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By M. Aizenman (Chief Editor)

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53 56 62 66 69 70 74 76 77 80 81 82 83 84 87 87 89 91 91 92 0. Introduction These notes are a natural outgrowth of our previous work on a local holomorphic formula for the dimension of a superselection sector [47] and were motivated by the purpose to give a geometrical picture to aspects of local quantum physics related to the superselection structure. They may be read from different points of view, in particular, guided by a similarity of the statistical dimension with the Fredholm index and a possible index theorem, already suggested in [15], we shall regard the DHR localized endomorphisms [16] as quantum analogs of elliptic differential operators.

Preprint MPRESS (1998) 10. Kleban, P. : A Farey Fraction Spin Chain. Commun. Math. Phys. 203, 635–647 (1999) 11. : On a ferromagnetic spin chain. Commun. Math. Phys. 153, 77–115 (1993) 12. : Elementary Number Theory. , 1958 13. : Topics in Multiplicative Number Theory. Lecture Notes in Mathematics 227, Berlin– Heidelberg: Springer-Verlag, 1971 14. : An Introduction to Algebraic Number Theory. New York, London: Plenum Press, 1990 15. : Momente der Klassenzahlen binärer quadratischer Formen mit ganzalgebraischen Koeffizienten.

London Math. Soc. 7, 199–208 (1876) 18. : Class Numbers of Indefinite Quadratic Forms. J. Number Theory 15, 229–247 (1982) 19. : The Theory of the Riemann Zeta-function. 2nd Edition, Oxford: Oxford Science Publications, 1986 Communicated by H. Spohn Commun. Math. Phys. 222, 45 – 96 (2001) Communications in Mathematical Physics © Springer-Verlag 2001 Notes for a Quantum Index Theorem Roberto Longo Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy.

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