigurationsraum und zweiten Quantelung, Zeits. f. Physik, 75 (1952), 622–647.Google Scholar
4. Friedrichs, K. O., Mathematical aspects of the quantum theory of fields, I-II, Commun. Pure Appl. Math., 4 (1951), 161–224.Google Scholar
5. Cook, J. M., The mathematics of second quantization, Thesis, University of Chicago, 1951; in part in Trans. Amer. Math. Soc, 74 (1953), 222–245.Google Scholar
6. Wightman, A. S., Quelques problèmes mathématiques de la théorie quantique relativiste, in Colloque [Lille, 1957) on the mathematical problems of quantum field theory (Paris, 1959) 1–38.Google Scholar
7. Kâllén, G. and Wightman, A. S., The analytic properties of the vacuum expectation values of a product of three scalar fields, K. Danske Vidensk. Selsk., mat-fys. Skr., 1 (1958).Google Scholar
8. Shohat, J. A. and Tamarkin, J. D., The problem of moments (New York, 1943).Google Scholar
9A. Kâllén, G., Lectures at the Institute for Theoretical Physics, Lund, Spring, 1959.Google Scholar
9B. Schwinger, J., Quantum electrodynamics. I, A covariant formalism, Phys. Rev., 74 (1948), 1439–1461.Google Scholar
10. Segal, I. E., Distributions in Hilbert space and canonical systems of operators, Trans. Amer. Math. Soc, 55 (1958), 12–41.Google Scholar
11. Segal, I. E., Direct formulation of causality requirements on the S-operator, Phys. Rev., 109 (1958), 2191–2198.Google Scholar
12. Segal, I. E., Foundations of the theory of dynamical systems of infinitely many degrees of freedom, I, Kgl. Danske Vidensk. Selsk., mat.-fys. Medd., 31 (1959), 1–38.Google Scholar
13. Segal, I. E., Postulates for general quantum mechanics, Ann. Math., 48 (1947), 930–948.Google Scholar
14. Murray, F. J. and von Neumann, J., On rings of operators, IV, Ann. Math., 44 (1943), 716–808.Google Scholar
15. Kaplansky, I., A theorem on rings of operators, Pac. J. Math., 1 (1951), 227–232.Google Scholar
16. Segal, I. E., A non-commutative extension of abstract integration, Ann. Math., 57 (1953), 401–457.Google Scholar
17. Segal, I. E., Tensor algebras over Hilbert spaces, I, Trans. Amer. Math. Soc, 81 (1956), 106–134.Google Scholar
18. van Hove, L., Sur certaines représentations unitaires d'un groupe infini de transformations, Acad. Roy. Belgique, Cl. Sci. Mém. Coll in 8°, 26 (1951), 102 pp.Google Scholar
19. Shale, D., On certain groups of operators on Hilbert space, Doctoral Thesis, University of Chicago, 1959.Google Scholar
20. Phillips, R. S., Integration in a convex linear topological space, Trans. Amer. Math. Soc, 47 (1940), 114–145.Google Scholar
21. Wigner, E., On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749–759.Google Scholar
22. Baker, G. A. Jr, Formulation of quantum mechanics based on the quasi-probability distribution induced in phase space, Phys. Rev., 109 (1958), 2198–2206.Google Scholar
23. Hochschild, G., Cohomology of restricted Lie algebras, Amer. J. Math., 76 (1954), 555–580.Google Scholar
24. Thomas, L. H., General relativity and particle dynamics, Phys. Rev., 112 (1958), 2129–2134. and to appear.Google Scholar