no. 4
The Tomita operator for the free scalar field
Annales de l'I.H.P. Physique théorique, Tome 51 (1989) no. 4, pp. 419-435. 
@article{AIHPA_1989__51_4_419_0,
     author = {Figliolini, Franca and Guido, Daniele},
     title = {The {Tomita} operator for the free scalar field},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     pages = {419--435},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {4},
     year = {1989},
     mrnumber = {1034596},
     zbl = {0715.46049},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1989__51_4_419_0/}
}
TY  - JOUR
AU  - Figliolini, Franca
AU  - Guido, Daniele
TI  - The Tomita operator for the free scalar field
JO  - Annales de l'I.H.P. Physique théorique
PY  - 1989
SP  - 419
EP  - 435
VL  - 51
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/item/AIHPA_1989__51_4_419_0/
LA  - en
ID  - AIHPA_1989__51_4_419_0
ER  - 
%0 Journal Article
%A Figliolini, Franca
%A Guido, Daniele
%T The Tomita operator for the free scalar field
%J Annales de l'I.H.P. Physique théorique
%D 1989
%P 419-435
%V 51
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/item/AIHPA_1989__51_4_419_0/
%G en
%F AIHPA_1989__51_4_419_0
Figliolini, Franca; Guido, Daniele. The Tomita operator for the free scalar field. Annales de l'I.H.P. Physique théorique, Tome 51 (1989) no. 4, pp. 419-435. http://www.numdam.org/item/AIHPA_1989__51_4_419_0/

[1] H. Araki, A Lattice of von Neumann Algebras Associated with the Quantum Field Theory of a Free Bose Field, J. Math. Phys., Vol. 4, 1963, pp. 1343-1362. | MR | Zbl

[2] G. Benfatto and F. Nicolo, The Local von Neumann Algebras for the Massless Scalar Free Field and the Free Electromagnetic Field, J. Math. Phys., Vol. 19, 1978. | MR

[3] J.J. Bisognano and E.H. Wichmann, On the Duality Condition for an Hermitian Scalar Field, J. Math. Phys., Vol. 16, 1975, p. 985. | MR | Zbl

[4] G.F. Dell'Antonio, Structure of the Algebras of Some Free Systems, Comm. Math. Phys., Vol. 9, 1968, pp.81-117. | MR | Zbl

[5] J.P. Eckmann and K. Osterwalder, An Application of Tomita's Theory of Modular Hilbert Algebras: Duality for Free Bose Field, J. Funct. Analysis, Vol. 13, 1973, pp. 1- 22. | MR | Zbl

[6] F. Figliolini and D. Guido, The type of Second Quantization Factors, Preprint. | MR

[7] R. Haag, N.M. Hugenoltz and M. Winnik, On the Equilibrium States in Quantum Statistical Mechanics, Comm. Math. Phys., Vol. 5, 1967, pp. 215-236. | MR | Zbl

[8] P. Hislop and R. Longo, Modular Structure of the Local Observables Associated with the Free Massless Scalar Field Theory, Comm. Math. Phys., Vol. 84, 1982. | MR | Zbl

[9] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1984.

[10] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary value problems and applications I, Springer-Verlag, Berlin, 1972. | Zbl

[11] G.V. Maz'Ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. | MR

[12] J.E. Roberts, P. Leyland and D. Testard, Duality for Quantum Free Fields, unpublished paper.

[13] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I and II, Academic Press, 1975.

[14] Segal and Goodman, Anti-Locality of Certain Invariant Operators, J. Math. Mech., Vol. 14, 1965. | Zbl

[15] G.L. Sewell, Relativity of Temperature and Hawking Effect, Phys. Lett., Vol. 79A, 1980, p. 23. | MR

[16] E.C. Zeeman, Causality Implies the Lorentz Group, J. Math. Phys., Vol. 5, 1964. | MR | Zbl