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Quantization of Nonlinear Systems
1.See notably W. Heisenberg, Revs. Modern Phys. 29, 269 (1957),
1.and S. Deser, Revs. Modern Phys. 29, 417 (1950) , Rev. Mod. Phys.
1.(and other articles, in addition to the last‐named, reporting the Chapel Hill Conference on Gravitation);
1.and especially articles by Heisenberg and Yukawa, Proc. Internatl. Conf. High‐Energy Nuclear Phys., Geneva, 1958.
2.E. Segal, Kgl. Danske Videnskab. Selskab, Mat.‐Fys. Medd. 31, No. 12, 1–39 (1959).
3.I. E. Segal, Ann. Math, (to be published).
4.R. E. Peierls, Proc. Roy. Soc. (London) A214, 143 (1952).
5.I. E. Segal, Report of Lille Conference on Quantum Fields (C.N.R.S., Paris, 1959), pp. 57–103.
6.L. Van Hove, Acad. roy. Belg. Classe sci. Mém. Collection in 8° 29, No. 6, 1–102 (1951).
7.(a) I. E. Segal, Trans. Am. Math. Soc. 81, 106 (1956);
7.(b) I. E. Segal, Ann. Math. 63, 160 (1956).
8.For convenience, it is assumed, as seems no essential loss of generality from a physical standpoint, that the manifold M is infinitely differentiable, i.e., that it is possible near each point to choose local coordinates in such a manner that whenever a point is assigned two sets of coordinates, then near the point the one set may be expressed as ininitely differentiable functions of the other set. It is known (virtually as a matter of definition) that the existence of a measure with a nowhere vanishing continuous density function is mathematically equivalent to the orientability of M, which will be assumed in the present section.
9.E. Schrödinger, Naturwissenschaften 22, 518 (1934).
10.The number of inequivalent such is an invariant of M closely related to its one‐dimensional cohomology in the following way: if ω is any closed first‐order differential form on M, then the equations define a Heisenberg system (in infinitesimal terms) which will be equivalent to the system if ω is exact, but not generally otherwise. Specifically, there is equivalence if, and only if, ω is logarithmically exact, in the sense that for some function F on M.
10.It follows from a study of the logarithmically exact forms (cf. a forthcoming paper by R. S. Palais; similar but less complete and unpublished results are due to E. Dyer and R. Swan) that on a manifold with first Betti number r, there is an r‐parameter family of inequivalent Heisenberg systems.
10.Mathematically it is interesting to weaken statement (4) by requiring only ergodicity: no nontrivial function of the P’s and Q’s commutes with all the P’s and Q’s. The analog of the Schrödinger representation with square‐integrable functions replaced by square‐integrable tensor fields is an example of a system satisfying but not (4). The foregoing connection with closed differential forms and cohomology can be extended, but some of the quantum‐mechanical invariants of M obtained in the indicated fashion may be new, depending in part on the extent to which the tensor field examples exhaust the possibilities, within unitary equivalence and the intervention of a closed form. This is a point having a certain differential‐geometric interest, and conceivably there is a physical role for the tensor, etc. representations in other physical connections, but in the present paper only the “scalar” Heisenberg representations given by Definition 1 are used.
11.L. Gross, Trans. Am. Math. Soc. 94, 404 (1960).
12.O. Veblen and J. H. C. Whitehead, Foundations of Differential Geometry (Cambridge University Press, New York, 1932).
13.E. T. Whittaker, Analytical Dynamics (Cambridge University Press, New York, 1959).
14.J. von Neumann, Math. Ann. 104, 570 (1931).
15.I. E. Segal, Can. J. Math, (to be published).
16.S. T. Kuroda, J. Math. Soc. Japan 11, 247 (1959).
17.C. N. Yang and D. Feldman, Phys. Rev. 79, 972 (1950);
17.G. Kälién, Arkiv Fysik 2, 33 (1950).
18.C. Ehresmann, Proc. Int. Congr. Math. 1950 (Providence, 1952).
19.I. E. Segal, Phys. Rev. 109, 2191 (1958).
20.Cf. the suggestive work of Dirac in a linear case in Proc. Roy. Soc. (London) A183, 284 (1945).
21.G. Källén and A. Wightman, Kgl. Danske Videnskab. Selskab. Mat.‐Fys. Skrifter 1, No. 6, 58pp (1958).
22.I. E. Segal, Trans. Am. Math. Soc. 88, 12 (1958).
23.L. van Hove, Physica 21, 901 (1955).
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