We give a complete proof of the twisted duality property of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.
REFERENCES
1.
Achieser, N. I. and Glasmann, I. M., Theorie der linearen Operatoren im Hilbert-Raum (Verlag Harri Deutsch, Thun, 1981).
2.
Araki
, H.
, “A lattice of von Neumann algebras associated with the quantum theory of a free bose field
,” J. Math. Phys.
4
, 1343
–1362
(1963
).3.
Araki
, H.
, “On quasifree states of CAR and Bogoliubov automorphisms
,” Publ. RIMS, Kyoto Univ.
6
, 385
–442
(1970/71
).4.
Araki, H., “Bogoljubov automorphisms and Fock representations of canonical anticommutation relations,” in Operator Algebras and Mathematical Physics (Proceedings of the summer conference held at the University of Iowa, 1985), edited by P. E. T. Jorgensen and P. S. Muhly (American Mathematical Society, Providence, RI, 1987).
5.
Avron
, J.
, Seiler
, R.
, and Simon
, B.
, “Charge deficiency, charge transport and comparison of dimensions
,” Commun. Math. Phys.
159
, 399
–422
(1994
).6.
Avron
, J.
, Seiler
, R.
, and Simon
, B.
, “The index of a air of projections
,” J. Funct. Anal.
120
, 220
–237
(1994
).7.
Baumgärtel, H., Operatoralgebraic Methods in Quantum Field Theory. A Series of Lectures (Akademie Verlag, Berlin, 1995).
8.
Baumgärtel
, H.
, Jurke
, M.
, and Lledó
, F.
, “On free nets over Minkowski space
,” Rep. Math. Phys.
35
, 101
–127
(1995
).9.
Baumgärtel, H. and Wollenberg, M., Causal Nets of Operator Algebras. Mathematical Aspects of Algebraic Quantum Field Theory (Akademie Verlag, Berlin, 1992).
10.
Bisognano
, J. J.
and Wichmann
, E. H.
, “On the duality condition for quantum fields
,” J. Math. Phys.
17
, 303
–321
(1976
).11.
Borac
, S.
, “On the algebra generated by two projections
,” J. Math. Phys.
36
, 863
–874
(1995
).12.
Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics 1 (Springer Verlag, Berlin, 1987).
13.
Davis
, C.
, “Separation of two linear subspaces
,” Acta Sci. Math. Szeged
19
, 172
–187
(1958
).14.
Dell’Antonio
, G. F.
, “Structure of the algebras of some free systems
,” Commun. Math. Phys.
9
, 81
–117
(1968
).15.
Dixmier
, J.
, “Position relative de deux variétés linéaires fermées dans un espace de Hilbert
,” Rev. Sci.
86
, 387
–399
(1948
).16.
Doplicher
, S.
, Haag
, R.
, and Roberts
, J. E.
, “Local observables and particle statistics I
,” Commun. Math. Phys.
23
, 199
–230
(1971
).17.
Doplicher
, S.
, Haag
, R.
, and Roberts
, J. E.
, “Local observables and particle statistics II
,” Commun. Math. Phys.
35
, 49
–85
(1974
).18.
Eckmann
, J. P.
and Osterwalder
, K.
, “An application of Tomita’s theory of modular Hilbert algebras: duality for free bose fields
,” J. Funct. Anal.
13
, 1
–12
(1973
).19.
Evans, D. E. and Kawahigashi, Y., Quantum Symmetries and Operator Algebras, Oxford Science Publications (Clarendon, Oxford, 1998).
20.
Foit
, J. J.
, “Abstract twisted duality for free Fermi fields
,” Publ. RIMS, Kyoto Univ.
19
, 729
–741
(1983
).21.
Haag, R., Local Quantum Physics (Springer Verlag, Berlin, 1992).
22.
Halmos
, P. R.
, “Two subspaces
,” Trans. Am. Math. Soc.
144
, 381
–389
(1969
).23.
Hislop
, P. D.
, “A simple proof of duality for local algebras in free quantum field theory
,” J. Math. Phys.
27
, 2542
–2550
(1986
).24.
Jurke, M., “Ergebnisse zu massiven, freien Netzen über dem Minkowskiraum,” Ph.D. thesis, Universität Postdam, 1997.
25.
Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras II (Academic, Orlando, 1986).
26.
Kato, T., Perturbation Theory for Linear Operators (Springer Verlag, Berlin, 1995).
27.
Leyland, P., Roberts, J. E., and Testard, D., Duality for Quantum Free Fields, preprint (CNRS Marseille, 1978).
28.
Lledó
, F.
, “Conformal covariance of massless free nets
,” Rev. Math. Phys.
13
, 1135
–1161
(2001
).29.
Osterwalder
, K.
, “Duality for free bose fields
,” Commun. Math. Phys.
29
, 1
–14
(1973
).30.
Rieffel
, M. A.
and van Daele
, A.
, “A bounded operator approach to Tomita-Takesaki theory
,” Pac. J. Math.
69
, 187
–221
(1977
).31.
Summers
, S. J.
, “Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with applications to quantum field model
,” Commun. Math. Phys.
86
, 111
–141
(1982
).32.
Wassermann
, A.
, “Operator algebras and conformal field theory III
,” Invent. Math.
133
, 467
–538
(1998
).
This content is only available via PDF.
© 2002 American Institute of Physics.
2002
American Institute of Physics
You do not currently have access to this content.
