Algebraic quantum field theory and noncommutative moment problems. I. (English) Zbl 0653.46071

Alcantara-Bode, J.; Yngvason, J.

Ann. Inst. Henri Poincaré, Phys. Théor. 48, No. 2, 147-159 (1988).

Let \({\mathcal S}\) denote Borchers’ test function algebra and \({\mathcal I}_ c\) the locality ideal. It is shown that the quotient algebra \({\mathcal S}/{\mathcal I}_ c\) admits a continuous \(C^*\)-norm and thus has a faithful representation by bounded operators on Hilbert space. This representation can be chosen to be Poincaré-covariant. Some further properties of the topology defined by the continuous \(C^*\)-norms on this algebra are also established.

Reviewer: C.Itzykson

MSC:

46N99	Miscellaneous applications of functional analysis
81T05	Axiomatic quantum field theory; operator algebras
46F05	Topological linear spaces of test functions, distributions and ultradistributions

Keywords:

algebraic quantum field theory; noncommutative moment problems; Borchers’ test function algebra; locality ideal; continuous \(C^*\)-norm; faithful representation by bounded operators on Hilbert space; Poincaré- covariant

Citations:

Zbl 0653.46072

References:

[1]	A.S. Wightman and L. Gårding , Fields as operator valued distributions in relativistic quantum field theory . Ark. f Fys. , t. 28 , 1965 , p. 129 . Zbl 0138.45401 · Zbl 0138.45401
[2]	R. Haag and D. Kastler , An algebraic approach to quantum field theory . J. Math. Phys. , t. 5 , 1964 , p. 848 . MR 165864 | Zbl 0139.46003 · Zbl 0139.46003
[3]	H. Araki , Einführung in die axiomatische Quantenfeldtheorie . ETH Zürich 1961 - 1962 .
[4]	H.J. Borchers and W. Zimmermann , On the self-adjointness of field operators . Nuovo Cim. , t. 31 , 1963 , p. 1047 . MR 161607 | Zbl 0151.44303 · Zbl 0151.44303
[5]	W. Driessler and J. Fröhlich , The reconstruction of local observable algebras from the Euclidean Green’s functions of relativistic quantum field theory . Ann. Inst. H. Poincaré , t. 27 , 1977 , p. 221 . Numdam | MR 484123 | Zbl 0364.46051 · Zbl 0364.46051
[6]	Fredenhagen and J. Hertel , Local algebras of observables and pointlike localized fields . Commun. Math. Phys. , t. 80 , 1981 , p. 555 . Article | MR 628511 | Zbl 0472.46051 · Zbl 0472.46051
[7]	W. Driessler , S.J. Summers and E.H. Wichmann , On the connection between quantum fields and von Neumann algebras of local operators . Commun. Math. Phys. , t. 105 , 1986 , p. 49 . Article | MR 847127 | Zbl 0595.46062 · Zbl 0595.46062
[8]	M. Dubois-Violette , A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. I . Commun. Math. Phys. , t. 43 , 1975 , p. 225 . Article | MR 383100 | Zbl 0362.46044 · Zbl 0362.46044
[9]	M. Dubois-Violette , A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. II . Commun. Math. Phys. , t. 54 , 1977 , p. 151 . Article | MR 440369 | Zbl 0365.46060 · Zbl 0365.46060
[10]	H.J. Borchers , On the structure of the algebra of field operators . Nuovo Cim ,. t. 24 , 1962 , p. 214 . MR 142320 | Zbl 0129.42205 · Zbl 0129.42205
[11]	A. Uhlmann , Über die Definition der Quantenfelder nach Wightman und Haag . Wiss. Zeitschr., Karl Marx Univ. , t. 11 , 1962 , p. 213 . MR 141413 | Zbl 0119.43804 · Zbl 0119.43804
[12]	G. Choquet , Lectures on Analysis , vol. 2 , New York , Benjamin , 1984 .
[13]	H. Araki and G.A. Elliott , On the definition of C*-algebras . Publ. RIMS Kyoto , t. 9 , 1973 , p. 93 . Article | MR 355611 | Zbl 0272.46033 · Zbl 0272.46033
[14]	J. Yngvason , On the locality ideal in the algebra of test functions for quantum fields . Publ. RIMS Kyoto , t. 20 , 1984 , p. 1063 . Article | MR 764348 | Zbl 0598.46047 · Zbl 0598.46047
[15]	H.J. Borchers , C*-algebras and automorphism groups . Commun. Math. Phys. , t. 88 , 1983 , p. 95 . Article | MR 691200 | Zbl 0524.46047 · Zbl 0524.46047
[16]	G.K. Pedersen , C*-algebras and their Automorphism Groups . London , New York , San Francisco , Academic Press , 1979 . MR 548006 | Zbl 0416.46043 · Zbl 0416.46043