Canonical commutation relations and interacting Fock spaces
Zied Ammari1
1 Université Cergy-Pontoise Site Saint Martin 95302 Cergy-Pontoise cedex
Journées équations aux dérivées partielles (2004), article no. 2, 13 p.

We introduce by means of reproducing kernel theory and decomposition in orthogonal polynomials canonical correspondences between an interacting Fock space a reproducing kernel Hilbert space and a square integrable functions space w.r.t. a cylindrical measure. Using this correspondences we investigate the structure of the infinite dimensional canonical commutation relations. In particular we construct test functions spaces, distributions spaces and a quantization map which generalized the work of Krée-Rączka [KR] and Janas-Rudol [JR1]-[JR3].

DOI: 10.5802/jedp.2
Zied Ammari 1

1 Université Cergy-Pontoise Site Saint Martin 95302 Cergy-Pontoise cedex 
@incollection{JEDP_2004____A2_0,
     author = {Zied Ammari},
     title = {Canonical commutation relations and interacting {Fock} spaces},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {2},
     pages = {1--13},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2004},
     doi = {10.5802/jedp.2},
     mrnumber = {2135357},
     zbl = {1067.35082},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.2/}
}
TY  - JOUR
AU  - Zied Ammari
TI  - Canonical commutation relations and interacting Fock spaces
JO  - Journées équations aux dérivées partielles
PY  - 2004
SP  - 1
EP  - 13
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.2/
DO  - 10.5802/jedp.2
LA  - en
ID  - JEDP_2004____A2_0
ER  - 
%0 Journal Article
%A Zied Ammari
%T Canonical commutation relations and interacting Fock spaces
%J Journées équations aux dérivées partielles
%D 2004
%P 1-13
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.2/
%R 10.5802/jedp.2
%G en
%F JEDP_2004____A2_0
Zied Ammari. Canonical commutation relations and interacting Fock spaces. Journées équations aux dérivées partielles (2004), article  no. 2, 13 p. doi : 10.5802/jedp.2. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.2/

[Am] Ammari, Z.: On canonical commutation relations and quantization in infinite dimension spaces, in preparation

[Ar] Aronszajn, N.:Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404 | MR | Zbl

[As] Asai, N.: Analytic characterization of one-mode interacting Fock space, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 4 (2001),409-415 | MR | Zbl

[AB] Accardi, L., Bożejko, M.: Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 1 (1998), 663-670 | MR | Zbl

[ADKS] Albeverio, S., Daletsky, Yu. L., Kondratiev, Yu. G., Streit, L.,: Non-Gaussian infinite-dimensional analysis, J. Funct. Anal., 138 (1996), 311-350 | MR | Zbl

[AN] Accardi, L., Nahni, M.: Interacting Fock spaces and orthogonal polynomials in several variables, 192-205 | MR | Zbl

[AKK] Asai, N., Kubo, I., Kuo, H. H.: Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures, Proc. Amer. Math. Soc., 131 (2003), 815-823 | MR | Zbl

[B1] Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math., 14 (1961), 187-214 | MR | Zbl

[B2] Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Part II. A family of related function spaces. Application to distribution theory, Comm. Pure Appl. Math., 20, (1967), 1-101 | MR | Zbl

[BSZ] Baez, J. C., Segal, I. E., Zhou, Z.F., Introduction to algebraic and constructive quantum field theory, Princeton Series in Physics, Princeton University Press, (1992) | MR | Zbl

[DX] Dunkl, C., Xu, Y.: Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge Univ. Press, 2001. | MR | Zbl

[JR1] Janas, J., Rudol, K.: Toeplitz operators on the Segal-Bargmann space of infinitely many variables, Linear operators in function spaces (Timişoara, 1988), Oper. Theory Adv. Appl., 43, 217-228, Birkhäuser | MR | Zbl

[JR2] Janas, J., Rudol, K., Toeplitz operators in infinitely many variables, Topics in operator theory, operator algebras and applications (Timişoara, 1994), 147-160, Rom. Acad., Bucharest, 1995 | MR | Zbl

[JR3] Janas, J., Rudol, K.: Two approaches to Toeplitz operators on Fock space, Quantization and infinite-dimensional systems (Bialowieza, 1993), 3-7, Plenum | MR | Zbl

[KR] Krée, P., Rączka, R.: Kernels and symbols of operators in quantum field theory, Ann. Inst. H. Poincaré Sect. A (N.S.), 28 (1978), 41-73 | Numdam | MR | Zbl

[KSWY] Kondratiev, Y. G., Streit, L., Westerkamp, W., Yan, J.,: Generalized functions in infinite-dimensional analysis, Hiroshima Math. J., 28 (1998), 213-260 | MR | Zbl

[M] Martens, F. J. L.: Spaces of analytic functions on inductive/projective limits of Hilbert spaces, Dissertation, Technische Universiteit Eindhoven, Eindhoven, 1988 | MR | Zbl

[vN] von Neumann, J.: Collected works, volume 2, edited by A.H. Taub, Pergamon Press (1961) | Zbl

[Re] Reeh, H.: A remark concerning canonical commutation relations, J. Math. Phys., 29 (1988), 1535-1536 | MR | Zbl

[Sl] Slawny, F.: On factor representations and 𝒞 * -algebra of canonical commutation relations, Comm. Math. Phys., 24 (1971), 151-170 | MR | Zbl

[St] Stone. M.H.: Linear transformations in Hilbert space, III: Operational methods and group theory, Proc. Nat. Acad. Sci. USA, 16 (1930), 172-175

Cited by Sources: