Abstract
This paper provides a brief introduction to how unitary representations of diffeomorphism groups can describe certain quantum systems having infinitely many degrees of freedom. It is a partial report of our joint work [1], based on the August 1994 talk by the first author at the Symmetries in Science VIII conference in Bregenz, Austria. We would like to express appreciation to the conference organizers, especially Professor Bruno Gruber, for the opportunity to present our results.
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References
G. A. Goldin and U. Moschella, Saclay preprint T94/029 (1994).
G. A. Goldin, “Current Algebras as Unitary Representations of Groups”, Ph. D. thesis, Princeton University (1969). G. A. Goldin and D. H. Sharp, in 1969 Battelle Rencontres: Group Representations, Lecture Notes in Physics 6, edited by V. Bargmann (Springer, Berlin, 1970), p. 300; G. A. Goldin, J. Math. Phys. 12, 462 (1971).
For more detailed reviews, see G. A, Goldin and D. H. Sharp, “Diffeomorphism Groups and Local Symmetries: Some Applications in Quantum Physics”, in Symmetries in Science III, edited by B. Gruber and F. lachello (New York: Plenum, 1989), p. 181; G. A. Goldin, “Predicting Anyons: The Origins of Fractional Statistics in Two-Dimensional Space”, in Symmetries in Science V, edited by B. Gruber, L. C. Biedenharn, and H.-D. Doebner (New York: Plenum, 1991), p. 259; G. A. Goldin and D. H. Sharp, Int. J. Mod. Phys. B5, 2625 (1991); and G. A. Goldin, Int. J. Mod. Phys. B6, 1905 (1992).
R. Dashen and D. H. Sharp, Phys. Rev. 165 (1968), 1867.
H.-D. Doebner and J. Tolar, in Symposium on Symmetries in Science, Carbondale, Illinois 1979, edited by B. Gruber and R. S. Millman (Plenum, New York, 1980), p. 475; B. Angermann, H.-D. Doebner, and J. Tolar, in Nonlinear Partial Differential Operators and Quantization Procedures, Lecture Notes in Mathematics 1037, edited by S. I. Andersson and H.-D. Doebner (Springer, Berlin, 1983), p. 171; H.-D. Doebner, H. J. Elmers, and W. Heidenreich, J. Math. Phys. 30 (1989), 1053.
G. A. Goldin, R. Menikoff, and D. H. Sharp, J. Math. Phys. 21, 650 (1980); J. Math, Phys. 22, 1664 (1981); Phys. Rev. Lett. 51, 2246 (1983); G. A. Goldin and D. H. Sharp, Phys. Rev. D28, 830 (1983).
J. M. Leinaas and J. Myrheim, Nuovo Cimento 37B, 1 (1977); J. M. Leinaas, Nuovo Cimento 4A, 19 (1978); Fort. Phys. 28, 579 (1980).
F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982); Phys. Rev. Lett. 49, 957 (1982).
H.-D. Doebner and G. A. Goldin, Phys. Letts. A162, 397 (1992); G. A. Goldin, Int. J. Mod. Phys. B6, 1905 (1992); H.-D. Doebner and G. A. Goldin, J. Phys. A: Math. Gen. 27, 1771 (1994).
G. A. Goldin, J. Grodnik, R. T. Powers, and D.H. Sharp, J. Math. Phys. 15, 88 (1974); R. Menikoff, J. Math. Phys. 15 (1974), 1138 and 1394; A. M. Vershik, I. M. Gelfand, and M. I. Graev, Dokl. Akad. Nauk, SSSR 232, 745 (1977).
I. M. Gelfand and N. Ya Vilenkin, Generalized Functions, Vol. 4 (New York: Academic Press, 1964).
M. Rasetti and T. Regge, Physica 80A, 217 (1975); J. Marsden and A. Weinstein, Physica 7D, 305 (1983); G. A. Goldin, R. Menikoff, and D. H. Sharp, Phys. Rev. Lett. 58, 2162 (1987); Phys. Rev. Lett. 67, 3499 (1991). For recent reviews of related results see G. A. Goldin and D. H. Sharp, Int. J. Mod. Phys. B5, 2625 (1991), and G. A. Goldin, Int. J. Mod. Phys. B6, 1905 (1992).
S. W. Hawking, “The Path-Integral Approach to Quantum Gravity”, in General Relativity: An Einstein Centenary Survey , edited by S. W. Hawking and W. Israel, Cambridge University Press (1979).
A. Ashtekar and J. Lewandowski, “Representation Theory of Analytic Holonomy C*-algebras”, in Knots and Quantum Gravity, edited by J. Baez, Oxford University Press (1994, in press); A. Ashtekar, D. Marolf, and J. Mourao, “Integration on the Space of Connections Modulo Gauge Transformations” (March 1994 preprint).
W. Feller An Introduction to Probability Theory and Its Applications, Vol. II Second Edition, Wiley, New York (1971).
G. Baker, Phys. Rev. 126, 2071 (1962); M. Kac, G. Uhlenbeck, and P. Hemmer, J. Math. Phys. 4, 216 (1963); F. Dyson, Commun. Math. Phys. 12, 1961 (1969).
E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961), Rev. Mod. Phys. 36, 856 (1964). See also J. M. Luck, J. Stat. Phys. 72, 417 (1993) and references therein.
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Goldin, G.A., Moschella, U. (1995). Diffeomorphism Groups, Quasi-Invariant Measures, and Infinite Quantum Systems. In: Gruber, B. (eds) Symmetries in Science VIII. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1915-7_13
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DOI: https://doi.org/10.1007/978-1-4615-1915-7_13
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