No Access

ON LOCALIZATION AND POSITION OPERATORS IN MÖBIUS-COVARIANT THEORIES

NICOLA PINAMONTI

NICOLA PINAMONTI

Department of Mathematics, Faculty of Science, University of Trento, Italy

Istituto Nazionale di Alta Matematica "F. Severi", unità locale di Trento, Italy

Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, via Sommarive 14, I-38050 Povo (TN), Italy

Search for more papers by this author

https://doi.org/10.1142/S0129055X07003024Cited by:0 (Source: Crossref)

Abstract

Some years ago it was shown that, in some cases, a notion of locality can arise from the symmetry group of the theory [1–3], i.e. in an intrinsic way. In particular, when the Möbius covariance is present, it is possible to associate some particular transformations with the Tomita–Takesaki modular operator and conjugation of a specific interval of an abstract circle. In this context we propose a way to define an operator representing the coordinate conjugated to the modular transformations. Remarkably this coordinate turns out to be compatible with the abstract notion of locality. Finally a concrete example concerning a quantum particle on a line is given.

Keywords:

AMSC: 22E70, 46L60, 47L90, 81R10, 81T40

References

R. Brunetti, D. Guido and R. Longo, Rev. Math. Phys. 14, 759 (2002), DOI: 10.1142/S0129055X02001387. Link, Web of Science, Google Scholar
B. Schroer, Nucl. Phys. B 499, 519 (1997), DOI: 10.1016/S0550-3213(97)00358-1. Crossref, Web of Science, ADS, Google Scholar
L. Fassarella and B. Schroer, J. Phys. A 35, 9123 (2002), DOI: 10.1088/0305-4470/35/43/311. Crossref, ADS, Google Scholar
E. P. Wigner, Ann. Math. 40, 149 (1939). Crossref, Web of Science, ADS, Google Scholar
J. Mund, J. Math. Phys. 44, 2037 (2003), DOI: 10.1063/1.1561592. Crossref, Web of Science, Google Scholar
J. Mund, B. Schroer and J. Yngvason, Phys. Lett. B 596, 156 (2004). Crossref, Web of Science, ADS, Google Scholar
D. Buchholz and G. Lechner, Ann. Henri Poincaré 5, 1065 (2004), DOI: 10.1007/s00023-004-0190-8. Crossref, Web of Science, Google Scholar
B. Schroer, Ann. Phys. 321, 435 (2006). Crossref, Web of Science, ADS, Google Scholar
M. Takesaki and M. Tomita , Theory of Modular Hilbert-Algebras and Its Application , Lecture Notes in Mathematics 128 ( Springer Verlag , 1970 ) . Crossref, Google Scholar
I. E. Segal and R. W. Goodman, J. Math. Mech. 14, 629 (1965). Web of Science, Google Scholar
G. Fleming, Philos. Sci. 67, S495 (2000), DOI: 10.1086/392841. Crossref, Web of Science, Google Scholar
H. Halvorson, Philos. Sci. 68, 111 (2001), DOI: 10.1086/392869. Crossref, Web of Science, Google Scholar
P. D. Hislop and R. Longo, Commun. Math. Phys. 84, 71 (1982), DOI: 10.1007/BF01208372. Crossref, Web of Science, ADS, Google Scholar
D. Buchholz, G. Mack and I. Todorov, Nucl. Phys. Proc. Suppl. 5, 20 (1988), DOI: 10.1016/0920-5632(88)90367-2. Crossref, Google Scholar
D. Buchholz and H. S. Mirbach, Rev. Math. Phys. 2, 105 (1990), DOI: 10.1142/S0129055X90000053. Link, Google Scholar
F. Gabbiani and J. Fröhlich, Commun. Math. Phys. 155, 569 (1993), DOI: 10.1007/BF02096729. Crossref, Web of Science, ADS, Google Scholar
D. Guido and R. Longo, Commun. Math. Phys. 181, 11 (1996), DOI: 10.1007/BF02101672. Crossref, Web of Science, ADS, Google Scholar
D. Guido, R. Longo and H. W. Wiesbrock, Commun. Math. Phys. 192, 217 (1998), DOI: 10.1007/s002200050297. Crossref, Web of Science, ADS, Google Scholar
S. Carpi, Commun. Math. Phys. 244, 261 (2004), DOI: 10.1007/s00220-003-0988-0. Crossref, Web of Science, ADS, Google Scholar
W. Pauli, Die Allgemeinen Prinzipen der Wellenmechanik V/1, ed. S. Flügge (Springer Verlag, Berlin) p. 60. Google Scholar
Y. Aharonovet al., Phys. Rev. A 57, 4130 (1998), DOI: 10.1103/PhysRevA.57.4130. Crossref, Web of Science, ADS, Google Scholar
W. G. Unruh and R. M. Wald, Phys. Rev. D 40, 2598 (1989), DOI: 10.1103/PhysRevD.40.2598. Crossref, Web of Science, ADS, Google Scholar
R. Giannitrapani, Internat. J. Theoret. Phys. 36, 1575 (1997), DOI: 10.1007/BF02435757. Crossref, Web of Science, ADS, Google Scholar
R. Brunetti and K. Fredenhagen, Phys. Rev. A 66, 044101 (2002), DOI: 10.1103/PhysRevA.66.044101. Crossref, Web of Science, Google Scholar
R. Giannitrapani, J. Math. Phys. 39, 5180 (1998), DOI: 10.1063/1.532564. Crossref, Web of Science, ADS, Google Scholar
M. Toller, Phys. Rev. A 59, 960 (1999), DOI: 10.1103/PhysRevA.59.960. Crossref, Web of Science, ADS, Google Scholar
M. Toller, Phys. Rev. D 70, 024006 (2004), DOI: 10.1103/PhysRevD.70.024006. Crossref, Web of Science, Google Scholar
M. A. Naimark, Izv. Akad. Nauk. SSSR Ser. Mat. 4, 277 (1940). Google Scholar
G. Ludwig , Foundations of Quantum Mechanics , Texts and Monographs in Physics ( Springer-Verlag , New York , 1983 ) . Crossref, Google Scholar
P. Busch , P. J. Lahti and P. Mittelstaedt , The Quantum Theory of Measurement , Lecture Notes in Physics ( Springer Verlag , Berlin , 1991 ) . Crossref, Google Scholar
P. Busch , M. Grabowski and P. J. Lahti , Operational Quantum Physics , Lecture Notes in Physics ( Springer Verlag , Berlin , 1995 ) . Crossref, Google Scholar
H. Narnhofer, M. Requardt and W. E. Thirring, Commun. Math. Phys. 92, 247 (1983), DOI: 10.1007/BF01210849. Crossref, Web of Science, ADS, Google Scholar
D. Guido and R. Longo, Commun. Math. Phys. 218, 513 (2001), DOI: 10.1007/s002200100416. Crossref, Web of Science, ADS, Google Scholar
V. Moretti and N. Pinamonti, Nucl. Phys. B 647, 131 (2002), DOI: 10.1016/S0550-3213(02)00940-9. Crossref, Web of Science, ADS, Google Scholar
V. Moretti and N. Pinamonti, J. Math. Phys. 45, 230 (2004), DOI: 10.1063/1.1626271. Crossref, Web of Science, ADS, Google Scholar
V. Moretti and N. Pinamonti, J. Math. Phys. 45, 257 (2004), DOI: 10.1063/1.1629396. Crossref, Web of Science, ADS, Google Scholar
E. Nelson, Ann. Math. 70, 572 (1959). Crossref, Web of Science, Google Scholar
G. L. Sewell, Ann. Phys. (NY) 141, 201 (1982), DOI: 10.1016/0003-4916(82)90285-8. Crossref, Web of Science, ADS, Google Scholar
B. Schroer, Int. J. Mod. Phys. A 18, 1672 (2003). Google Scholar
D. Guidoet al., Rev. Math. Phys. 13, 125 (2001), DOI: 10.1142/S0129055X01000557. Link, Web of Science, Google Scholar
L. Pukansky, Math. Ann. 156, 96 (1964). Crossref, Web of Science, Google Scholar
P. J. Sally, Bull. Am. Math. Soc. 72, 269 (1966), DOI: 10.1090/S0002-9904-1966-11489-1. Crossref, Web of Science, Google Scholar