On the use of modular groups in quantum field theory
Annales de l'I.H.P. Physique théorique, Volume 63 (1995) no. 4, p. 331-382
@article{AIHPA_1995__63_4_331_0,
     author = {Borchers, H. J.},
     title = {On the use of modular groups in quantum field theory},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {63},
     number = {4},
     year = {1995},
     pages = {331-382},
     zbl = {0838.46059},
     mrnumber = {1367142},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1995__63_4_331_0}
}
Borchers, H. J. On the use of modular groups in quantum field theory. Annales de l'I.H.P. Physique théorique, Volume 63 (1995) no. 4, pp. 331-382. http://www.numdam.org/item/AIHPA_1995__63_4_331_0/

[BW1] J. Bisognano and E.H. Wichmann, On the duality condition for a Hermitian scalar field, J. Math. Phys., Vol. 16, 1975, pp. 985-1007. | MR 438943 | Zbl 0316.46062

[BW2] J. Bisognano and E.H. Wichmann, On the duality condition for quantitum fields, J. Math. Phys., Vol. 17, 1976, pp. 303-321. | MR 438944

[Boa] R.P. Boas, Entire Functions, Academic Press, New York, 1954. | MR 68627 | Zbl 0058.30201

[Bch1] H.-J. Borchers, Translation Group and Modular Automorphisms for Local Regions, Commun. Math. Phys., Vol. 132, 1990, pp. 189-199. | MR 1069208 | Zbl 0717.46057

[Bch2] H.-J. Borchers, The CPT-Theorem in Two-dimensional Theories of Local Observables, Commun. Math. Phys., Vol. 143, 1992, pp. 315-332. | MR 1145798 | Zbl 0751.46045

[Bch3] H.-J. Borchers, On Modular Inclusion and Spectrum Condition, Lett. Math. Phys., Vol. 27, 1993, pp. 311-324. | MR 1219504 | Zbl 0810.46074

[Bch4] H.-J. Borchers, When does Lorentz Invariance imply Wedge-Duality?, Lett. Math. Phys., Vol. 35, 1995, pp. 39-60. | MR 1346044 | Zbl 0833.46054

[BY] H.-J. Borchers and J. Yngvason, From Quantum Field to local von Neumann Algebras, Rev. Math. Phys., Special Issue, 1992, pp. 15-47. | MR 1199168 | Zbl 0785.46062

[BR] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Springer Verlag, New York, Heidelberg, Berlin, 1979. | MR 611508 | Zbl 0421.46048

[BOT] H.J. Bremermann, R. Oehme and J.G. Taylor, Proof of dispersion relation in quantized field theories, Phys. Rev., Vol. 109, 1958, pp. 2178-2190. | MR 97252 | Zbl 0081.43601

[BEGS] J. Bros, H. Epstein, V. Glaser and R. Stora, Quelques aspects globaux des problèmes d'Edge-of-the-Wedge, In: Hyperfunctions and Theoretical Physics (Nide Rencontre 1973), Lecture Notes in Mathematics, Vol. 449, Springer-Verlag, Berlin, Heidelberg, New York, 1975. | MR 419831 | Zbl 0317.32013

[BGL1] R. Brunetti, D. Guido and R. Longo, Modular structure and duality in conformal quantum field theory, Commun. Math. Phys., Vol. 156, 1993, pp. 201-219. | MR 1234110 | Zbl 0809.46086

[BGL2] R. Brunetti, D. Guido and R. Longo, Group cohomology, modular theory and space-time symmetrics, to appear in Rev. Math. Phys. | MR 1310766 | Zbl 0837.46058

[Bu] D. Buchholz, On the Structure of Local Quantum Fields with Non-trivial Interactions, In: Proceedings of the International Conference on Operator Algebras, Ideals and their Applications in Thepretical Physics, Leipzig, 1977, Teubner-Texte zur Mathematik, 1978, pp. 146-153. | MR 528269 | Zbl 0404.46054

[BDL1,2] D. Buchholz, C. D'Antoni and R. Longo, Nuclear Maps and Modular Structures, I. General Properties, Jour. Func. Analysis, Vol. 88, 1990, pp. 233-250.II. Application to Quantum Field Theory, Commun. Math. Phys., Vol. 129, 1990, pp. 115-138. | MR 1038440 | Zbl 0705.46033

[BJ] D. Buchholz and P. Junglas, On the existence of equilibrium states in local quantum field theory, Commun. Math. Phys., Vol. 121, 1989, pp. 255-270. | MR 985398 | Zbl 0673.46045

[BSM] D. Buchholz and H. Schulz-Mirbach, Haag-duality in conformal quantum field theory, Rev. Math. Phys., Vol. 2, 1990, p. 105. | MR 1079298 | Zbl 0748.46040

[BuSu1] D. Buchholz and S.J. Summers, An Algebraic Characterization of Vacuum States in Minkowsky Space, Commun. Math. Phys., Vol. 155, 1993, pp. 442-458. | MR 1231637 | Zbl 0788.46074

[BuSu2] D. Buchholz and S.J. Summers, Geometric modular action and representations of the Poincaré group, in preparation.

[BuWi] D. Buchholz and E.H. Wichmann, Causal independence and the energy-level density of states in local quantum field theory, Commun. Math. Phys., Vol. 106, 1986, pp. 321-344. | MR 855315 | Zbl 0626.46064

[Ep] H. Epstein, Some Analytic Properties of Scattering Amplitudes in Quantum Field Theory, in 1965 Brandeis Summer Institute, Gordon and Breach, New York, London, Paris, 1966.

[Fre] K. Fredenhagen, Generalization of the Theory of Superselection Sectors, In: The Algebraic Theory of Superselection Sectors. Introduction and Recent Results, World Scientifique 1990, p. 379. | MR 1147469

[GF] F. Gabbiani and J. Fröhlich, Operator algebras and conformal field theory, Commun. Math. Phys., Vol. 155, 1993, pp. 569-640. | MR 1231644 | Zbl 0801.46084

[Ha] R. Haag, Local Quantum Physics, Springer verlag, Berlin, Heidelberg, New York, 1992. | MR 1182152

[HHW] R. Haag, N. Hugenholtz and M. Winnink, On the equilibrium state in quantum statistical mechanics, Commun. Math. Phys., Vol. 5, 1967, pp. 215-236. | MR 219283 | Zbl 0171.47102

[HL] P.D. Hislop and R. Longo, Modular structure of the local algebra associated with a free massless scalar field theory, Commun. Math. Phys., Vol. 84, 1982, pp. 71-85. | MR 660540 | Zbl 0491.46060

[Jo] R. Jost, Eine Bemerkung zum CTP Theorem, Helv. Phys. Acta, Vol. 30, 1957, pp. 409-416. | MR 89720 | Zbl 0085.43305

[Ka] R.V. Kadison, Derivations of operator algebras, Ann. of Math., Vol. 83, 1966, pp. 280-293. | MR 193527 | Zbl 0139.30503

[KR] R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras II, New York: Academic press, 1986. | MR 859186 | Zbl 0601.46054

[Lo] R. Longo, Algebraic and modular structure of von Neumann algebras in physics, Proc. Symp. Pure Math., Vol. 38, 1982, pp. 551-566. | MR 679537 | Zbl 0504.46050

[OT] A.I. Oksak and I.T. Todorov, Invalidity of the TCP-Theorem for Infinite-Component Fields, Commun. Math. Phys., Vol. 11, 1968, p. 125. | MR 239828 | Zbl 0162.28802

[RS] H. Reeh and S. Schlieder, Eine Bemerkung zur Unitäräquivalenz von Lorentzinvarianten Feldem, Nuovo Cimento, Vol. 22, 1961, p. 1051. | MR 137513 | Zbl 0101.22402

[Sak] S. Sakai, Derivations of W*-algebras, Ann. of Math., Vol. 83, 1966, pp. 273-279. | MR 193528 | Zbl 0139.30601

[Str] R. Streater, Local Fields with the Wrong Connection Between Spin and Statistics, Commun. Math. Phys., Vol. 5, 1967, pp. 88-96. | Zbl 0165.58202

[Ta] M. Takesaki, Tomita's Theory of Modular Hilbert Algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer verlag, Berlin, Heidelberg, New York, 1970. | MR 270168 | Zbl 0193.42502

[TMP] I.T. Todorov, M.C. Mintchev and V.B. Petkova, Conformal invariance in quantum field theory, Publ. Scuola Normale Superiore, Pisa 1978. | MR 533358 | Zbl 0438.22011

[To] M. Tomita, Quasi-standard von Neumann algebras, Preprint, 1967. | MR 284822

[Wie1] H.-W. Wiesbrock, A comment on a recent work of Borchers, Lett. Math. Phys., Vol. 25, 1992, pp. 157-159. | MR 1182035 | Zbl 0773.46034

[Wie2] H.-W. Wiesbrock, Half-Sided Modular Inclusions of von Neuman Algebras, Preprint, FU Berlin, 1992. | MR 1244859

[Wie3] H.-W. Wiesbrock, Symmetries and Half-Sided Modular Inclusions of von Neumann Algebras, Lett. Math. Phys., Vol. 28, 1993, pp. 107-114. | MR 1229893 | Zbl 0794.46057

[Wie4] H.-W. Wiesbrock, Conformal Quantum Field Theory and Half-Sided Modular Inclusions of von Neumann Algebras, Commun. Math. Phys., Vol. 158, 1993, pp. 537-543. | MR 1255426 | Zbl 0802.46089

[Win] M. Winnink, An Application of C* -Algebras to Quantum Statistical Mechanics of Systems in Equilibrium, Thesis groningen, 1968.

[Yng] J. Yngvason, A Note on Essential Duality, Lett. Math. Phys., Vol. 31, 1994, pp. 127-141. | MR 1280391 | Zbl 0807.47061

[Ze] E.C. Zeeman, Causality Implies the Lorentz Group, J. Math. Phys., Vol. 5, 1964, pp. 490-493. | MR 162587 | Zbl 0133.23205